Coherent Dynamics of a Bose ... - Universidad de Chilerepositorio.uchile.cl/tesis/uchile/2011/cf-troncoso_rc/pdfAmont/cf... · Coherent Dynamics of a Bose-Einstein Condensation of

Universidad de Chile

Facultad de Ciencias Físicas y Matemáticas

Departamento de Física

Coherent Dynamics of a Bose-Einstein Condensation ofMagnons at Room Temperature

TESIS PARA OPTAR AL GRADO DE DOCTOR EN CIENCIAS

MENCIÓN FÍSICA

Roberto Enrique Troncoso Coña

Santiago de Chile

Diciembre, 2011

Universidad de Chile

Facultad de Ciencias Físicas y Matemáticas

Departamento de Física

Coherent Dynamics of a Bose-Einstein Condensation ofMagnons at Room Temperature

TESIS PARA OPTAR AL GRADO DE DOCTOR EN CIENCIAS

MENCIÓN FÍSICA

Roberto Enrique Troncoso Coña

PROFESOR GUÍA

Alvaro Nuñez Vásquez

MIEMBROS DE LA COMISIÓN

Rodrigo Arias Federici

Rembert Duine

Pedro Landeros Silva

Sergio Rica Mery

Santiago de Chile

Diciembre, 2011

The reductionist hypothesis does not by any means imply a

constructionist one: The ability to reduce everything to

simple fundamental laws does not imply the ability to start from

those laws and reconstruct the universe.

-P. W. Anderson

II

Summary

A study for the Bose-Einstein condensation of magnons phenomena on ferromagnetic thin

lms at room temperature is presented from a fenomenological and microscopic perspective

based on quantum eld theory. It is shown that, despite magnon decay processes, ongoing

during the condensate formation, the system manifests spontaneous quantum coherence with

a pseudo-spin degree of freedom originated from the presence of doubly degenerate valleys in

momentum space. It is also found that the condensate consists of two magnon components

lying in the vicinity of the two points of minimum energy in momentum space where the mag-

netic interactions introduce a coupling between them. In addition, a real space description of

the condensed state is provided, revealing the condensate state of magnons as a spin density

wave(SDW), where their wavelength is equal to that of the lowest energy magnons.

The collective dynamics of the spinor magnon condensate wavefunction is completely charac-

terized by a Gross-Pitaevskii equation, described by a set of physical eective parameters which

are obtained from the microscopic theory. We show the existence of vortex-like structures in the

condensed phase that corresponds to an edge dislocation in the SDW with spatial anisotropy as-

sociated to the discrepancy between longitudinal and transverse masses of magnons. This result

has recently been conrmed in the S. Demokritov's group by a direct measurement of the local

magnon density. Additionally is discussed the nature of quantum interference between magnon

clouds, highlighting the close relation between such phenomena and the well known Joseph-

son eects. Using those ideas a detailed calculation of the Josephson oscillations between two

magnon clouds, spatially separated in a magnonic Josephson junction, is provided.

The theoretical aspects developed in this work constitutes the base for the comprehension of

magnon condensation and provides an appropriate framework to study novel quantum phenom-

ena of magnons on ferromagnetic lms . Direct applications can be performed to investigate new

trends in the development of magnonics eld, either storage, manipulation and transmission of

information on magnetic systems.

III

Resumen

Se presenta un estudio del fenómeno de condensación de Bose-Einstein de magnones en

películas ferromagnéticas delgadas a temperatura ambiente desde una perspectiva fenomenológ-

ica y microscópica basada en la teoría cuántica de campos. Se demuestra que, en paralelo a los

procesos de decaimiento de magnones, durante la formación del condensado, el sistema mani-

esta coherencia cuántica espontánea con grados de libertad de pseudo-spin originados a partir

de la presencia de una doble degeneración de valle en el espacio de momentum. Se encuentra

también que el condensado consiste en dos componentes de magnones pertenecientes a la vecin-

dad de los dos puntos de energía mínima en el espacio de momentum, donde las interacciones

magnéticas introducen un acoplamiento entre éstos. Adicionalmente se provee una descripción

en el espacio real, revelando al estado condensado de magnones como una onda de densidad de

espín (ODE), donde su longitud de onda es igual a la de los magnones de energía más baja.

La dinámica colectiva de la función de onda espinorial del condensado de magnones es

completamente caracterizada por una ecuación de Gross-Pitaevskii, descrita por un conjunto de

parámetros físicos efectivos los cuales son obtenidos desde la teoría microscópica. Se muestra la

existencia de estructuras tipo vórtice en la fase condensada, la cual corresponde a una dislocación

en la ODE con una anisotropía espacial asociada a la discrepancia entre las masas longitudinales

y transversales de los magnones. Este resultado ha sido conrmado experimentalmente por el

grupo de S. Demokritov mediante una medición directa de la densidad local de magnones.

Adicionalmente es discutida la naturaleza de la interferencia cuántica entre nubes de magnones,

destacando la estrecha relación entre dichos fenómenos y el conocido efecto Josephson. Se

proporciona un cálculo detallado, utilizando aquellas ideas, de las oscilaciones de Josephson

entre dos nubes de magnones, separadas espacialmente en junturas de Josephson magnónicas.

Los aspectos teóricos desarrollados en este trabajo constituyen la base para la comprensión de

la condensación de magnones y proporciona un marco adecuado para estudiar nuevos fenómenos

cuánticos en peliculas ferromagnéticas.

IV

Agradecimientos

Felizmente hemos arribado a buen puerto. Por ello desearía manifestar mis agradecimientos

a quienes fueron parte importante de este viaje.

Quisiera partir por agradecer a mi familia por el amor brindado y el apoyo constante en mi

primera etapa de formación como cientíco, por allá desde que comencé este largo camino.

Desearía compartir mi más profundo agradecimiento a Alvaro, quien me recibió con su

tremenda calidez humana. Haber trabajado y compartido con El, signicó un gran crecimiento

en lo cientíco.

La Clau, mi hermosa y amada mujer, mi compañera de viaje, gracias por tu paciencia durante

todo este periodo (de tesis). Podría escribir eternamente para expresar todo lo que quiero, pero

solo me resumiré a decir. . . te amo.

Este trabajo va dedicado a los amigos, Pablo y Jano, con quienes desarrollé una gran amistad,

asi como a Gabriel y Ernesto que estuvieron presente desde la lejanía. Agradezco tambien a

Sebax, Priscilla, Santa!, Pancha, Eli, Chuky, Paula y Nacho. Y por si alguien se quedó fuera,

también se lo agradezco.

Finalmente agradezco a sus valiosos impuestos que han contribuido a nanciar mi estancia en

el programa de Doctorado desde el año 2007, por medio de, Comision Nacional de Investigacion

Cientíca y Tecnológica (CONICYT), Proyecto Fondecyt 11070008 y 1110271, Núcleo Cientíco

Milenio P10061−F y Proyecto Basal FB0807-CEDENNA.

V

Contents

1 Introduction 1

1.1 Condensation of magnons and the aim of this thesis . . . . . . . . . . . . . . . . 2

1.2 Plan of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Spin wave physics on ferromagnetic thin lms 8

2.1 Classical spin wave theory on ferromagnetic thin lms . . . . . . . . . . . . . . . 8

2.2 Microscopic theory of spin waves; magnons . . . . . . . . . . . . . . . . . . . . . 13

2.3 An experimental brushstroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 YIG materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Microwave excitation of magnons . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Brillouin light scattering spectroscopy . . . . . . . . . . . . . . . . . . . . 23

2.3.4 Schematic structure of spin waves experiments on YIG thin lms . . . . 26

2.4 Bose-Einstein condensation of magnons at room temperature: experimental mo-

tivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

VI

3 Condensation of a Magnon Gas 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Hamiltonian Approach for Magnons Systems . . . . . . . . . . . . . . . . . . . . 34

3.3 Phenomenological Description of the Magnon Condensate . . . . . . . . . . . . . 38

3.4 Many-Body Scattering Theory for magnon-magnon Interactions . . . . . . . . . 42

3.4.1 Two-body scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.2 Many-body T-Matrix and ladder approximation . . . . . . . . . . . . . . 46

3.5 Many-body theory for nonequilibrium Magnons gas . . . . . . . . . . . . . . . . 48

3.5.1 Two components, double condensate . . . . . . . . . . . . . . . . . . . . 60

3.6 Semiclassical Theory of Bose-Einstein Condensation of Magnons . . . . . . . . . 61

4 Collective Dynamics of Magnons Condensate 67

4.1 Semiclassical interpretation of the condensed stage . . . . . . . . . . . . . . . . . 67

4.2 Two-components Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . 71

4.3 Magnon BEC as a Hopf-Andronov Bifurcation . . . . . . . . . . . . . . . . . . . 76

4.4 Topological excitations within the condensade phase . . . . . . . . . . . . . . . . 81

4.4.1 Vortex like structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Josephson eect in the condensate of magnons 87

5.1 Microscopic fundamentals for the magnon Josephson eect . . . . . . . . . . . . 90

5.1.1 The Nonlinear Two-Mode Approximation . . . . . . . . . . . . . . . . . . 91

5.2 The Magnon Josephson Junctions equations . . . . . . . . . . . . . . . . . . . . 94

VII

5.2.1 AC magnon Josephson oscillations . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 Macroscopic quantum self-trapping of magnons . . . . . . . . . . . . . . 100

5.3 Dynamical behavior of the spin density wave . . . . . . . . . . . . . . . . . . . . 104

5.4 Internal Josephson's oscillations and asymmetric current of magnons . . . . . . . 108

5.4.1 Internal Josephson eect . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.2 Magnon asymmetric current . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Conclusions and Outlook 117

Appendix A Eective parameters within the many-body T-matrix approxima-

tion 120

A.0.3 Eective interaction between Magnons; γ1 and γ2 parameters. . . . . . . 120

A.0.4 Determination of the Self-Energy . . . . . . . . . . . . . . . . . . . . . . 122

A.0.5 Determination of the Anomalous Self-Energy . . . . . . . . . . . . . . . . 124

VIII

List of Figures

1.1 Measurements of decay time of BLS intensity in the lowest energy level εm as a

function of the pumping power. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Schematic illustration of a magnon gas and their transition to Bose-Einstein

condensed state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Sketch of the geometry for the spin wave propagation through the magnetic sample. 9

2.2 Dispersion relation for spin waves on ferromagnetic thin lm at various angles

between the direction of propagation and the in-plane magnetic eld. . . . . . . 11

2.3 Illustration of spectrum for spin waves, emphasizing the relevance of each energy

contribution, with propagation direction parallel to the in-plane magnetic eld,

i.e. ϕ = 0 and θ = π/2, in terms of wavevectors. The spectrum is calculated for

typical experimental values on YIG samples, see below. . . . . . . . . . . . . . . 12

2.4 Crystalline structure of Yttrium Iron Garnet material, YIG. . . . . . . . . . . . 20

2.5 Elementary scattering process for the parametric excitations of magnons. . . . . 22

2.6 Sketch of the nature of Brillouin scattering processes. . . . . . . . . . . . . . . . 24

2.7 basic sketch of the experimental setup for the investigation of spin waves on

YIG-based samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

IX

2.8 Normalized BLS intensity measured for dierent two-dimensional magnon wave

vectors k =(k||, k⊥

). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9 Redistribution of magnons injected by a short-pulses of parametric pumping.

BLS spectra for dierent delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10 Decay rate of magnon density, represented by the normalized BLS intensity, at

the lowest level in the spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Feynmann diagrams depicted for the splitting and conuence magnons scattering

processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Spectrum of magnons in a YIG thin lm, with an in-plane eld of H ∼ 1kOe and

lm thickness d ∼ 5µm, for momentum parallel to the eld. . . . . . . . . . . . . 38

3.3 Feynmann diagrams for the bare interaction of two-magnon scattering process. . 44

3.4 Diagrammatic representation of the Dyson Equation. . . . . . . . . . . . . . . . 48

3.5 Two-particle connected Green's function determined within the many-body T-

matrix approximation. The one-particle Green's function renormalized . . . . . 49

3.6 The one-particle Green's function renormalized, due to the presence of magnon

gas, by means of many-body T-matrix in the ladder approximation. . . . . . . . 52

3.7 Illustration of the Feynman diagram for the anomalous scattering as a result of

matching of two magnon-splitting process. . . . . . . . . . . . . . . . . . . . . . 56

3.8 Renormalization of the anomalous magnon scattering by interactions with the

many-body system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.9 Diagrammatic illustration of the anomalous one-particle Green's function renor-

malized by the many-body system eect. . . . . . . . . . . . . . . . . . . . . . . 59

3.10 Diagramatic representation of T-matrix or ladder approximation. The interaction

vertex V ABCD is renormalized by the interaction vertices V AB

C GEBV DEF + V A

BCGCDV DEF . 60

X

3.11 Illustration of the macroscopic wave function pair for the magnon gas . . . . . . 61

3.12 Numerical calculation of eective parameters calculated within the T-matrix or

ladder approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.13 The "free-energy" of the system as a function of the order parameter. The critical

density of the condensed state as a function of applied magnetic eld. . . . . . . 64

3.14 The critical pumping as a function of in-plane magnetic eld, in unity of energy. 65

4.1 Spin density wave pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Experimental evidence of the spin density wave for dierent continuous pumping

power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Microscopic illustration of the Spin Density Wave. . . . . . . . . . . . . . . . . . 71

4.4 Dissipation characteristic for the magnon gas in the condensate state, calculated

from the many-body T-matrix approach. . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Cartoon of the magnons population around of each valley when the net interaction

γ2 > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6 Temporal evolution of the magnon density under parametric pumping above the

threshold value Pc = µ− ν. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7 Illustration of the Hopf-Andronov bifurcation . . . . . . . . . . . . . . . . . . . 79

4.8 Portrait phase space for the global magnon phase trajectories in the γ < 1 regime,

i.e. in the condensate stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.9 Illustration of the potential energy associated with the trajectories followed by

the global phase Θ(t) in phase space. . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10 Sketch of the deformation for a closed loop over a surfaceM. . . . . . . . . . . 83

XI

4.11 Normalized density prole of the vortex and healing length as a function of in-

plane magnetic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.12 Vortex structure with an elliptic cross section of aspect ratio γ =√m||/m⊥ ∼ 5. 85

5.1 Sketch of the experimental setup for the magnon Josephson's eect realization . 88

5.2 Cartoon of the two-mode approximation over the full macroscopic wave function

of the condensate state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Prole of the potential barrier produced by a wire conductor . . . . . . . . . . . 94

5.4 Phase diagram which delimits two, qualitatively dierent, Josephson's oscillation

regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5 Dynamical behavior for the ac Josephson's eect . . . . . . . . . . . . . . . . . . 98

5.6 Dynamical behavior for the macroscopic quantum self trapping phenomena. . . . 102

5.7 Phase-space portrait for the libration oscillation. . . . . . . . . . . . . . . . . . . 103

5.8 Dynamical evolution for the spin density wave at each side of the barrier. . . . . 105

5.9 Behavior of the Spin Density Wave in the macroscopic quantum self trapping

oscillation regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.10 Magnon asymmetric-current Josephson eect between two cloud condensates . . 108

5.11 Internal Josephson eect. The sliding modes. . . . . . . . . . . . . . . . . . . . . 112

5.12 Asymmetric Josephson eect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.13 Small amplitude oscillations for the asymmetric Josephson eect. . . . . . . . . 115

5.14 Schematic illustration of spin pumping setup by Josephson oscillations in the

ferromagnet and voltage signal induced in the Pt metal . . . . . . . . . . . . . . 116

XII

A.1 Behavior of anomalous vertex for magnons with the bare interactions as a function

of temperature and in-plane magnetic eld. . . . . . . . . . . . . . . . . . . . . . 125

XIII

Chapter 1

Introduction

One of the most relevant concepts in the eld of condensed matter physics, is the idea about

condensation which has long been recognized to play a key role in the behavior of macroscopic

quantum phenomena like superconductivity, superuidity or quantum Hall eect.

The Bose-Einstein condensate (BEC) is a macroscopic quantum state of matter which ex-

ploits the ability of a system of bosons to occupy simultaneously the ground state[1, 2]. This

is manifested as a sudden apparition of a macroscopic occupation in the lowest energy level

and the emergence of long range order due to the quantum coherence. The rst evidence for

condensation was performed in a remarkable series of experiments on rubidium vapor [32] and

sodium [33] where the atoms were conned in magnetic traps and cooled down to extremely

low temperatures, the order of microkelvins. A simplied picture of the transition toward the

condensation is provided by the following analysis. At ordinary temperatures the wavelength

of classical objects is extremely small and the wavelike properties are imperceptible. But as the

temperature is decreased, the quantum nature of the matter become prominent. Once the tem-

perature T is near to critical temperature Tc, the atoms become delocalized since the de Broglie

wavelength is comparable to mean distance between them. At even lower temperatures, the

wave functions of many atoms overlap with each other. They loose their individual identities,

and together they form a Bose-Einstein condensate. It is worth mentioning that is necessary to

considerer weakly interacting bosons to avoid phase transition to solid state.

Recently the BEC transition in non-equilibrium systems has attracted a great deal of atten-

tion. Although it has been predicted that any integer-spin quasiparticle with an eective mass,

i.e. with a gap of energy, should be able to undergo a transition to spontaneous coherence ex-

actly analogous to BEC it has been observed just in the last decade in dierent physical systems

of quasiparticles such as excitons [37][38][39][40], phonons [41], polaritons[42] have reported ex-

perimental observation of the onset of coherence at low temperature in CdTe structures and

recently photons in an optical microcavity[43]. Unlike the rest of quasiparticles condensates,

1.1 Condensation of magnons and the aim of this thesis

the condensation in magnonic systems is reached at room temperature[29]. The study of the

last phenomena constitutes the heart of this work.


The recent discovery of condensation at room temperature in a magnon gas has helped to

enlarge our understanding of the complex quantum states of matter. This is essentially due

to the lack of conservation of the quasiparticles participating in the condensate. The main

ndings took place on Yttrium Iron Garnet(YIG) thin lms samples, ordered magnetic crystal

of ferrimagnet type.

The concept of spin waves as the lowest lying magnetic states above the ground state of

a magnetic medium, originally introduced by Bloch[3], considered an excitation consisting of

many spins deviating slightly from their equilibrium orientation. These disturbances propagate

as waves through the medium, forming spin waves. The naive idea of magnons, and sucient

to keep in mind, is conceived as elementary spin waves excitations of a magnetically ordered

system, i.e. quasi-particles with a quanta of energy which carries integer spin angular mo-

mentum. Therefore magnons obeys Bose-Einstein statistics and behave as weakly interacting

quasi-particles that under particular conditions should demonstrate Bose-Einstein condensa-

tion. Several groups have reported observations of eld-induced BEC of magnetic excitations

in dierent quantum low-dimensional magnets like quantum antiferromagnets TlCuCl3 [5, 6],

Cs2CuCl4 [7, 8] and BaCuSi3O3 [9]. In these kind of materials, the phase transition occurs if

the applied magnetic eld is strong enough to overcome the antiferromagnetic exchange cou-

pling. Although such transition is interpreted as BEC of magnetic excitations these can hardly

be considered as magnons, since(explain). However the last point is questioned since there is

no satisfactory theoretical progress to describe the BEC transition on antiferromagnets(citar).

BEC also has been observed in ensemble of nuclear spins in 3He at very low temperatures in

the range of millikelvin [10, 11].

Broadly speaking, following Bloch, at zero temperatures the ground state congurations

correspond to the spins of the system fully aligned. As the temperature increases, spins deviate

more and more from the common direction, thus increasing the amplitude of spin waves and

reducing the net magnetization. Nevertheless the physical properties of magnons in YIG thin

lms structures, vastly studied in the classical regime[22, 24, 25], is slightly more complex

and results from the dominant coupling either exchange and dipole-dipole interactions. The

2


dipole-dipole contributions dominates for low k-wavevectors (k < 104[cm−1]) since the dipolar

is a long-range interaction, which introduce a energetic penalty for uniform spin waves modes,

k = 0. By that reason the magnon spectrum has a gap, i.e. εm > 0. On the other hand, for

relatively large wavevectors (k > 106[cm−1]) the magnetic moments tend to be aligned and the

dynamics is entirely determined by exchange magnetic interactions. However, in the interval

104cm−1 < k < 106cm−1 neither of these interactions can be neglected and the corresponding

excitations are denominated as dipole-exchange magnons.

In thermal equilibrium the magnon gas is a system of quasi-particles with nite lifetime thus

the number particles N is variable, therefore the chemical potential is zero µ = 0. Moreover due

to the existence of a gap, BEC of magnons in equilibrium is not possible. To expect that this

1 2 3 4 5 6

pumping power (W)

BLS

intens

ity (

arbitra

ry u

nits)

deca

y tim

e (n

s)

100

150

200

250

10

10

10

2

3

Figure 1.1: Circles show the measured decay time of the BLS intensity in the lowest energy levelεm as a function of the pumping power. Squares show the maximum detected BLS intensity atthe frequency ωm versus the pumping power as measured by S. Demokritov, et al. [31]. Forpumping over the critical value ∼ 2.5[W] the overpopulation of magnons in the bottom state ofenergy εm is observed and interpreted as a BEC transition.

system show coherence eects, and eventually displays a BEC transition, the magnon gas must

be excited externally to remove it far from the equilibrium. A successful mechanism to excite

spin waves is the parametric pumping, which excites two magnons with opposite momenta and

the middle of frequency of pumping.

3


In experiments, at room temperature, performed on YIG thin lms with thickness ∼ 5[µm]

and magnetized uniformly in a eld of 700−1000[Oe] oriented in the plane of the lm, the exter-

nal injection of magnons is provided by means of continuous parallel parametric pumping with

a frequency 8.0− 8.1[GHz]. For pumping power amplitudes over critical values, near to 2.5[W],

the magnons overpopulate the lowest level of energy. This observation has been associated with

BEC transition on the magnon gas[29]. From a thermodynamically point of view is not evident

that the BEC transition can be reached, essentially due to the quasiparticles are characterized

by a nite lifetime which is usually compared with the relaxation time. However experiments

about thermalization have been carried out to establish that the spin-lattice (magnon-phonon)

mediated processes are associated with a typical magnon lifetime of 1µs [12, 49]. while the re-

laxation processes leading to thermalization are in the nanosecond scale[36], namely the magnon

gas is decoupled from the crystalline lattice. Direct observations of coherence properties have

been performed measuring the life-time of the magnon gas concentrated at the lowest energy

level as a function of pumping power and compared with the maximum detected BLS intensity

at the energy εm, as displayed in the Fig. (1.1).

Classical Gas Quantum Gas BEC

?

density

kBEC

Figure 1.2: Schematic illustration of a magnon gas and their transition to Bose-Einstein con-densed state. At low density the system of magnons is well described as a classical gas. As thedensity increase to values near to a critical density dened by N

2/3c = mKT/2π2~2, the magnon

gas enter in a region where the quantum eects take place. Above the critical density the BECtransition can be reached at any temperature, provided that the density of quasi-particles ishigh enough. Obtaining a detailed understanding of the system's evolution to magnon BECstate is an important goal of this thesis.

Although the Bose-Einstein condensation is a well understood issue, at least in alkali-gases,

this new experiments on magnetic medium, together with the rest of observation of condensation

4


on quasiparticles, have challenged our comprehension of condensation phenomena. Our eorts

are concentrated, as the goal of this work, in obtaining a detailed understanding of the highly

dense magnon system, the formation of the magnon BEC state and their characterization. The

focus of this thesis is mainly motivated by the experiments realized about parametric pumping

of magnons and their behavior at high densities on YIG-based samples[30, 31, 36].

There are several properties that set the magnons condensate apart from standard BECs.

The rst is related to the no conservation of particles, which implies a explicit broken of global

phase U(1). By that reason is necessary to keep in the theory contributions which describe

anomalous scattering processes∗. Both the collective dynamics and redistribution of magnons

occurs in a out of equilibrium regime, this eects are considered and in particular we are inter-

ested in their consequences when the magnon gas reach the quasiequilibrium state. Moreover

the collective interactions between magnons must be treated adequately, in fact the huge in-

crease of magnons on the bottom state and quantum coherent phenomena can not be explained

without taking into account the many-body eects. This is particularly important since may

lead to established incorrectly the instability of magnon BEC like in [56]. Despite some attempts

to describe both the formation [64], emergence and characterization of the magnon condensate

[61, 62], a consistent quantum eld theory that reveals the microscopic origin and physical mech-

anism is still lacking. Following these ideas we adopt a quantum eld theory formulation for an

accurate description of collective dynamics between magnons far from equilibrium. With the

aim of demonstrate the BEC transition in the magnon gas we show the spontaneous emergence

of quantum coherence where the many-body properties are incorporates in the two-magnon

physics by summing all the ladder Feynman diagrams of the microscopic theory. In the semi-

classical and low energy limit the magnon gas manifest a BEC transition where the macroscopic

quantum state (magnon BEC) consist of two components because of the doubly degeneracy of

lowest energy level, which has been recently conrmed [81]. A description of quantum interfer-

ence phenomena between magnon condensates is performed to unravel the macroscopic phase

coherence.

∗The anomalous scattering processes correspond to certain class of magnon-magnon interactions which do

not conserve the number of particles

5

1.2 Plan of work

1.2 Plan of work

Following the ideas developed in the last sections this thesis is organized as follows. In the

Chapter 2 we introduce the physics of spin waves in magnetic media with special emphasis on

ferromagnetic thin lms. Any results exposed is original of our work, just we intend to pro-

vide some sense of self-consistency and is not pretended to be a review of the subject. At the

beginning of that chapter we clarify the concept of spin waves from classical models, giving a

introduction of basic terminology and phenomenology that will help to have in mind the typol-

ogy of spin waves on ferromagnetic thin lms throughout this work. The microscopic theory

of spin waves is outlined to dene one of the central concepts in this thesis, the magnons, and

determine the started point in the theory developed below. We conclude with a experimental

motivation about the Bose-Einstein condensation of magnons that will guide us and lays the

groundwork for our future investigation. In Chapter 3 we start with a examination about of the

microscopic physics of magnons on YIG ferromagnetic thin lms. Emphasizing in the eective

interactions that determine the magnon dynamics we argue the microscopic scattering processes

that dominate in terms of the k-dependence of magnon-type in question, i.e. their location on

the spectrum. That discussion establishes the basis to develop the eective quantum eld theory

for a magnon gas externally excited. In particular to determine the eective interaction between

magnons, the many-body eects are taken into consideration by summing all the relevant con-

tributions within the many-body T-matrix approximation[46], assuming that the magnon gas

is suciently dilute. Within this framework the set of microscopic physical parameters that

describe the eective theory are calculated using the above approximation. The main physical

result is twofold; a semiclassical eective theory in the low-energy limit that explain the Bose-

Einstein condensation of magnons phase transition and a physical picture of the condensate

state in terms of magnetization variables. Numerical calculations are implemented to determine

the eective parameters that characterize the condensate stage. It is used in chapter 4 which is

devoted to the complete characterization of the collective dynamics of the magnon condensate

phase, whose central focus is related to the analysis of the structure of Gross-Pitaevskii equa-

tion. Here we show the precise meaning of order parameter, the macroscopic wave function,

that characterize the quantum phase transition. In particular, the magnon condensate consists

of two condensate components which coherent phase is the phase locking between the compo-

nents. Although in a more qualitative sense, we give a nice interpretation of the appearance

of magnon condensate state in terms of Hopf-Andronov bifurcation. The topological structures

like vortices, are studied and their manifestation on the spin density wave. In chapter 5 our

6

1.2 Plan of work

semiclassical picture, GP equation, to describe the magnon condensate stage is applied to in-

vestigate the spontaneous occurrence of the quantum coherence. To unravel this property we

perform a study of the macroscopic interference eect between magnon condensates, i.e. the

magnon Josephson eect. We derive the magnon Josephson equations (MJJ) and discuss several

novel phenomena associated with the spinorial degrees of freedom of the magnon condensate.

Moreover we propose a experimental realization for the magnon Josephson junction version. We

nish in the Chapter 6 with our conclusions.

7

Chapter 2

Spin wave physics on ferromagnetic thin

lms

The concept of spin waves as low-energy excitations on magnetic media, is one of the cru-

cial notions in the study of magnetization dynamics study [3, 4]. To illustrate this point let

us consider a magnetic system where a magnetic moment has been removed from its equi-

librium conguration the local magnetic interactions will change and to restore the original

conguration, such local disturbance propagates through of magnetic media. That perturba-

tion correspond to a spin wave. From a quantum mechanics point of view the spin waves can

be treated in a convenient way introducing the concept of magnon, as a elementary excitation

on magnetic materials, from which one could derive kinetic or thermodynamics properties, such

as transport properties, magnetization, specic heat, etc.. The importance of spin waves in

magnetic research lies in the fact that their properties reveal the nature of the interactions that

produce the magnetic ordering. In this chapter we shall be mainly concerned with spin waves

on conned geometries, like thin lms.

2.1 Classical spin wave theory on ferromagnetic thin lms

Typically the magnetization behavior on magnetic ordered crystals are essentially described

by dierent energetic contributions involving individual interactions between spins either short

or long range. We may distinguish some of these as, exchange and dipolar interactions in

Ferro(Anti)-magnetic lms [12], Dzyaloshinskii-Moriya interactions for more exotic magnetic

structures like helimagnets [13] or spin orbit interactions in metallic systems. The competition

among those interactions determine the dynamic and equilibrium properties of collective exci-

tations giving rise interesting phenomena as magnetic vortex, skyrmions structures or spin hall

eect, just to mention a few. A good description of magnetic phenomenon in ferromagnetic

2.1 Classical spin wave theory on ferromagnetic thin films

thin lms at macroscopic scale can be provided by a classical phenomenological theory, where

exchange and dipolar eects are well captured by the Maxwell's and Landau-Lifshitz equations

[14]. In these micromagnetic approach the magnetization is assumed as continuous vector eld

where the eects of geometry are included by appropriate boundary conditions.

The rst step to exploit the concept of spin waves is to analize the properties of linear,

i.e. non interacting, spin waves propagation through the lm. In fact the questions that are

highlighted, what is the role of the dipolar and exchange interactions on the dynamics or how

does the geometry of the sample on the spin wave propagation. As an example, the nite

size eect break the translation symmetry and the momentum is not conserved, resulting in a

discretization of spectrum and a dependence of the spin waves frequency on the magnitud of the

wave vector. The propagation of the spin waves over the plane YIG lm, even more constrained,

can lead to spin wave connement giving rise to selection rules. To begin our understanding of all

YIG film

ξ

η

ζ

x

y

z

M0

φ

θd

Figure 2.1: Sketch of the geometry and coordinate system to describe the spin wave propagationthrough the magnetic sample. The system is a ferromagnetic YIG thin lm, where the thicknessis nite and of the order of micrometers.

those properties mentioned above and to analyze the typology of spin wave modes as a function

of interaction strength and the geometry, we need to nd the spectrum of magnons. This section

is devoted to review the classical theory to describe the properties of the propagation of spin

waves on ferromagnetic thin lms [15]. Such approach takes into account both dipolar and

exchange interactions, hence the elementary excitations are called dipole-exchange spin waves,

and the problem is formulated as follows. Let us assume the geometry conguration, as shown

in the Fig. (2.1), for the magnetic thin lm where by convenience the z-direction is along the

magnetization saturation eld. Writing the integral-dierential equation, which is equivalent to

the initial system of equations consisting of the linearized Landau-Lifshitz equation of motion

for the magnetization and to the Maxwell equations in the magnetostatic limit with the usual

9


boundary conditions.

Fm(~ξ) = hd(~ξ) (2.1)

where m(~ξ) is the vector Fourier amplitude of the plane spin wave, F is the linear matrix-

dierential operator which holds information of exchange contribution, and hd(ξ) is the dipole

magnetic eld and whose expression is given by

hd(~ξ) =

∫ d/2

−d/2G(~ξ, ~ξ′)m(~ξ′)d~ξ′, (2.2)

with G(~ξ, ~ξ′) the tensorial Green function inside the ferromagnetic lm in the (ξ, η, ζ) coordinate

system Fig. (2.1). The expression for the spectrum ω(kζ) is derived as a eigen-value problem

for m(~ξ) in the integral-dierential equation. The frequencies of spin waves obeys

ω2n =

(ωH + αωMk

2n

) (ωH + αωMk

2n + ωMFnn

)(2.3)

where it was introduced the functions Fnn dened by

Fnn = Pnn +

(1− Pnn

(1 + cos2 ϕ

)+ ωM

Pnn (1− Pnn) sin2 ϕ

ωH + αωMk2n

)sin2 θ (2.4)

where ωH = γH, ωM = 4πγM and α = γD, with γ the gyromagnetic factor and D the exchange

stiness. The wavevector of spin waves are k2n = k2

ζ + κ2n, where integer n represent the band

index of the wavevectors along of perpendicular direction due to the nite thickness of lm. The

expression for Pnn can be explicitly determined by the boundary conditions applied, in fact for

unpinned∗ surface spins these obeys

Pnn =k2ζ

k2n

−k4ζ

(1 + δ0n) k4n

2

kζd

[1− (−1)ne−kζd

]. (2.5)

It is worth noting that we pointed out the unpinned case to connect with the derivation of Eq.

(2.3) from a microscopic approach. The ϕ,θ-angles in the Eq. (2.4) are dened in the Fig. (2.1)

and determine the direction of the spin wave propagation respect to the bias magnetization

eld. Below we analyze some illustratives cases for ϕ and θ.

According to the Eq. (2.3) it is clear how both dipole-dipole and exchange interactions

∗The pinned surface spins boundary conditions refers to every spin in the surface of lm are not allowed to

precess.

10


10 3 10 4 10 5 10 6 10 3 104 10 5 10 6

Wavevector, [cm ] Wavevector, [cm ]

Fre

quen

cy, [

GH

z]

Freq

uenc

y, [

GH

z]

4

8

12

3

5

7

6

φ = φ = φ = 0

ππ

2

φ = φ = φ = 0

ππ

2

d=0.1 d=5μm μm

-1 -1

Figure 2.2: Dispersion relation for spin waves on ferromagnetic thin lm at various anglesbetween the direction of propagation and the in-plane magnetic eld. Both graphs are calculatedfor dierent thickness of YIG lm, 0.1[µm] and 5[µm], and H = 1[KOe].

inuences the spin wave spectrum. Due to the long range nature of the dipole-dipole inter-

action introduces an energetic penalty for the uniform modes k = 0, and thus determines the

slope of spin waves dispersion curves in the relatively long wavelenght part of the spectrum,

k < 104[cm−1]. On the other hand, i.e. for relatively high wavevectors k > 106[cm−1], the ex-

change interaction is dominant and determines the propagation of spin waves. These properties

encourage the distinction of three this kinds of spin waves k-vector dependents which will be

described below. The short scale of the lm thickness imposes selection rules that manifest in

dierent branches on the spectrum indexed by n. It should be noted that the energy to excite

the n-branches increase as the thickness is reduced.

In the magnetostatic limit, for wavenumber k < 104[cm−1], we refer to these modes as

magnetostatic modes or Damon-Eshbach spin waves [14]. The spin waves behavior for large

wavelenght, can be detached from the dispersion relation Eq. (2.3) where we make the distinc-

tion for in-plane or perpendicular magnetized lms, i.e. θ = 0 and θ = π/2 respectively, see

Fig. (2.1). Therefore in the magnetostatic limit and considering a in-plane magnetized lm the

dispersion relation take the form

ω2(k) = ωH

(ωH + ωM

(1− e−kzd

kzd

))(2.6)

for spin waves propagating parallel, i.e. φ = 0, to the magnetization saturation eld in the

fundamental mode n = 0, appropriate for thin lms. Some observations can be regarded about

the nature of this magnetostatic modes. Although has been shown only the fundamental mode

in the Eq. (2.6), can be said that the all modes have the same cuttof frequency and there is

11


no a frequency range where only a single mode propagates. The frequency decrease with the

increment of kz, which implies a negative group velocity vg = ∂ω/∂kz < 0, while the phase

velocity vp = ω/kz, is positive. A wave with this property is called a backward wave. Due to

the versatility of those modes, either the excitation and observation, have been largely studied.

The excitations of spin system with wavevectors k > 106[cm−1], i.e. near to the exchange

lenght λ ∼ 10[nm], are determined by the exchange interaction which is dominant where the

boundary eects and the long-range interactions are negligible. In that sense such kind of spin

waves are called exchange magnons. From a microscopic perspective and since the dipole-dipole

interaction is neglected at such spatial scales, the elementary scattering processes conserve

the number of quasiparticles and the angular momentum is conserved. As a consequence the

time-relaxation will be dierent for magnons with dierent wavevectors, either dipolar and

dipolar-exchange modes. In that sense the spin-spin relaxation is k-dependent and the ex-

change magnons can be considered as a subsystem from the full magnon gas. The exchange

magnons plays an important role in the spin pumping eect on alloys YIG/Pt too [16]. Finally,

10 2 10 4 105 10wavevector, [cm ]

Fre

quen

cy, [

GH

z]

3.0

5.0

7.0

10 3

-1

Dipole

Dipole-

Exch

ange

Exch

ange

6

k

H

Figure 2.3: Illustration of spectrum for spin waves, emphasizing the relevance of each energycontribution, with propagation direction parallel to the in-plane magnetic eld, i.e. ϕ = 0 andθ = π/2, in terms of wavevectors. The spectrum is calculated for typical experimental valueson YIG samples, see below.

in the range 105[cm−1] < k < 106[cm−1], neither the dipolar and exchange interaction can be ne-

glected for themselves. The excitations with those wavevectors are denominated dipole-exchange

12

2.2 Microscopic theory of spin waves; magnons

magnons which are in the vicinity of the lowest energy level. Obviously dipole-exchange magnons

will be important for us since the physical processes toward the condensation state take place

on the bottom of spectrum.

In this section we have reviewed the basic elements in the theory of spin waves on ferro-

magnetic thin lms. The approach unies the description both short and long wavelenghts,

including exchange and dipolar interactions, on magnetized lms with an arbitrary direction

and resulting in an appropriate description to dipole-exchange modes. In particular, such results

were applied to quantify and describe the characteristic dispersion relation for magnons on YIG

magnetic materials, either in perpendiculary or tangentially magnetized lms. Some limiting

cases were recovered, for long-wavelenght in the magnetostatic limit and exchange spin waves

for high wavevectors. The presented results can be used to make a detailed study of the spin

wave excitations and provided us an appropriate scheme for the future investigations.


Although the classical approach, introduced in the last section, is highly advantageous for

an appropriate macroscopic description of the dynamical properties of magnetization in a fer-

romagnetic media, it is no longer valid when exploring phenomena whose nature is intrinsically

quantum, for example spin-pumping [17] or spin Seebeck eects [18] among others on insulat-

ing magnetic media, or the condensation of magnons phenomena treated in this work. Even

more, that phenomenological model cannot be rigorous since the nature of the exchange inter-

action is purely quantum, or the dipolar strength, causing of asymmetry in the propagation of

longitudinal or transvers excitations in the magnetization.

In order to obtain the microscopic Hamiltonian one should take the expressions for the

interactions between magnetic moments, represented by the spin operator Si in the i-site on the

lattice and sum over all of them, driven by the exchange and dipole-dipole interaction and by the

Zeeman energy of a single spin in a external magnetic eld[19]. The eective spin Hamiltonian

H = HZ +Hex +Hdip +Hp, (2.7)

13


contain the Zeeman interaction

HZ =∑i

H · Si, (2.8)

where H is the in-plane magnetic eld, applied parallel to the magnetization excitation. The

exchange interaction

Hexc = −J∑〈ij〉

Si · Sj (2.9)

where the sum run over the nearest neighbors and J > 0 is the eective exchange strength.

Finally the long-range interaction represented by the dipolar energy is

Hdip = −1

2

∑〈ij〉,i 6=j

µ2

|Rij|3[3(Si · rij

)(Sj · rij

)−(Si · Sj

)](2.10)

where the sums runs over the sites Ri of the lattice and rij = Rij/ |Rij| are unit vectors in the

directions Rij ≡ Ri −Rj. The constant µ = gµB is the magnetic moment associated with the

spins, where g is the eective g−factor and µB = e~/(2mc) is the Bohr magneton. The last

term in the right-hand side of Eq. (2.7) correspond to a external source that excite spin waves

on the ferromagnet, but that will be discussed later. The above system of spin operator obeys

the angular-momentum commutation relations[Sα, Sβ

]= i~εαβγSγ, (2.11)

where α ∈ (x, y, z) and the sum over repeated indices is assumed. The spin Hamiltonian,

Eq. (2.7), provided an accuracy quantum mechanical description of the low-energy excitations

associated to the broken symmetry mode in a ferromagnetic media. However an alternative,

but equivalent, way to describe the low-energy modes may be done introducing, for instance,

a second quantization language. In those approach the low-energy modes will be treated as a

set of discret elementary excitations, the quanta of spin waves. It is worth mentioning than

that due to the dipole-dipole contribution the low-energy excitations posses a energy gap and

the broken symmetry modes will not be Goldstone modes. In the course of this work we will

note that the small but non-vanishing mass acquired by the elementary excitations does play a

essential role in the condensation phenomena.

The spin operators system obeys the Lie algebra SU(2) dened by the relation Eq. (2.11),

14


where the SU(2) stands for spin rotations. It should be noted that the commutation relation

Eq. (2.11) determine a irreducible representation for the involved algebra, nevertheless this is

not unique[21]. An irreducible representation of the corresponding group can be then carried

by a subspace of the boson Fock space. In mathematical terms the spin operator algebra admit

an irreducible representation in terms of a single-Boson operator. Introducing a single-bosonic

operator, b, it can be shown than that a mapping which preserves the commutation rules and

at the same time satisfy the rules for bosonic elds,[ai, a

†j

]= δij, are given by the mapping

S+i =

(√2S − ni

)ai, (2.12)

S−i = a†i

(√2S − ni

), (2.13)

Szi = −ni + S. (2.14)

with ni = a†i ai and where in this representation the spin raising (resp. lowering) operator is

associated with the annihilation (resp. creation) of a bosonic excitation S+i ∼ ai (resp. S

−i ∼ a†i ).

That mapping is denominated Holstein-Primako transformation[20]. That bosons operators

acting on a subspace of the innite Fock space which is spanned by the basis of the eigenstates

|n〉 = (n!)1/2 (a†)n |0〉 . (2.15)

where the state |0〉 represent the vacuum, i.e. there is no excitation. From physical considera-

tions, one can see that the maximum number of bosons created must be n = 2S, corresponding

to the spin value available. This is in agreement with the form of Eq. (2.12), since reminds us

that it is dened only in the subspace n ≤ 2S and therefore the new bosonic operators acting

on a physical subspace of the innite and unphysical boson Fock space.

The advantage of introducing the Holstein-Primako representation lies in that we can unveil

the bosonic behavior of the low-energy excitations in a magnetic media. Formally the concept of

magnon can be considered from here, in fact that approach allows a correct quantum mechanical

treatment of several physical phenomena, in addition to a consistent building of a Feymann's

diagram rules of microscopic processes.

The next step is to investigate the consequences of Holstein-Primako transformation on the

spin Hamiltonian Eq. (2.7). Due to the highly nonlinear character of the mapping Eq. (2.12)

it is necessary to explore the mapping of the Hamiltonian expanding the operator expression in

15


Taylor series (1− aia

†i

2S

)1/2

= 1− aia†i

4S+ · · · (2.16)

where the expansion converges quickly because of the value of S, allowing us to restrict ourselves

with the rst term only. For the YIG samples it can be considered as a ferromagnet with the

magnet moment 10µ (µ is Bohr magneton), per unit cell in the low energy limit. In such way

the eective spin is quite large, so that the expansion in powers of 1/S is justied. Thus the

Holstein-Primako transformation can be regarded as

S+i ≈ ~

√2S

(ai −

a†i aiai4S

)+ · · ·

S−i ≈ ~√

2S

(a†i −

a†i a†i ai

4S

)+ · · · . (2.17)

Substituting the last expansions in Eq. (2.7) one gets for the Hamiltonian of the thin lm

ferromagnet a series of contributions as a function of order of the operators a†i , ai.

H = H(0) +H(2) +H(3) +H(4) + · · · (2.18)

where H(0) doesn't contain a†i , ai, while the rest of terms H(2), H(3) and H(4) are proportional

to second, third and forth order in a†i , ai. It should be noted that H(1) is not present in the

expansion Eq. (2.18) because it is associated to the equilibrium condition, which correspond

to complete alignment along z-axis, see Eq. (2.12). As already made clear, these operators

represent the creation(annihilation) in each site i of the lattice. However, by means of Fourier

transformed we can pass a new kind of collective variables, which relates an excitation in all

places at once. The performed transformation is

aj =1

N1/2

∑k

eik·rj ak

a†j =1

N1/2

∑k

e−ik·rj a†k (2.19)

where N is the number of lattice sites. In these knew variables the cuadratic contribution takes

16


the form

H(2) =∑

k

[Aka

†kak +

1

2

(Bkaka−k +B∗ka

†−ka

†k

)](2.20)

with the coecients

Ak =γ[H +Dk2 + 2πM (1− Fk) sin2 θk + 2πMFk

](2.21)

Bk =γ[2πM (1− Fk) sin2 θk − 2πMFk

](2.22)

Fk =(1− e−kd

)/kd. (2.23)

where γ,M,D were dened in the last subsection and the angle θk correspond to ϕ in the Fig.

(2.1), see also Fig. (2.2). In order to diagonalize the quadratic Hamiltonian, we perform a

unitary transformation, called Bogoliubov transformation

bk = ukck + vkc†−k (2.24)

b†k = u∗kc†k + v∗kc−k (2.25)

which ensures that [bk, b†k′ ] = δk,k′ with the condition |uk|2 − |vk|2 = 1, where

|uk|2 =Ak + ω(k)

2ω(k)(2.26)

|vk|2 =Ak − ω(k)

2ω(k). (2.27)

The Hamiltonian acquires the diagonal form Ha =∑

k ωkb†kbk with the frequency given by

ω(k)2 = A2k − |Bk|2. The explicit form that adopts the frequency for elementary excitations on

the wavevector in the plane,

ω(k) = γ[(H +Dk2 + 4πM (1− Fk) sin2 θk

) (H +Dk2 + 4πMFk

)]1/2. (2.28)

Indeed the quasiparticles with energy ~ω(k) as eigen-excitations of a ferromagnet. The

equation for excitations spectrum is the same as the one obtained for the lowest lying branch

(n = 1) for the dipole-exchange modes, described in the last section. In that sense the correspon-

dence spin wave-magnon is established as well as electromagnetic waves-photon or vibrational

modes-phonon, i.e. the magnon quasiparticle as a quanta of the collective excitation spin waves.

17


The rest of contributions of higher order in the Hamiltonian which describe the interaction

between magnons, are

H(b) =∑

V(2,1)k1,k2,k3

bk1b†k2

b†k3+ h.c., (2.29)

which represents the splitting and conuence contributions, namely the annihilation of a magnon

and consecutive creation of a pair of magnons and the Hermitian conjugate process for the split-

ting process. It is worth noting that this scattering contribution, coming from the dipole-dipole

energy, doesn't conserve the number of magnons and neither the total spin. From a classical

point of view, those nonlinear terms inuence the damping of the dynamic magnetization. By

other side the fourth order contribution to the Hamiltonian

H(c) =∑

V(2,2)k1,k2,k3,k4

bk1bk2b†k3

b†k4+ h.c., (2.30)

which represents the magnon-magnon scattering processes, i.e. the four-magnon scattering. For

high k wavevectors the exchange interaction is stronger and mainly dominates that processes

and, therefore, the total spin is conservative, since the number of magnon doesn't change in

the time. It should be noted that the forth strength of interaction V (2,2), implicitly contains

contributions coming from the dipole-dipole interaction which results in of scattering processes

like conuence-splitting of magnons. Finally a term that represent the parametric excitation of

magnons

H(d) =∑

ρke−iωptb†kb†−k + h.c., (2.31)

in the next section we discuss in more detail that mechanism. Summarizing all the above,

the microscopic scheme for the interacting magnons system can be congured as follow. A

microscopic Hamiltonian, expanded to forth order, which is composed by

H = Ha +Hb +Hc + · · · (2.32)

where the quadratic contribution dene the spectrum of magnons and the rest of terms determine

the interaction between them. Actually that result will be our starting point in the next Chapter

3, to develop a many-body theory for interacting magnons.

18

2.3 An experimental brushstroke


In this section we review some basic elements from the typical experimental setups, which

is useful for the broad understanding of spin-wave physics in ordered magnetic crystal like YIG

ferromagnetic thin lms and makes our discussion of highly dense magnon gas self-consistent.

2.3.1 YIG materials

The experimental investigations about magnetization dynamics has challenged the realiza-

tion of techniques to growth suitable magnetic materials to realize those experiences. The

success in the spin waves physics issue has experienced a strong development, fundamentally in-

spired by the discovery of Yttrium Iron Garnet (YIG) magnetic material[22]. Indeed the marvel

properties of YIG materials have meant a vertiginous progress in the eld of spin waves physics

for forty years and mainly during the last two decades. For this reason the understanding of

the spin wave, and recently the concept of magnonics, are setting new trends in the design of

YIG-based devices for the transmission, storage and processing of information using spin waves.

Among all key properties one of which has done to this magnetic crystal, a unique material,

is their narrowest ferromagnetic resonance line, generally smaller that 0.5[Oe], which result in

a lowest spin-wave damping and then a magnon lifetime of a few hundred nanoseconds. Unlike

in pure iron and in the commonly used polycrystalline alloy permalloy, Ni81Fe19, the magnon

lifetime is of the order of nanoseconds, which if is combined together with slow magnon speeds

(approximately four orders of magnitude slower than the speed of light) produce a spin-wave

mean free paths typically less than 10[µm]. In that sense, the low damping in YIG lms provides

a spin-wave propagation to be observed on macroscopic scales.

The prominent role of the YIG magnetic materials due to their physical properties has

made this material, the king of the low-damping magnetic crystal and that has meant a great

progress in the spin waves investigation. Just to mention a few novel phenomena [23], formation

of solitions, dierent types of parametric instability, spin Seebeck eect, self-oscillating systems

and Bose-Einstein condensation of magnon at room temperature.

The Yttrium Iron Garnet has a complex cubic crystal structure, like bcc crystalline lattice,

where each unit cell houses 80 atoms and takes up a half of a cube with the lattice constant

19


Dodecahedral siteOctahedral site

Tetrahedral site

Fe3+

Y Fe O3 5 12

Y 3+

O 2-

Figure 2.4: Crystalline structure of the unit cell of Yttrium Iron Garnet material. The Y3+ ionsare distributed in the crystal over each dodecahedral site. The unit cell contains twenty Fe3+

ions being distributed over two antiferromagnetically coupled octahedral, 8 ions, and tetrahedral12 ions sub-lattices. The O2− ions are represented by little white spheres located in the vertexof the gure.

20


a = 12.4A. Each one of such cells contain twenty magnetic Fe3+ ions being distributed over

two antiferromagnetically coupled octahedral, 8 ions, and tetrahedral 12 ions sub-lattices, while

the Y3+ ions occupy all the dodecahedral sites, see Fig. (2.4). Only Fe ions have magnetic

moments in YIG and due to their inequivalent position result in a magnetic(ferrimagnetic)

behavior of this material with a curie temperature TC = 560[K]. The YIG materials have a

magnetization saturation, where the material behaves as a ferromagnet, of 1.7× 103[Gauss] at

room temperature. For more detail about their crystallographic properties can be reviewed

[22] and as well the optimal conditions for YIG fabrication [24, 25]. This material manifests a

high transparency to visible light which has made it possible the direct observation of dierent

class of phenomena by optical technique, like Brillouin Light Scattering spectroscopy method.

Within our aim in this work, the YIG lm is a crucial piece in the theoretical development of

quantum properties of the magnon gas.

2.3.2 Microwave excitation of magnons

In order to investigate the spin waves physics, dierent mechanisms are implemented to

excite traveling spin waves. For example antennas based on microstrip lines [15], which create

in the bulk of the lm a nonuniform alternating magnetic eld, exciting spin waves. In this

section we focused in the physical mechanism of excitations of magnons on magnetic materials.

As well as the spectrum is sensitive to the geometry, dened by the magnetization saturation

eld and the direction of spin wave propagating, the excitation too. When the amplitude of

this pumping mode exceeds certain threshold value, the energy acquired by spin waves from

this mode compensates spin-wave losses, the instability is activated. When the lm is normally

magnetized, i.e. θ = π/2 see Fig. (2.1), this instability phenomena is denominated transverse

pumping. While the spin waves excited by a magnetic eld tangential to the steady eld, is

called parallel pumping [12].

The applied dynamic microwave magnetic eld is oriented parallel to the direction of the

in-plane static magnetic eld H0. The spatially uniform pumping eld produces a modulation of

the longitudinal component of the magnetization with frequency 2ωp. This modulation produce

a parametric excitation of two spin waves with frequency ωp at equal but opposite wavevectors

±kp.

From a microscopic point of view the above process can be considered as the creation of

two magnons by a photon of the pumping eld. The wave vectors and frequency of the cre-

21


p

photon

magnon

(2ω )p

(k, ω )p

(-k, ω )

magnon

(k, ω )p(-k, ω )p

(0,2ω )p

Figure 2.5: Elementary scattering process involved in the parametric excitations of magnons.In the mechanism illustrated the photon creates a pair of magnons with opposite momenta andthe middle of the pumping frequency.

ated magnons are ±kp and ωp, respectively. Writing the microwave pumping eld as h(t) =

h cos (ωpt) z, the interaction between these with the magnetic system can be expressed as a

Zeeman interaction, whose hamiltonian for a ferromagnetic lm in terms of magnon operators

Eq. (2.24) is given by

Hp =~2

∑k

hρke−iωptc†kc

†−k + h.c. (2.33)

with ρk = γωM((1− Fk) sin2 θk − Fk

)/4ωk and where the form of Eq. (2.33) comes from to

perform the aforementioned Bogoliubov transformation. The microscopic process associated to

the parametric pumping of magnons is depicted in the Fig. (2.5). The Heisenberg equation of

motion for the bosonic operators, with the Hamiltonian H = H0 +Hp, is

dckdt

=1

i~[ck,H] , (2.34)

can be easily solved, having dening 〈n〉 = 〈c†kck〉. The formal expression for the pumped

magnon population is

〈nk(t)〉 = 〈nk(0)〉 exp(2[(hρk)

2 −∆ω2k

]1/2 − ηk) t (2.35)

where ηk is phenomenological parameter which model the damping in the systen and 〈nk(0)〉 is

22


assumed to be the thermal magnons. The parallel pumping excitation is expressed by the Eq.

(2.35), i.e. a magnon pairs with frequency ωk = ωp/2 and momentum k are driven parametrically

and their number grow exponentially when the pumping amplitude exceeds the critical value,

hc =[(ηk)

2 −∆ω2k

]1/2. (2.36)

The huge increase of the number of magnons not only remove the gas from the thermody-

namical equilibrium, but also change the eective nonlinear interactions among them.

2.3.3 Brillouin light scattering spectroscopy

The growing attention devoted to spin waves dynamics on dierent magnetic media has

demanded increasingly accurate techniques for a wide comprehension of dynamic properties of

magnetic media. For a variety of reasons, either from the fundamental physics to understand

the spin wave behavior or applications to the development of magnetic devices, the essential

point is provide a precise tool for detecting spin waves.

The most widely experimental mechanism used to investigate collective excitations on mag-

netic material, with frequencies in the GHz regime, is the optical spectroscopic Brillouin Light

Scattering (BLS) method[26]. The BLS technique has many advantages over microwave spec-

troscopy respect to the rest of spectroscopic methods. The BLS technique allows:

• to investigate spin waves with dierent values and orientations of their wavevectors.

• the possibility to study a wide dynamic range from low-amplitude thermal spin waves to

high-amplitude spin waves excited by external microwave eld.

• high spatial resolution dened, for the rst case, by the size of the laser beam focus, which

is 30− 50µm in diameter and high temporal resolution which is of the order of ∼ 1[ns].

The last point has been enhanced with the development of the micro-BLS technique [27].

The microscopic nature of the BLS method is illustrated in the Fig. (2.6) where the scattering

mechanism between magnons and photons is depicted. Photons with energy ~ωi and momentum

~ki interact with magnons characterized by (~ωm, ~km). The scattered photon increases, or

decreases, his energy and momentum:

23


magnon

photon

(k , ω )mm

(k , ω )ii

photon

(k , ω )ss

photon

photon

magnonmagnon

(k , ω )mm

magnon

photon

(k , ω )ss

photon

(k , ω )ii

photonphoton

Figure 2.6: Sketch of the nature of Brillouin scattering processes. The inelastic scatteringbetween one photon and a magnon is displayed in two cases. In both cases the frequency of thescattered photon presents a shift that represent the absorption or emission of a magnon. Thatprocesses are so-called anti-stokes and stokes, respectively, where the momentum and frequencyfor both mechanism are (ks, ωs) = (km ± ki, ωm ± ωi).

~ωs = ~ωi ± ~ωm (2.37)

~ks = ~ki ± ~km

where the ± refers to annihilation(creation) of a magnon. Those processes are denominated

stokes and anti-stokes, respectively, and correspond to shift in the spectral density observed,

where for nite temperature both processes have almost the same probability. From a classical

approach the scattering processes can be understood as the magneto-optic eect due to the

interaction between the magnetic system and the incident light. That interplay result in a local

change, both the space and time, of the permittivity tensor ~ε of the crystal. Indeed the presence

of a spin wave of the form ~m ∼ ei(ωmt−kmr) results in the modulation of the permittivity tensor.

For a quantitative description of the scattering processes involved in the BLS method, the

idea of dierential light scattering cross section becomes handy. The intensity of light is pro-

portional to the number of photons scattered into the solid angle dΩ in the frequency interval

between ωs and ωs + dωs per unit incident ux density. It can be written as

24


d2σ

dΩdωs∝ 〈δε∗ (ki − ks) δε (ki − ks)〉ωi−ωs (2.38)

where δε is the uctuating term of the dielectric permittivity and 〈· · · 〉 represent a statistical

average. That dynamic contribution to the dielectric permittivity comes from the spin waves

due to magneto-optical eects that gives rise to the scattering, it is proportional to the dynamic

part of the magnetization of the spin wave. In fact that relation can be made explicit, expressing

the correlation function in Fourier space

〈δε∗ (k) δε (k)〉ω =

∫d(t2 − t1)d(r2 − r1) exp [−iωt− ik(r2 − r1)] 〈δε∗ (r1, t1) δε∗ (r2, t2)〉

∝∫d(t2 − t1)d(r2 − r1) exp [−iωt− ik(r2 − r1)] 〈m∗ (r1, t1) m (r2, t2)〉.

(2.39)

The expression found in Eq. (2.39) establishes a relationship between the BLS intensity,

given by of the Eq. (2.38), and the magnetization components. In fact since the light scattering

cross section depend on the magnetization square, the frequency of the BLS intensity patterns

will be twice than that for the magnetization. A detailed studied presented for the Brillouin light

scattering based on Green-function theory including dipole-dipole and exchange contributions

can be found in [28].

In summary, the BLS technique can access a high degree of information since it allows, at

the same time, the measurement of inelastically scattered light under a certain angle allows cal-

culation of the k-vector and, by measuring the energy shift, the spectral distribution frequency.

It should be noted this method has experienced signicant progress increasing their perfection

over the last years, allowing it to be applied to approaching 200nm-sized structures, extension

denominated µBLS, and even reaching resolutions below 55nm. The µ-BLS technique is the

great interest for us, because was used for the rst observation of Bose Einstein condensates of

magnons, see references about them. For all these reasons, the BLS spectroscopic method, is

the most powerful method to access the dispersion and spin-wave band structure of a magnetic

material directly.

25


2.3.4 Schematic structure of spin waves experiments on YIG thin lms

In this subsection we give some basics insights of the techniques commonly employed in the

experimental study of spin waves on YIG-based devices[24][25].

Optically transparent single-crystalline YIG thin lms with crystallographic orientation

(111) and typical thickness of ∼ 5µm are used in experiments regarding of nonlinear magnetiza-

tion dynamics, for example; spin waves dynamics on magnonic crystal, soliton propagations or

Bose-Einstein condensation of magnons. The YIG lms are epitaxially grown on a gadolinium-

gallium-garnet (GGG) substrate. The lattice constant of GGG (12.383A) is very well matched

to YIG materials, enabling the fabrication of high quality, defect-free, unstressed lms. The

lm sample are mounted onto a microstrip resonator with resonant frequency in the GHz regime

providing a pumping eld. In Fig. (2.7) two types of basics experimental setups are is displayed.

In the rst gure the microstrip antenna are placed over the extremum of the sample where

the spin wave excited by the input antenna is captured, by the inverse process, on the output

antenna. In the second illustration the spin waves are excited by the microwave pumping eld,

but the dynamics are measured by the BLS spectroscopy method. Variants of that scheme

change the spatial and temporal resolution.

GGGYIG

HInput antenna

GGGYIG

H

MW pumping Microstrip antenna

Output antenna

MW pumping

Figure 2.7: Illustration of basic sketch of the experimental setup for the investigation of spinwaves on YIG-based samples.

In order to study magnetization dynamic properties of ferromagnetic system, the sample

is located in spatially uniform static magnetic eld H. As discussed above, the application of

magnetic eld determine the typology of the spin waves. To investigate dierent spin wave

behaviors the previous schemes are modied, either changing locally the magnetization by in-

troducing wire conductors to study tunneling or interference of spin waves, or attaching a metal

26

2.4 Bose-Einstein condensation of magnons at room temperature:experimental motivation

grating on YIG lms producing magnonic crystals to study transport of magnons.

2.4 Bose-Einstein condensation of magnons at room tem-

perature: experimental motivation

The recent discovery of Bose-Einstein condensation of magnons at room temperature[29]

has challenged our understanding concerning to the concept of condensation and consecutively

the way we think about quantum phenomena. Indeed the questions that naturally arise come

from physical mechanism behind the formation of condensate state, what does it mean the

concept of condensation?, when does the magnon gas reaches this stage, is the gas in a state of

equilibrium?, what are the superuid properties of the magnon gas?, how manifest themselves?,

or being even more fundamental, how legitimate is the quantum state observed and if is it even

possible to talk of condensate state.

All these naive questions, are dicult to answer, we will try to clarify in the course of this

work. However, for the time being we can outline some key ideas. Is widely known that magnons

in thermal equilibrium have zero chemical potential, since these doesn't preserve their total

population. For this reason the condensation of magnons is forbidden and to circumvent this

diculty one would needs to increase the chemical potential increasing the density of magnons,

for example by parametric pumping.

The experiment was performed on ferromagnetic YIG thin lms where the kinetics and

thermodynamics of magnons were investigated using advanced time and spatial resolved BLS

technique, while the external magnon injection was provided by the microwave microstrip res-

onator. It was showed that for high enough pumping powers the relaxation of the driven magnon

gas results in a quasiequilibrium condensation state. An increase of the pumping power above

a critical value is reported by an observation of magnon over-accumulation at the lowest energy

level. In the Fig. (2.8) several graphs are depicted with the magnon distribution in the momen-

tum space in the form of constant-frequency contours for a amplitude of pumping power above

the threshold value[30]. Each line correspond a dierent isofrequency, while the cross indicate

the point of bottom state in the dispersion spectrum. The colors represents the measured nor-

malized BLS intensity of magnon density distributed in the k-space. Once the pumping pulse

is started, the magnons appears at ∼ 20[ns] occupying dierent k|| and k⊥ states corresponding

to the constant frequency of pumped fp/2 = 4.1[GHz]. The magnons redistribute over the

27


100 150 200

Delay, ns

-2

-1

0

1

2

e) =200 nsτ e) =700 nsτ

e) =60 nsτ e) =100 nsτ

a) =20 nsτ b) =40 nsτk

, 10

cm

4

-1

0 1 2 3 4

k , 10 cm 4 -1

0 1 2 3 4

k , 10 cm 4 -1

-2

-1

0

1

2

k ,

10

cm

4-1

-2

-1

0

1

2

k ,

10

cm

4-1

Figure 2.8: Set of graphs showing the magnon spectrum at constant frequency (isofrequency)and the magnon density depicted by normalized BLS intensity measured for dierent two-dimensional magnon wave vectors k =

(k||, k⊥

). Each map represents the distribution of

magnons in the k-space, for dierent delays, after the pumping power was swithed-o. Thelatter maps reect the overpopulation of the bottom state, with the cross indicates their posi-tion in the magnon spectrum. The exerimental parameters were H = 1[KOe], fp = 8.24[GHz],P = 8[GHz] and pumping pulse τ = 30[ns]. V.E. Demidov, et al., Phys. Rev. Lett. 101 257201(2008).

28


momentum space in such way that the maximum occupation is continuously shifted towards

the point corresponding to the bottom state in the magnon spectrum. From the consecutive

graphs this tendency is is clear and after the stage of redistribution processes the magnons

mainly occupy states in the proximity of the bottom of the spectrum. At the nal of this stage,

∼ 700[ns], a narrow distribution form a peak in the minimum of the energy. That observed

peaklike distribution, is associated with the Bose-Einstein condensation of magnons.

In the same way that before, but now in the spirit of investigate the quantum coherent

properties, the approach is addressed to prove the spontaneous emergence of coherence of the

magnons at the lowest level when the density exceeds some critical level[31]. The magnon gas is

driven out of thermodynamical equilibrium injecting magnons by parametric pumping method.

When the magnon gas is pumped with the amplitude above the threshold Pc, as was shownearlier, the magnons collecting the bottom state and suddenly emerge a narrow distribution of

them in the dispersion spectrum. The evolution of the peak magnon distribution, exhibited

in the Fig. (2.9), is shown as a function of the delay time with respect to the start of the

pumping pulse, where the attention is placed on rate decay of the magnon gas in the minimum

of the spectrum. Albeit the over population of the bottom state is allowed suddenly, and by

suddenly we thinck in the concept of spontaneous, the question that keep in mind is is that

physical process coherent?. The experimental approach to answer that question is to compare

the decay rate of the magnon gas in two cases, either below and above of the critical value

of the pumping. It was found that the for the same dacay of the magnon density, the BLS

intensity decays twice as fast for coherent magnons than that for incoherent magnons. Indeed

there is a dierent qualitative behavior if the magnon gas is driven to the minimum energy

state with a pumping amplitude above or below the critical value, see Fig. (2.10). In fact it is

concluded that for pumping powers P > 2.5[W] the magnons at the bottom state start to form

a coherent Bose-Einstein condensate and the contribution of the condensate to BLS dominates

for P > 4[W]. In such way the experimental data clearly show that the magnons accumulated

at the lowest energy level are coherent at high enough pumping powers, and this coherence

emerges spontaneously if the density of magnons exceeds a certain critical value.

This nding gives the direct experimental evidence of the BEC transition in a magnon

gas at room temperature. In the course of this work we will try to understand the precise

meaning of those quantum coherence. Indeed the quantum nature of those phase will be unravel

throughout of a many-body microscopic theory for a order parameter, that contains all of

coherence properties displayed in this section.

29


3.0

3.5

4.0

4.5

0 50 100 150 200

Delay, [ns]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.0 3.5 4.0 4.50.0

0.1

0.2

0.3

0.4

0.5

Frequency, [GHz]

3.0 3.5 4.0 4.5

Frequency, [GHz]

Nor

mal

ized

BL

S in

tens

ity

Fre

quen

cy, [

GH

z]

Figure 2.9: Redistribution of magnons injected by a short-pulse of parametric pumping withpower P = 3[W]. The spectral density of magnons, represented by the colored distribution ofthe BLS intensity over the spectrum, is depicted as a function of the delay time respect to thestart of the pumping pulse. b) BLS spectra for dierent delays. The inset shows the t of themeasured magnon distribution using the Bose-Einstein statistic. V.E. Demidov, et al., Phys.Rev. Lett. 100 047205 (2008).

30


400

Delay, [ns]500 600 700 800 900 1000

1

10

100

Nor

mal

ized

BL

S in

tens

ity P = 2 W

P = 2.5 WP = 4 WP = 6 W

Figure 2.10: Graphs of decay rate of magnon density, represented by the normalized BLSintensity, at the lowest level in the spectrum. After the formation of the overpopulated state inthe frequency ωm, the magnon gas leave that state with mainly two rates of decay for dierentvalues od the pumping power. The decay rate for coherent magnons is double respect to themagnon gas when the pumping power is smaller than the critical value. V.E. Demidov, et al.,Phys. Rev. Lett. 100 047205 (2008).

31


Before concluding, it is worth noting some properties that characterize this system and

that addresses our approach. The most fundamental requirement is related to remove the

magnon gas from their thermodynamical equilibrium and achieve high densities to eventually

reach the condensation phase transition. Due to the dynamic of nonequilibrium magnons,

the magnon-magnon scattering time scale is lesser than the magnon decay time, so that the

magnon gas reaches the thermalization. On these grounds the description of emergence of a long

range order in the magnon gas, i.e. quantum coherence, require a accurate description where

both nonequilibrium behavior and many-body eects must be taken into consideration. The

next chapter is devoted to develop the precise formalism to describe the phenomena previously

mentioned.

32

Chapter 3

Condensation of a Magnon Gas

In this chapter we develops a eective theory for the out of equilibrium magnon gas highly

dense on a ferromagnetic ordered crystal. Starting from a standard hamiltonian model will be

determined the eect of the interacting magnon gas over the two-magnon scattering by means

of many-body T-matrix approach within the ladder approximation. Collecting the main con-

tributions of the elementary scattering processes we arrive, nally, to the semiclassical eective

action within the low-energy limit.

3.1 Introduction

In the last Chapter we have established the basic physics of the spin waves over ferromagnetic

thin lms of YIG. To deduce the dispersion relation for spin waves was assumed a microscopic

approach. Furthermore we have introduced the concept of magnon as an elementary excitation

of spin waves or in a more sophisticated language, a irreducible representation of a SU(2) algebra

of spin operators in terms of bosonic operators.

For decades, the phenomenon of Bose-Einstein condensation (BEC) has dazzled as a remark-

able manifestation of the quantum nature of matter. Although it was achieved experimentally

long after its prediction [33], the phenomenon had already became a leading theme in the eld

of physics of quantum collective phenomena. Indeed, its closed ties with superuidity and su-

perconductivity provided enough motivation to make the study of the physical behavior of BEC

to grow into a large and subtle subject of theoretical physics [34].

In this Chapter we address, for the interesting phenomena revealed by several experiments

on the formation of room temperature BEC of magnons on magnetic thin lms [30, 31, 36],

the issue of the dynamical behavior of the condensate and the physical nature of the quantum

coherence that is displayed by these systems. The experiments are performed by externally

3.2 Hamiltonian Approach for Magnons Systems

exciting a large population of magnons on an yttrium iron garnet (YIG) thin lm and tracking

their subsequent behavior through optical techniques. The understanding of these features lies

right at the frontier of the knowledge of quantum behavior of non-equilibrium systems. Non-

equilibrium BEC has attracted a great deal of attention in dierent physical systems such as

excitons [37, 38, 39, 40], phonons [41] and polaritons [42].

Several works have been devoted to understand the condensation process [44, 45]. Neverthe-

less a precise description of the collective dynamics of the condensed magnons is still lacking.

The reasons behind this are found on the intrinsic non-equilibrium nature of the system and

the lack of conservation of the excitations that manifest itself as a lack of gauge symmetry in

the microscopic models of the system.

Based upon a quantum functional kinetic theory [46], we will provide a systematic account

of the phenomenology of magnon condensation and further clarify the role played on it by the

processes that violate the conservation of magnons. Therefore the theory we propose consistently

accounts for the transient nature of the condensate. Our eort is crowned by the derivation of a

semiclassical eective action that described the low-energy collective dynamics of the magnons

in condensate state [47].

The question that arise and the bosonic behavior of the magnons drives the system, eventu-

ally, into a condensate as will be shown in the following sections. In fact the following section

is devoted to the justication and specication of the basic reduced model that will be used to

describe the magnon dynamics. In section 3.4 we proceed to derive the eective action of the

magnon system through the diagrammatic representation of the microscopic processes and their

collective eects within the Schwinger-Keldysh non-equilibrium formalism. Section 3.6 explores

the trail of consequences left behind by the reduced eective action and the role that symmetry

breaking plays in the condensed phase, where some technical details on the evaluation of certain

diagrams are discussed in the appendix.


Once the magnon gas were excited through a process of parametric parallel pumping the

ferromagnetic relaxation mechanism take place over the dynamical evolution and the magnons

are driven, solely by their magnetic interactions, into states of lower energy [29]. In momentum

space there are two opposite points that correspond to the lowest energy states, see Fig. (3.2)

34


[15, 44]. According with the double degeneracy of the spectrum, the magnon population grows

in the vicinity of those points. To answer the question previously established, rst we have to

adopt a microscopic approach, introducing rigorously the magnon concept, since the condensate

magnon state has intrinsically quantum nature. From a microscopic point of view the precession

of spin placed in each site of the lattice is described by the Heisenberg equation

i~dSidt

=[H, Si

](3.1)

where the Hamiltonian, in terms of the spin operators Si at each site i of the lattice, is con-

stituted by adding up the contributions of the Zeeman energy, dipolar and exchange magnetic

interactions and external microwave pumping,

H[Si] = HZ +Hex +Hdip +Hp. (3.2)

However the description of our interest is in terms of the magnon system and can be found

introducing a representation of the SU(2) algebra of spin operators in terms of a single Bose

operator. In other words we unveil the bosonic behavior of the magnons using the socalled

Holstein-Primako representation of the spin operators [19, 48]:

S+i = (

√2S − nib)bi,

S−i = b†i

√2S − nib,

Szi = −nib + S. (3.3)

where in this representation the spin raising (resp. lowering) operator is associated with the

annihilation (resp. creation) of a bosonic excitation S+i ∼ bi (resp. S

−i ∼ b†i ). It is worth note

that the new bosonic operators acting on a physical subspace of the innite boson Fock space,

which this subspace is dened for nib ≤ 2S.

The task of reducing the original Hamiltonian to its bosonic representation has been pursued

by other authors in several contexts as the works on ferromagnetic resonance and parametric

excitations. There is a wide consensus [19, 15, 22] on the basic phenomenology that dominates

the dynamical processes, namely: (a) a dipolar interaction-renormalized dispersion relation,

that shifts the states of minimum energy away from the k = 0, that is expected solely on

account of the exchange term, to k = ±k0; (b) a so called 3-magnon conuence (resp. splitting)

term that reduces (resp. increases) the magnon number. These processes are consequence of the

35


long wave length contributions of the dipolar energy; (c) a magnon-magnon scattering term that

comprises contributions of both the exchange and the dipolar interactions. Expanding the spin

operator Eq. (3.3) in Taylor series and substituting in Eq. (3.2) one gets for the Hamiltonian

of the ferromagnet

H = H(0) +H(2) +H(3) +H(4) + · · · (3.4)

where each part of the Hamiltonian H(2),H(3),H(4) represent the proportional terms to second,

third and fourth order in b,b† and describe the interactions between spins at dierent lattice

sites. Now if we perform a Fourier transform we can pass from the bi operator in i site to the

operator bk such that represent the collective behavior of the all set of spins.

Finally we can cast those eects into a model Hamiltonian that simply reads as,

H = H(a) +H(b) +H(c) +H(d) (3.5)

where is implicit the diagonalization by means of the Bogoliubov transformation. Thus, we

summarize the most essential part from the Eq. (3.2) as follow

H(a) =∑

k

~ωkb†kbk, (3.6)

corresponds to the free magnon contributions,

H(b) =∑

V(2,1)k1,k2,k3

bk1b†k2

b†k3+ h.c., (3.7)

represents the splitting and conuence contributions,

H(c) =∑

V(2,2)k1,k2,k3,k4

bk1bk2b†k3

b†k4+ h.c. (3.8)

representing the magnon-magnon scattering processes, and nally,

H(d) =∑

ρke−iωptb†kb†−k + h.c., (3.9)

being the external pumping mechanism. In writing Eq. (3.5), we have neglected 3-magnon

and higher processes of conuence and splitting. With this assumption we are accepting that

the cross section associated with the process of one magnon decaying into three is considerable

smaller than the one with the process of one magnon decaying into two [12]. This approach is

36


linked to the fact that the parametric parallel pumping allow us excite only small precession

angles. Additionally, we consider the magnon gas decoupled from the crystalline lattice. This

approximation is justied since the spin-lattice (magnon-phonon) mediated processes are associ-

ated with a typical magnon lifetime of 1µs [12, 49] while the processes leading to thermalization

of the magnon system under consideration are associated with the nanosecond scale [50].

We remark the explicit global U(1) symmetry breaking induced by H(b) on this model.

This behavior reects the fact that the full dipolar interaction term does not conserve the net

magnon number. It will be shown in the nal section of this chapter, that in spite this fact we

still can understand the system's behavior in terms of spontaneous coherence phenomena. We

emphasize that, we are focused to describe the dynamics of the system once the condensation

has been achieved, where the pumping has switched o. Nevertheless to give an account of

the mechanism of formation of magnon condensate the pumping processes, into our model

Hamiltonian Eq. (3.5), plays a fundamental role to drive the system out of thermodynamic

equilibrium state.

Figure 3.1: Feynmann diagrams depicted for the splitting and conuence scattering processesrespectively. The triangle is a pictorial representation of interaction mechanism mentionedabove. The coupling, denoted by U (2,1)(U (1,2)) for splitting(conuence) process, of each one ofthese processes stem from the dipole-dipole interaction between the spins in the lattice crystalYIG.

In agreement with the above discussion, the magnons will meet mostly around the bottom

of the dispersion relation

~ω(k) = γ[(H +Dk2 + 4πM (1− Fk) sin2 θk

) (H +Dk2 + 4πMFk

)]1/2(3.10)

and then can be written, in the vicinity of the base states, in terms of eective masses,

~ω(k) = ~ω0 +~2

2m||q2|| +

~2

2m⊥q2⊥, (3.11)

37

3.3 Phenomenological Description of the Magnon Condensate

where q = k± k0. This dispersion relation is represented by dashed lines in Fig. (3.2).

!1.0 !0.5 0.0 0.5 1.0

4.0

8.0

12.0

k|| [106 cm-1] ω[G

Hz]

Φ1 Φ2

1 2 3 4 52.5

3.0

3.5

4.0

4.5

5.0

H [kOe]

m [1

0-3 m

He] !H

k!

k|| d

50 m!

m||

Figure 3.2: Spectrum of magnons in a YIG thin lm, with an in-plane eld of H ∼ 1kOe andlm thickness d ∼ 5µm, for momentum parallel to the eld. The continuous line correspondto the spectrum as presented in [44]. Magnons accumulate in the vicinity of the two minimumenergy states around which the dispersion is accurately described by a quadratic form depictedby dashed lines. Inset.- Eective masses around the minimum energy states as a function ofthe external magnetic eld. The system anisotropy manifest it self in distinct masses for spinwaves with momenta along and perpendicular to the external magnetic eld. The masses dierby a factor of about 102 rendering the magnon system as highly anisotropic.

In addition to this interactions, magnon coupling to other degrees of freedom (such as lat-

tice dynamics) could be represented by an eective reservoir plus system model in the form

suggested by Caldeira-Leggett[51]. Precise details of the physical nature of this reservoir are

not expected to be relevant for the physical behavior of the low energy regime.

3.3 Phenomenological Description of the Magnon Conden-

sate

In the following sections we will discuss theoretically the existence of phase coherence start-

ing from the standard microscopic description of the magnon gas dynamics. Our theoretical

38


description predicts the existence of such coherence when the magnon density reaches a certain

critical density. Interactions between magnons turned out to be essential in the creation of

such coherence. In this section we pursue a phenomenological approach, based on the basic

microscopic features that gave rise to our next treatment. The basic conclusions reached at the

end of both treatments are essentially equivalent. The basic processes that have been found

[19, 15] to dominate magnon dynamics are:

1. a dipolar interaction-renormalized dispersion relation, that shifts the states of minimum

energy away from the k = 0, that is expected solely on account of the exchange term, to

k = ±k0, this degenerate minima is depicted in Fig.(3.2);

2. a so-called 3-magnon conuence (resp. splitting) term that reduces (resp. increases) the

magnon number. These processes are consequence of the long wave length contributions

of the dipolar energy;

3. a magnon-magnon scattering term that comprises contributions of both the exchange and

the dipolar interactions,

4. parametric excitation of magnons, through a pumping eld that creates magnons at a

rate, P . Magnon condensation, in the form of macroscopical occupation of the lowest

energy state, is observed when P exceeds a critical value, Pc.

We note that magnon excitations can be treated eectively as bosonic excitations. Indeed,

we can use bosonic operators directly related to the magnetization through the well known

Holstein-Primako transformation [20, 48]. In this representation the spin ladder operators are

mapped into bosonic creation and annihilation operators. In this way the spin raising operator

is associated with the annihilation of a bosonic excitation S+i ∼ b, while the spin lowering

operator is correspondingly associated with the creation of a bosonic excitation S−i ∼ b†. The

dispersion relation

We remark the explicit global U(1) symmetry breaking induced by magnon decay processes

on this model. This peculiar behavior reects the fact that the full dipolar interaction term

does not conserve the net magnon number. It will be shown that this fact does not pose any

obstacle to a proper interpretation of the system's behavior in terms of spontaneous coherence

phenomena. This fact is in direct analogy with the case of the magneto-crystalline anisotropy

in ferromagnets. As in that case, the weak anisotropy is of relevance only after a condensate

state is achieved.

39


To take into account quantum coherence over macroscopic length scales an envelope wave-

function approach can be envisaged. From this picture, the system is described in terms of the

two collective wave-functions associated with each minima. Using the collective eld Φσ(x, t),

whose absolute value corresponds to the local density of magnons in states σ = −1, 1, see

Fig.(3.2), while its phase correspond to the local collective phase. The energy associated with

this state can be written in a compact form by using the following notation, (x, t, σ) labels will

be summarized in a single subindex. For an homogeneous system the energy can be expressed

in terms of the series expansion

E =∑

M=m+n

Γσ,x,tη,y,τ Φσ1(x1, t1) · · ·Φσn(xn, t)Φ∗η1

(y1, τ1) · · ·Φ∗ηm(ym, τm) (3.12)

Terms unbalanced in the eld and it conjugate are explicit violation of overall U(1) symmetry

associated with conservation of the number of particles. In general, this argument forces them

to cancel. However, the microscopic dynamics of magnons does not manifest invariance under

such symmetry, reecting the inherent lack of magnon conservation, and in principle such terms

must be considered. By restricting our attention to low momentum, we need to focus only

in those terms for which (σ1 + · · ·+ σn) = (η1 + · · ·+ ηm). In particular we can discard from

the contribution to the energy, terms proportional to odd powers of the elds. Such reduction

is as far as one can get due to the U(1) symmetry-breaking terms. The anomalous-terms are

restricted to valley-mixing terms. Requiring that: (1) in the limit of vanishing density the

system recovers the magnon spectrum Eq. (3.11), (2) the net momentum of the magnons be

zero, and (3) the system is symmetric with respect to valley indices; it is possible to simplify

the energy into:

E [Φ,Φ∗] =

∫dr((

Φ†1~ω(∂r)Φ1 + Φ†2~ω(∂r)Φ2

)+ µ

(Φ†1Φ1 + Φ†2Φ2

)+ νΦ†1Φ†2 + νΦ1Φ2

+γ1

2

(Φ†1Φ1 + Φ†2Φ2

)2

+γ2

2

(Φ†1Φ1 − Φ†2Φ2

)2)

(3.13)

where ν, γ1 and γ2 are phenomenological parameters that should be determined from the ex-

periment. Despite the explicit breakdown of the U(1) symmetry, as reected by the terms

proportional to ν, this energy is invariant under the residual symmetry transformation:

Φ1 → eiδΦ1 and, Φ2 → e−iδΦ2. (3.14)

40


The parameters µ, ν, γ1 can be obtained from experimental data as follows. First we set

ν = νeiψν If γ2 > 0 the energy is easily minimized by equally populating both valleys. Let

Φσ =√neiψσ , the energy density becomes

EA

= 2n (µ+ ν cos (ψ1 + ψ2 + ψν)) + 2γ1n2 (3.15)

From the last equation, we nd a condensation transition at µ − ν < 0. We identify this

symmetry breaking transition with the transition towards a macroscopically occupied lowest

energy state reported in the experiments [29, 30, 31, 81]. We can use this fact to associate

µ− ν = λ (Pc − P) , (3.16)

for a positive value phenomenological parameter λ. The stationary density of magnons in such

regime is:

nBEC = 2n =λ

γ1

(P − Pc) . (3.17)

Remarkably, this behavior, linear in (P − Pc) is a natural consequence of the phenomenological

approach together with the assumption in Eq. (3.16). Its agreement with experimental data

[50] can be veried readily. Additionally, following [34] a healing length can be calculated:

ζ2 =~2

2√m||m⊥λ

(P − Pc)−1 . (3.18)

with a measurement of Pc, nBEC and ζ the phenomenological parameters can be determined.

The main result of this section is to provide a phenomenological picture of the collective

dynamics of the magnons. The dissipation mechanisms can be encoded, within the phenomeno-

logical approach, in terms of a Rayleigh dissipation function [35]. In principle this function

must be expanded in powers of ∂tΦi, this expansion to lowest order becomes:

R = α

∫dr(|∂tΦ1|2 + |∂tΦ2|2

). (3.19)

where α characterizes the damping constant as a phenomenological parameter.

The condensate consists, roughly speaking, of two magnon condensates lying the vicinity of

the two points of minimum energy in momentum space, and magnetic interactions introduce a

41

3.4 Many-Body Scattering Theory for magnon-magnon Interactions

coupling between them. The basic phenomenological description of the dynamics is obtained,

using the energy functional E together with a kinetic term

S =

∫drdt

(Φ†1i~∂tΦ1 + Φ†2i~∂tΦ2

)− E [Φ,Φ∗]. (3.20)

In the magnon condensate, the equations of motion correspond to the Euler-Lagrange equations:

δSδΦ†i

=δR

δ(∂tΦ

†i

) . (3.21)

Straightforward calculations lead to the conclusion that the dynamics of the two condensates

follow can be described by the following generalized pseudo-spin GPE:

i~(1 + iα)∂t|Ψ〉 = − ~2

2m||∇2|||Ψ〉 −

~2

2m⊥∇2⊥|Ψ〉+µ|Ψ〉+ νσx|Ψ∗〉

+γ1|Ψ|2|Ψ〉+ γ2〈Ψ|σz|Ψ〉σz|Ψ〉 (3.22)

wher the pseudo spin |Ψ〉 = (Φ1,Φ2)t, refers to valley degeneracy in momentum space, while α,

m, µ, ν and γi are real parameters characterizing the dynamics. Before closing this section we

comment on other phenomenological approaches that have been taken in the literature. Gross-

Piatevskii equations have been constructed to describe the dynamics of magnon condensates

in the works of [60, 61]. We emphasize that this form of the equation is essentially dierent

than the phenomenological ones proposed in those works, since Eq. (3.22) has a dierent form

for the dissipation term, an explicitly gauge symmetry breaking term and a spinor nature. An

equation with such features have been used independently in an unpublished experimental work

[81]. The task of determining the main parameters from such experiments will be tackled in a

following work.

3.4 Many-Body Scattering Theory for magnon-magnon In-

teractions

In this section we will discuss in detail the scattering theory for many-body systems, widely

studied in the context of Nuclear Physics and of course in Condensed Matter. Starting from the

most straightforward description of scattering theory, i.e. considering the two-body scattering

42


and within the Born's approximation limit, we will arrive to a most accurate and sophisticated

description of the eective interactions in a many-body system [52].

More specically, is imperative to obtain a suitable renormalization of the magnon propaga-

tion and the scattering of these, because the phenomena of Bose-Einstein condensation involve

collective as well as quantum coherent properties, where the many-body eects are important.

We will use the knowledge gained in the microscopic hamiltonian developed in the Chapter 2

with the quantum eld theory formalism to describe in a realistic way the quantum coherent

properties arising in a high density magnon system.

3.4.1 Two-body scattering theory

The rst step to build a eective many-body scattering theory for magnons is pick up the

main physics of the most basic blocks in the magnon-magnon scattering process. This will

lead us to understand the microscopic origin and characterize the relevant parameters that

describe the physics of magnons system. The answers to this goal can be extracted with a

phenomenological focus based in the microscopic Hamiltonian, Eq. 3.5, or from the functional

form of the theory that we will deduce in the following section 3.5. Now we will adopt the

phenomenological and heuristic approach to nd the elementary two-magnon scattering and

their rst renormalization due to the dipole-dipole interaction eect.

We have showed that the microscopic Hamiltonian consists of several parts in terms of

the order of bosonic operators, whose parameters are a relationship between the strength of

exchange, dipolar and Zeeman interaction. It turns out that for experimental conditions, see

Chapter 2, the momentum dependence of interacting potential is neglected and we can write∑V

(2,2)k1,k2,k3,k4

b†k4b†k3

bk2bk1 =∑(

V(2,2)b†k4b†k3

bk2bk1 + U (1,2)U (2,1)b†k4b†k3

bk2bk1

), (3.23)

where the rst term correspond to the usual two-body scattering, while the second contribu-

tion can be seen as the joining splitting-conuence scattering process. Namely one magnon

with momentum K, coming from the splitting of a magnon with momentum k1, interact with

another with momentum k2, through the conuence mechanism and giving rise one magnon

with momentum k3, see the Fig. (3.3). In other words the bare interactions of two-magnon

scattering is renormalized by the contribution derived from the dipole-dipole interaction which

43


can be written as

Γ0(k1,k2,k3,k4; τ, τ ′) = Vδ (k1 + k2 − k3 − k4) δ (τ, τ ′)

+ U2 [G0(k1 − k4; τ, τ ′) + G0(k4 − k1; τ, τ ′)] (3.24)

includes two parameters V and U , where the superindex are suppressed. The free Green's

function G0 carriers the momentum that is transferred from the splitting to conuence process.

The temporal dependence of the propagator is evaluated at the time dierence τ − τ ′, i.e. thisprocess is nonlocal and unlike the rst instantaneous scattering. However, in the low-energy

limit we can consider such process as instantaneous.

= + +

k1 k2

k4 k3

k1 k2

k4 k3

k1 k2

k4 k3

V U U U U

Figure 3.3: Representation in terms of Feynmann diagrams of the bare interaction of two-magnon scattering process. The two-magnon scattering is renormalized due to the dipole-dipolecontribution and is proportional to U2. The usual two-body interaction strength, proportionalto V , is showed by the light gray square.

Within a more general context we can give a formal treatment of the scattering problem

described above. Strictly speaking, the trouble to nd the quasiparticle excitation energies

for a system composed by two particles and under the inuence of an interaction potential V .

Starting from the Schrodinger equation(H0 + V

)|ψ〉 = E|ψ〉 (3.25)

where we have that H0 = −~2∇2/2m for the kinetic-energy operator. In scattering problems

we have interested in solutions which asymptotically represent an incoming wave plane and an

outgoing spherical wave. Since in the absence of potential V there is no scattering and the

solutions are plane waves |k〉, the Eq. (3.25) can be formally solved by introducing scattering

states |ψ(+)k 〉 that satisfy the following recursive relation

|ψ(+)k 〉 = |k〉+ G0(2εk)V |ψ(+)

k 〉 (3.26)

44


with the noninteracting propagator of the particle dened as

G0(2εk) ≡(

2εk − H0 + iε)−1

, (3.27)

in the limit ε→ 0+, where the simbol (+) it means that for t→∞ the solutions were aected

by the potential, i.e. an outgoing spherical waves. In scattering theory a central role is played

by the operator dened through V |ψ(+)k 〉 ≡ T 2B(2εk)|k〉 and that determine the scattering

amplitude directly. If we multiply the Eq. (3.26) by V is obtained a equation for the two-body

T-matrix

T 2B(2εk)|k〉 =(V + V G0(2εk)T 2B(2εk)

)|k〉. (3.28)

This equation is known as Lippmann-Schwinger equation and can also be generalized to an

operator equation, whose recursive expansion gives rise to the Born series

T 2B(z) = V + V G0(z)T 2B(z)

= V + V G0(z)V + V G0(z)V G0(z)V + . . . (3.29)

As an example we can see that for spherically symmetric interaction potentials, in position

representation, the scattered state take the form

ψ(+)k (r) = eik·r + f(k,k′)

eikr

r(3.30)

where the rst term, on the right hand side, obviously correspond to the plane wave 〈r|k〉 ∼exp (ik · r/~). The second term represent the inuence of the potential once that the particle

have interacted, with the scattering amplitude given by

f(k,k′) = − m

4π~2〈r′|T 2B(2ε+k )k〉. (3.31)

For our purposes is more convenient to write the Lippmann-Schwinger equation in the following

form

T 2B(k,k′; τ, τ ′) =V (k,k′)δ(τ, τ ′)

+i

~

∫C∞dτ ′′

∫dk′′

(2π)2V (k,k′′)G0(k′′; τ, τ ′′)G0(k′′; τ, τ ′′)T 2B(k′′,k′; τ ′′, τ ′)

(3.32)

45


where the two-body T-matrix is directly evaluated at the momentum of the particle and the time

is dened over the Keldysh contour [53]. The potential V (k,k′) in the Eq. (3.32) correspond to

the bare interaction, Eq. (3.24), for the magnons on the ferromagnetic lm and the two-body

T-matrix is associated to the renormalization of bare interaction V (k,k′) due to the consecutive

self-interactions for the pair of interacting magnons.

Viewed another way, we have written the interactions among spins in the YIG lattice that

determine the spin wave dynamics, in a microscopic language where the elementary constitutes

are the magnons, where the interaction between these are described by the two-body T-matrix.

In that sense, it should be noted that from microscopic point of view is more accuracy to

give a description in terms of the two-body T-matrix T 2B, since experimentally is a quantity

physically more realistic. These approach is appropriate to describe the realization of magnons

at low magnon density, for example excitations of Backward Volume Magnetostatics Waves

(BVMSW) [14], i.e. spin waves which wave vector is parallel to the magnetization saturation.

However, the coherent properties of interest or the long-range order achieved by the magnon

gas, emerges as a consequence of collective interactions of a great number of constitutes and we

have to leave this approximation. Then is necessary consider not only the scattering between

two magnons but the possibility to scatter with the rest of gas and the many-body eect must

be taken in account as we will explain with accuracy in the next section.

3.4.2 Many-body T-Matrix and ladder approximation

In the previous subsection the bare magnon-magnon interaction was given, emphasizing its

internal structure, consisting of the usual two-body process and the combination of splitting-

conuence diagram. Moreover the concept of two-body transition matrix was introduced, which

provide the possibility to access to information on the properties of interacting systems. How-

ever, knowing only the features of the two-body system using the T-matrix approach is of course

unsatisfactory to describe accurately an interacting magnon gas. We generalize this procedure

to the case of an interacting many-body systems, where the inuence of the medium is taken

into account [52].

Before to introduce this ideas we would wish to give some more intuitive picture helped by

the standard diagrammatic theory. The many-body T-matrix is a generalization to the case

presented in the subsection 3.2.1, where we now think in the elemental scattering between two

magnons aected by the presence of the surrounding medium. As a result the nal scattered

46


states are occupation bosonic dependent where the many-body T-matrix satises

Γ(k,k′,K; τ, τ ′) = V (k,k′)δ(τ, τ ′)

+i

~

∫C∞dτ ′′

∫dk′′

(2π)2V (k,k′′)G(K/2 + k′′; τ, τ ′′)G(K/2 + k′′; τ, τ ′′)Γ(k′′,k′,K; τ ′′, τ ′), (3.33)

equation so-called the Bethe-Salpeter equation. Iterating this equation, we immediately note

that the many-body T-matrix indeed sum all of possible collisions between two magnons. Fur-

thermore, the Green's function G(K/2 + k′′; τ, τ ′′) describe the propagation of an magnon with

momentum ~ (K/2 + k′′) from time τ ′′ to time τ . Hence, we see that the many-body T-matrix

incorporates the eect of the surrounding gaseous medium on the propagation of the magnons

between the two collisions. This is unlike the two-body T-matrix theory presented in the last

subsection where we think the scattering as a isolated process. In other words the involved

energy in the magnon scattering is renormalized due to the possibility to the interact with the

rest of the many-magnon system.

If we write the Eq. (3.33) in the Fourier space, for the time coordinate, we note the explicit

dependence on the occupation number

G(K/2 + k′′; ε)G(K/2 + k′′; ε) ∝ 1 +N (K/2 + k′′) +N (K/2− k′′)

ε± − ε′ (K/2 + k′′)− ε′ (K/2− k′′)(3.34)

where the factor 1 + N (K/2 + k′′) + N (K/2− k′′) reect the fact to the scattering take

place in a medium. This term arise as the net dierence between two particles scattering

into and scattering out, which are proportional to (1 +N (K/2 + k′′)) (1 +N (K/2− k′′)) and

N (K/2 + k′′)N (K/2− k′′). Such considerations can be summarized from diagramatically

point of view, see Fig. (3.5). The Bethe-Salpeter equation can be written as a function of

two-body T-matrix as follow

Γ(k,k′,K; τ, τ ′) = Γ2B(k,k′; τ, τ ′)

+i

~

∫C∞dτ ′′

∫dk′′

(2π)2Γ2B(k,k′; τ, τ ′)G(K/2 + k′′; τ, τ ′′)G(K/2 + k′′; τ, τ ′′)Γ(k′′,k′,K; τ ′′, τ ′).

(3.35)

Once the foundations of the many-body T-matrix have been established the main eects of

the magnon gas surrounded are considered and we can determine and treat in a more realistic

way all scattering processes arising in the interacting many-body magnon system, resulting in

47

3.5 Many-body theory for nonequilibrium Magnons gas

= + ΣG

G

Figure 3.4: Diagrammatic representation of the Dyson Equation showing their general structure.The dressed Green's function by the interactions is depicted by the double arrow and a singleline denoting the free Green's function.

a renormalization of the energy of each magnon and the strength of the interaction of them.

To deepen even more this analysis let us introduce the self-energy Σ by means of the Dyson

equation

Gc = G0 + G0ΣGc (3.36)

where Gc is the one-particle connected Green's function dressed by interaction. This recursive

form to Gc is similar to Eq. 3.28. The Dyson's equation summarizes in a particularly compact

form the various contributions to the exact one-particle connected Green's function, i.e. the

sum of all topologically inequivalent and connected diagrams, in terms of the non-interacting

Green's function plus an irreducible part. see Fig. 3.4.

The one-particle Green's function is obtained by contraction of the two-particle propagator

Fig. 3.5, giving as a result the analytic expression for the self-energy

Σ(ka,kd; τ, τ′) =

i

~

∫dkb

(2π)2[Γ(ka,kb,kb,kd; τ, τ

′) + Γ(ka,kb,kd,kb; τ, τ′)]G(kb; τ

′ − τ), (3.37)

where the integration is over all possible momentum ~kb carried by the propagator. The time

evaluation τ, τ ′ is upon the Keldysh contour since we are mainly interested in the out of equilib-

rium physics of the magnon gas, a more precise and detailed discussion take place in the next

section.


In this section we derive the many-body eective theory in the low-energy limit for the

Bose-Einstein condensation of magnons parametrically excited. The eective theory, in the

semiclassical level, is determined by a certain number of parameters which are calculated within

48


G = +

G = +

Figure 3.5: (a)Two-particle connected Green's function determined within the many-body T-matrix approximation. The arrows crossing means the two possibilities of the nal scatteredstate. (b) The one-particle Green's function renormalized, due to the presence of magnon gas,by means of many-body T-matrix in the ladder approximation. To rst order in the interactionstrength the Hartre-Fock approximation is recovered.

the many-body T-matrix approach. We deduce, as the main result, that these physical param-

eters satisfy a instability conditions for the phase transition toward the spontaneous quantum

coherence. This last is directly linked to the experimental observations made in ferromagnetic

thin lms of YIG, where is established a critical pumping to the magnon condensate formation.

Up to now we have prepared the fundamentals tools to treat a typical many-body interacting

systems. Knowing this basic properties we may have in mind a heuristic scheme the elementary

structure of the magnon interactions. Now we will deduce these from a functional formulation

where our main challenge is to formulate and derive a nonequilibrium eective theory for the

order parameter 〈ψ(x, t)〉 of the phase transition toward the magnons condensate state [53]. We

adopt the formulation in a functional form since is physically more intuitive and provides us a

natural evaluation of low-energy limit to nd the eective action for the magnon gas.

Starting from the model Hamiltonian in Eq. (3.5) we proceed to derive the eective action

that describes the low energy dynamics of the magnon gas [46, 54]. In this process we benet

from a diagrammatic representation of the dierent microscopic processes involving the interac-

tion terms. The action associated with this Hamiltonian can be written in a compact form by

using the following notation [55] Hereafter index σ correspond to the spinor component in Be-

liaev space Ψ(k, t) = (ψ†(k, t), ψ(k, t)). (k, t, σ) labels will be summarized in a single subindex.

The conjugation operation correspond to an interchange of the elements of Ψ, this is achieved

49


by multiplying by σx, the resulting spinor is denoted with a super-index. Finally Einstein's

summation convention will be used unless explicitly stated otherwise, where the summation is

understood as corresponding integrals over the labels (k, t) and sum over σ. The integral over

time runs over a Keldysh contour.

The action of interacting magnon systems consists of a part

S0[ψ, ψ] =

∫Cdτ

∫dxψ∗(x, τ)

(i~∂

∂t+ ~ω (∂x)

)ψ(x, τ) (3.38)

representing an ideal magnon gas and where in the mass-eective approximation the term

~ω → ~2/2m||∇2|| + ~2/2m⊥∇2

⊥, view the section 3.1, and the other part describing the most

relevants interactions between magnons.

SI [ψ, ψ] = −1

2

∫Cdτ

∫dx

∫dx′[ψ∗(x, τ)ψ∗(x′, τ)V(2,2)(x− x′)ψ(x′, τ)ψ(x, τ)

+V(2,1)(x,x′,x′′)ψ∗(x, τ)ψ(x′, τ)ψ(x′′, τ) + V(1,2)(x,x′,x′′)ψ∗(x, τ)ψ∗(x′, τ)ψ(x′′, τ)]

(3.39)

it is worth noting the presence of the term V(2,1) (V(1,2)) that does not conserve the number of

particles explicitly sine we are favoring the conuence and splitting processes respectively.

One of the central objects in the functional formulation is that it allows the accuracy evalu-

ation of all Green's function through the generating functional of correlation functions. In fact

if we introduce the exteral currents J(x, τ) and J∗(x, τ) as a sources the generating functional

for Keldysh Green's functions is dened, following customary use [46], as:

Z[J, J∗] =

∫D[ψ]D[ψ∗] exp

(i

~S[ψ, ψ∗] + i

∫Cdτ

∫dx (ψ(x, τ)J∗(x, τ) + J(x, τ)ψ∗(x, τ))

)(3.40)

where both the elds ψ(ψ∗) and the sources J(J∗) are dened over the Keldysh contour, see

Appendix X. The denition provided by Eq. (3.40) can be formally written as

Z[J, J∗] = exp

− i

~SI[

δ

iδJ(x, τ),

δ

iδJ∗(x, τ)

]Z0[J, J∗], (3.41)

where SI represent all interaction terms from S[ψ, ψ∗], given by 3.39, and

Z0[J, J∗] = exp

(−i∫Cdτ

∫dx

∫C′dτ ′∫dx′J∗(x, τ)G0(x, τ ; x′, τ ′)J(x′, τ ′)

)(3.42)

50


with G0(x, τ ; x′, τ ′) the noninteracting Green's function. This last allows the derivation of a set

of consistent Feynman rules [54] for the correct evaluation of the Green's functions. However, is

most convenient to introduce a new generating functional dened by W [J, J∗] = −i lnZ[J, J∗]

and analogous to the free energy of statistical mechanics. Due to the linked cluster theorem that

all set of diagrams generated by W are connected in the sense of graphs, i.e. each connected

diagram could not be factorized into a product of two disjoint pieces. The advantage lies in

several aspects, for example in a broad context the fact that the connected propagators unlike

the spurious propagators deduced by Z. In our case is useful to discuss many-body interacting

properties in Bose-Einstein condensation of magnons since we can pick up in a physically more

clear sense, the microscopic scattering structure. On the other hand, the evaluation of W can

be done in terms of a perturbation series where, as we will see later, the renormalization of

the bare two-magnon scattering due to the anomalous process, appears to second order in the

coupling energy U of splitting (conuence) process.

Obviously we are primarily focused in consider the eect of the surrounding medium and the

renormalization should be adding all terms in the perturbative series ad innitum. Fortunately

not all are relevant and just is necessary replicate the rst diagram by blocks, this approximation

is named ladder summations and consists in the formal expression that denes the many-body

T-matrix approach.

Let us start expanding the generating functional of connected Green's functions W [J, J∗]

in Taylor's series. Developing the above mentioning series up to second order in JJ∗, the rst

contributions is found to be

W [J, J∗] =W0[J, J∗]− 1

~Z−1SI

[δ

iδJ,δ

iδJ∗

]Z0[J, J∗]

+i

2~2

[Z−1S2

I

[δ

iδJ,δ

iδJ∗

]Z0[J, J∗]−Z−2

(SI[δ

iδJ,δ

iδJ∗

]Z0[J, J∗]

)2]

(3.43)

where W0[J, J∗] = −i lnZ0. Collecting all terms proportional to JJ∗ we can obtain the rst

contribution to the free propagator due to the interactions between magnon as well as the terms

proportional to (JJ∗)2 provide the renormalization to two-particle Green's function, to rst

order in the interaction V and second order in U , that will lead to the expression Eq. (3.28). Of

course the presence of splitting and conuence terms will be relevant to the anomalous terms

arising in the series expansion Eq. (3.43) and proportional to Jn (J∗)m, with n 6= m. As an

example we derive the one-particle Green's functions using the above result, where we distinguish

the Hartree and Fock contributions. As a result, from the introduction of the functional W the

51


propagator is obtained by

G(1,1)c (x, τ ; x′, τ ′) =

δ2

δJ∗(x, τ)δJ(x′, τ ′)W [J, J∗]|J=J∗=0 , (3.44)

then retaining only the terms porptortional to JJ∗ in the expansion Eq. (3.43) we obtain,

nally, the one-particle Green's function up to rst order,

G(1,1)c (ζ; ζ ′) =

∫dζ1dζ2 V(ζ1, ζ2) [G0 (ζ; ζ2)G0 (ζ1; ζ1)G0 (ζ2; ζ ′) + G0 (ζ; ζ1)G0 (ζ1; ζ2)G0 (ζ2; ζ ′)]

+

∫dζ3U2(ζ1, ζ2, ζ3) [G0 (ζ; ζ1)G0 (ζ1; ζ2)G0 (ζ2; ζ ′)G0 (ζ2; ζ ′)

+G0 (ζ; ζ1)G0 (ζ1; ζ2)G0 (ζ2; ζ ′)G0 (ζ2; ζ ′)] (3.45)

where we introduce the short notation ζ ≡ (x, τ) for the space-time coordinates and U2 =

U (2,1)U (1,2) to represent the product of interaction potential of the conuence and splitting scat-

tering. The various terms in Eq. (3.45) can be depicted by Feynmann diagram, representing

each part of the Green's function and displayed in the Fig. 3.6. We noticed that up to rst order

appears two contributions from dierent origin, the rst coming from the interaction potential

V and are recognized, the rst two diagrams, as the typical Fock and Hartree contributions

respectively. The other new diagrams are related to the joining of conuence and splitting scat-

tering process. We now introduce a new generating functional called the generating functional

G = +

+= + +

Figure 3.6: The one-particle Green's function renormalized, due to the presence of magnon gas,by means of many-body T-matrix in the ladder approximation. To rst order in the interactionstrength the Hartre-Fock approximation is recovered.

of vertex functions or vertex functional that generates the one-particle irreducible Feynmann

diagrams, i.e. diagrams that cannot be disconnected by cutting only one line. It should be

52


noted that the advantage of this treatment lies in the fact that is physically more transparent to

formulate a nonequilibrium theory for the order parameter 〈ψ(x, t)〉. It is obtained by applying

a Legendre transform to the generating functional of connected correlation functions W [J, J∗],

Γ[φ, φ∗] ≡∫Cdτ

∫dx (φ∗(x, τ)J(x, τ) + J∗(x, τ)φ(x, τ))−W [J, J∗] (3.46)

whith the elds φ(x, τ) and φ∗(x, τ) as new elds-variables

φ(x, τ) ≡ δW

δJ∗(x, τ)= 〈ψ(x, τ)〉 (3.47)

φ∗(x, τ) ≡ δW

δJ(x, τ)= 〈ψ∗(x, τ)〉 , (3.48)

where the new functional Γ depends on φ explicitly as well as implicitly via J(x) by Eq. (3.47)

and 〈·〉 stands for thermal average. By the Eq. (3.47) the new eld φ correspond to the order

parameter, in other words we have mapped the problem for the ψ(x, τ) variables to the order

parameter 〈ψ(x, τ)〉.

The vertex functional Γ[φ, φ∗] generate the n-point vertex functions or one particle irreducible

(1PI) function which are dened as

Γ (x1, . . . ,xn; x1, . . . , xn) ≡ δn+n

δφ(x1) · · · δφ(xn)δφ∗(x1) · · · δφ∗(xn)Γ[φ, φ∗]. (3.49)

where in the particular case we can identify

δΓ

δφ∗(x, τ)= J(x, τ) (3.50)

δΓ

δφ(x, τ)= J∗(x, τ). (3.51)

These expressions, Eq. (3.50-3.47), together to the equation Eq. (3.49) established the basic

dictionary between a physical representation in terms of connected and vertex diagrams. From

a diagrammatic point of view the vertex functions represent the skeleton or the inner structure

of each diagram, i.e. in each diagram we cut their legs left devoid the inner part, see the Fig..

53


In fact if we take the functional derivative of Eq. 3.50 respect to J(x) and using Eq. 3.47

δ

δJ(x)

(δΓ[φ, φ∗]

δφ(y)

)=

∫δφ(z)

δJ(x)

δ2Γ[φ, φ∗]

δφ(z)δφ(x)(3.52)

=

∫δ2W [J, J∗]

δJ(x)δJ∗(z)

δ2Γ[φ, φ∗]

δφ(z)δφ(x)= δ(x− y) (3.53)

we obtain the Dyson's equation, see Eq. (3.36),∫G(1,1)(x, τ ; y, τ)Γ(1,1)(y, τ ; z, τ ′) = δ(x− y)δ(τ − τ ′). (3.54)

Here we establish a relationship between the interacting two-point connected Green's function

and the two-point vertex function, in other words the connection bewteen connected and one-

particle irreducible diagrams.

Now, we return to analyze the vertex functional Γ[φ, φ∗] distinguishing two main interpre-

tations. First, from denition Eq. (3.46) we see that −~Γ is the eective action S[φ, φ∗] of the

magnon gas. Secondly, the time evolution of both the amplitude and phase of the order param-

eter are specied by the system of equations (3.50), in the limit when J, J∗ → 0. In agreement

with one of the main motivations, due to the smallness solid angle of precession of magnetic mo-

ments in the lattice, produced and characteristic of the parametric pumping method, it is very

unlikely for three or more magnons to interact with each other simultaneously and therefore be

within the range of interaction obviously this argument hold for the process that does not con-

serve the number of magnons and just we need to considerer conuence or splitting process of

two-particle at most. Then, at most the two-body processes will be involved in the microscopic

interactions at the magnon gas and just we need to take account the following expression for

the vertex functional

Γ[φ, φ∗] = ΓAφA + ΓAφA+ΓABφAφB + ΓABφ

AφB + ΓABφAφB

+ ΓABC φAφBφC + ΓCABφ

AφBφC + ΓABCDφAφBφCφD + · · · (3.55)

where was introduced a compact notation form by using the following rule[55], the labels of

each eld, (k, t), will be summarized in a single subindex with the conjugation operation cor-

responding to an interchange of the subindex to super-index. Finally Einstein's summation

convention will be used unless explicitly stated otherwise, where the summation is understood

as corresponding integrals over the labels (k, t). By the mentioned above arguments the reduced

vertex functional take the form outlined by Eq. 3.55, where was stressed the role of magnon

54


gas diluted. Now we will perform another approach over the set of anomalous process, noting

their bearing on the low-energy dynamics.

We remark momentum conservation makes the direct eect of terms with odd number of

elds, such as Γ(2,1) and its complex conjugate in the action, irrelevant. Magnon momentum

conservation forbids the conuence or splitting of two magnons lying in the valleys, i.e. their

explicit eects in the low-energy dynamics. For example, if we think in the splitting process case

which pair of magnons have a momentum ~k and −~k, the nal state, with null momentum,

have higher energies respect to the minimum energy, where the overpopulation mechanism occur.

From the spectrum, Fig. 3.2, we have ~ω(0) ~ω(km).

ΓABC φAφBφC

≡∫Cdτ

∫C′dτ ′∫Cdτ ′′

∫dk

(2π)2

dk′

(2π)2

dk′′

(2π)2φ∗(k′′, τ ′′)Γ(2,1)(k, τ,k′, τ ′; k′′, τ ′′)φ(k, τ)φ(k′, τ ′)

(3.56)

The same argumet applies to the diagrams represented by the vertex function Γ(1,0)(k, t) and

its complex conjugate, where the momentum carried by this type of diagrams is exactly zero.

ΓAφA + ΓAφA ≡

∫Cdτ

∫dk

(2π)2

[Γ(1,0)(k, τ)φ(k, τ) + Γ(0,1)(k, τ)φ∗(k, τ)

](3.57)

However, this statement does not imply that we can simply neglect the contribution of the

three-magnon interactions from the action, as done in [56], since three-magnon processes are

present implicitly in the expressions for the remaining terms in the action. In fact, the only

remaining terms and that does'nt conserve the number of magnons are the anomalous processes.

This kind of contribution consists in the spontaneous creation of two magnons with opposite

wave vectors and mediated by a splitting-splitting interactions and the opposite for the complex

conjugate case, as shown in Fig. (3.9) and whose analytical expression is given by

ΓABφAφB + ΓABφAφB ≡

∫Cdτ

∫C′dτ ′∫

dk

(2π)2

dk′

(2π)2

[Γ(2,0)(k, τ ; k′, τ ′)φ(k, τ)φ(k′, τ ′)

+φ∗(k, τ)φ∗(k′, τ ′)Γ(0,2)(k, τ ; k′, τ ′)]

(3.58)

where the inner structure, up to rst order in U2, for the anomalous vertex diagram is displayed

in the Fig. (3.7). In this picture, on one side is showed schematically the anomalous vertex

as the matching of two splitting process. On the other side of the gure, as a consequence of

55


k -k -k k

q

q -k

Figure 3.7: Illustration of the Feynman diagram for the anomalous scattering as a result ofmatching of two magnon-splitting process. Due to the momentum conservation both the pairof magnons created have opposite wavevector ±k and the momentum q transfered is arbitrary.

momentum conservation, the pair of magnon created have opposite momenta k and −k, with

each propagator within the vertex diagram carrying a momentum q and q − k respectively,

where the wave vector q is arbitrary. The anomalous vertex function acquires, up to rst order

in U2 in the perturbative scheme that has been explained above, the following form

Γ(0,2)0 (k,−k; τ, τ ′) = iU2

∫dk′

(2π)2G0(k′ − k; τ ′ − τ)G0(k′; τ − τ ′). (3.59)

For the moment we have not considered the many-body eects on the anomalous magnon

scattering. The next step is to proceed to renormalize the microscopic anomalous magnon-

magnon coupling. By using of the perturbative approach as before, we nd that the eective

anomalous vertex is given dressing each one of the internal free one-particle Green's functions

by the interaction with the many-body systems, i.e.

G0(k; t, t′)→ G(k; t, t′), (3.60)

where the dressed Green's function is provided by means of the Dyson equation, Eq. (3.36),

and is illustrated diagrammatically in Fig. (3.9). Now our eorts are focused in determine the

retarded component because we are mainly interested in to describe the dynamical inuence of

these scattering processes without considerer the quantum uctuations. Writing the retarded

component of the vertex function Eq. (3.59) and then their Fourier transformed

Γ(+)(0,2)(k,−k; ε)

= iU2

∫dk′

(2π)2

∫dε′

2π~(G(+)(k′; ε′)G<(k′ − k; ε′ + ε) + G<(k′; ε′)G(−)(k′ − k; ε′ + ε)

)(3.61)

where the retarded and the advanced dressed propagator are obtained projecting the Dyson

equation over the Keldysh contour by means of the Langreth theorem and writing in the Fourier

56


+ +... = Γ MB +

= + +

Figure 3.8: Renormalization of the anomalous magnon scattering by interactions with the many-body system. The solid lines correspond to one-particle free propagators where each one of thesejoin two splitting (or conuence) process. The triangle correspond to the interactions betweentwo magnons and resulting in the propagation of one magnon. The wiggly lines mean the bareinteractions Γ0.

space, see Appendix, given as a result

G(±)(k; ε) = ~(ε− ε(k)± iη + ~Σ(±)(k; ε)

)−1, η → 0 (3.62)

We note that the many-body eect over the retarded(advanced) one-particle propagator

is the renormalization of the energy of propagation for a magnon by the retarded(advanced)

self-energy. While that the lesser and greater Green's functions obeys the socalled Boltzman

equation

G< = G+Σ<G− (3.63)

using the steady-state approximation, i.e. the fact that we are far away from the initial condi-

tions and the transients have fall o to be neglected. remembering that

G<0 (k; ε) = −2πiN(k)δ(ε− ε(k)) (3.64)

G>0 (k; ε) = −2πi (N(k) + 1) δ(ε− ε(k)), (3.65)

with N(k) the bose distribution for the magnons in equilibrium. Using the results Eq. (3.63)

and (3.62) in the expression Eq. (3.61) it is found that the renormalized vertex anomalous

process, in terms directly of the many-body T-matrix and the bosonic occupation, yields the

57


nal result

Γ(0,2)(k,−k; ε)

=2πiU2

~

∫dε12π~

∫dk′

(2π)2

∫dk′′

(2π)2

[G+(k′; ε1)V<(k′,k′′,k′′,k′; ε(k′′)− ε1) [1 +N(k′′)]

×G−(k′; ε1)G(−)(k′ − k; ε1 + ε) + G(+)(k′; ε1)G+(k′ − k; ε1 + ε)

×V<(k′ − k,k′′,k′′,k′ − k; ε(k′′)− ε1 − ε) [1 +N(k′′)]G−(k′ − k; ε1 + ε)]

(3.66)

where the lesser component of the many-body T-matrix was determined in the Section 3.3. A

numerical analysis is showed in the Fig. (3.12). The anomalous vertex Fig. (3.7) correspond to

the density-density response function of the thermal magnons. Their behavior is calculated as

a dependence of temperature and in-plane magnetic eld, Eq. (3.66). In fact this anomalous

contribution is zero as the temperature tends to zero as seen in the Fig. (A.1) for several

values of magnetic eld, which is calculated neglecting the higher order contributions in the

interactions. In other words at low temperature there is no thermal magnons in agreement with

the Bogoliubov approach [52].

In summary there are two eects of the splitting and conuence mechanisms that do play a

relevant role in the condensate dynamics. On one hand, second order scattering processes, such

as those represented by the vertex ∝ V(1,2)G(2,2)V(2,1), renormalize the two-body interactions

vertices V (2,2). The net four points vertex can be written as:

γABCD ≡ V ABCD + V A

CEGEF V FBD + V AE

C GFEV BFD. (3.67)

On the other hand, the three magnon interaction also generate a contribution in the anoma-

lous interaction vertex ΓAB and ΓAB, this vertex can be approximated in the lowest level by

ΓAB = V AEF GFCGDE V CB

D . It will be shown that these anomalous vertex play a relevant role in the

condensate dynamics. They convey the eect of the processes where two magnons disappear or

are created. Clearly, momentum conservation only allow this processes when the two magnons

lie at valleys with opposite momentum.

The manipulations described so far are summarized in the diagrammatic representation used

in Fig. (3.5-3.8). In this representation we symbolize the interaction vertices V (2,1) (resp. V (1,2))

as an triangle with two (resp. one) incoming and one (resp. two) outgoing bosonic lines. The

interaction vertex V (2,2) is represented by an square with two incoming and two outgoing lines,

see Fig. (3.10). This is the basic element for the construction of the T-matrix approximations

58


G (0,2)

= Γ MB

Figure 3.9: Diagrammatic illustration of the anomalous one-particle Green's function renormal-ized by the many-body system eect. The gray square box, in the right-hand side, representthe renormalization mechanism of the spontaneous emission of two magnons.

indicated in Fig. (3.10). The T-matrix Γ obeys the Lippmann-Schwinger equation written in

the momentum space:

ΓABCD = γABCD + γEHCDGFEGGHΓABFG. (3.68)

All these considerations signicantly reduce the number of terms leading to the following

vertex generating functional:

Γ[φ, φ∗] = ΓABφAφB + ΓABφAφB + ΓABφAφ

B + ΓABCDφAφBφCφD + · · · , (3.69)

where each term is calculated using the expression for the many-body T-matrix ΓABCD given by

Eq. (3.68), and where both the self-energy and the anomalous self-energy are determined by,

ΓAB = ΓACBDGDC + ΓCABDGDC (3.70)

and the Eq. (3.66), respectively. From Eq. (3.69) we can obtain the action that characterize

the dynamics and the dissipative part of the equations of motion. The rst of those tasks is

the subject of the following section while the dissipation mechanisms are going to be discussed

afterwards. A precise calculation for each term in the action Eq. (3.69) will be given in the

next section.

59


T = +

(2,2)(2,2)

0

(2,2)(2,2) (2,2)(2,2)

0

(a)

(b)

==

== ++

++++

(2,2)(2,2)

(2,2)(2,2)

0

VV (2,2)(2,2)

VV (2,1)(2,1)

VV (1,2)(1,2)

VV (1,2)(1,2)

VV (1,2)(1,2)

Figure 3.10: Diagramatic representation of T-matrix or ladder approximation. The interactionvertex V AB

CD is renormalized by the interaction vertices V ABC GEBV D

EF + V ABCGCDV DE

F .

3.5.1 Two components, double condensate

In the last sections we have established the started point to the description of the emergent

quantum coherence properties of a interacting magnon gas highly dense. This focus is deter-

mined by the main interactions and the treatment of those in a more realistic way introducing

the concept of many-body T-matrix for interacting many-body system.

We will discuss briey in this section one of the main characteristic of the magnon conden-

sate state. This property is related to the double valley degeneracy of the magnon frequency

spectrum and means a twofold and symmetric occupation for the magnons around of the min-

imum energy. Eventually, when the overpopulation of magnons upon the ground state takes

place, the magnon gas achieve the condensation in two regions of the phase space. This feature

motivates us to consider the condensate state as a bi or two-component condensate, represented

by a spinorial wave function Ψ(k, t) = (φk, φ−k)t. The characterization of both order parame-

ters, φk and φ−k, is made separating in the momentum integral the contributions in a vicinity

of k = ±km, ∫dk

(2π)2φ(k) =

∫dq

(2π)2φ(q− km) +

∫dq

(2π)2φ(q + km) (3.71)

=

∫dq

(2π)2φkm(q) +

∫dq

(2π)2φ−km(q) (3.72)

As an example, let us see how this idea applies, more specically, in the term Γ(1,1)(k, τ) where

60

3.6 Semiclassical Theory of Bose-Einstein Condensation of Magnons

-k(=

ΦkΦkF

requ

ency

, [G

Hz]

Magnon distribution

Wavevector, [cm ] -1

10 6-106 0.0

12

8.0

4.0

Figure 3.11: Illustration of the macroscopic wave function pair for the magnon gas. In the picturethe shadowed region represent the population distribution of magnons over the spectrum in thebackground.

the time-dependence is omitted fro simplicity,∫dk

(2π)2φ∗(k)Γ(1,1)(k)φ(k)

=

∫Ωm

dq

(2π)2

(φ∗km(q)φ∗km(q)

)(Γ(1,1)(k, τ) 0

0 Γ(1,1)(k, τ)

)(φkm(q)

φ−km(q)

)(3.73)

with Ωm representing the phase space region of integration where the separation Eq. (3.71) has

validity.

3.6 Semiclassical Theory of Bose-Einstein Condensation of

Magnons

Joining the main results from the previous section, i.e. the evaluation of many-body T-

matrix approximation Eq. (3.68) and the respective renormalization of the self-energy and

anomalous processes with the suitable decomposition on the Keldysh contour, and considering

states in the vicinity of the k-space energy minima, we nd that the interacting magnon gas

61


out of equilibrium is described by the semiclassical low-energy eective action

−1

~Scl[φ, φ

∗] =

∫drdt

(〈Ψr|i~

∂

∂t+ ~ω (∂r) + µ|Ψr〉+

γ1

2|〈Ψr|Ψr〉|2 +

γ2

2|〈Ψr|σz|Ψr〉|2

+1

2

(ν + P [t]eiωpt

)〈Ψr|σx|Ψ∗r〉+

1

2

(ν∗ + P [t]e−iωpt

)〈Ψ∗r|σx|Ψr〉

)(3.74)

where the pseudo-spinor eld |Ψ〉 = (Φ1,Φ2)t, with Φi the single-wave function of each conden-

sate in the ±km. The eective parameters obeys

µ = Γ(1,1)(km,km; εm),

ν = Γ(2,0)(km,−km; εm),γ1

2= Γ(2,2)(km,−km,km,−km; εm) + Γ(2,2)(−km,km,km,−km; εm),

γ2

2= Γ(2,2)(km,km,km,km; εm)− γ1

2, (3.75)

while P [t] represent the pulsed pumping eld at frequency ωp. More precisely the semiclassical

and low-energy limit in the eective action is obtained from the action − 1~S[φ, φ∗] = Γ[φ, φ∗],

with Γ given in Eq. (3.55), by projecting the elds from the Keldysh time contours into real

time, neglecting the quantum uctuations (dar justicacion). In such limit the integral over k,

in the eective action, can be separated in integrals over the vicinity of bottom spectrum, see

Fig. (3.11), and denoting the elds φ(±k0 +q) as Φ1,2(q) respectively. Therefore, each eective

parameter in Eq. (3.75) are evaluated in the valley ±km and at minimum energy, εm.

In the low energy limit and considering the magnon gas near to the bottom state ±km the

many-body T-matrix obeys

Γ+(+ + ++; ε) =Γ2B

1− Ξ−1 [ε]Γ2B

Γ+(+−+−; ε) = Γ+(+−−+; ε) =Γ2B

1− 2Ξ+1 [ε]Γ2B

with the expression for the two-body T-matrix

Γ2B(+ + ++; ε) =Γ0

1− Ξ−0 [ε]Γ0

Γ2B(+−−+ ε) = Γ2B(+−+−; ε) =Γ0

1− 2Ξ+0 [ε]Γ0

62


where we have dened

Ξ±ν [ε] = P∫

dq

(2π)2

[[N(q + km) +N(q± km)]ν

ε− ε(q + km)− ε(q± km) + i0+

].

for ν = 0, 1. With the combination of last result and with Eq. (3.75) we nd a simple form for

the self-energy,

µ = Γ2B

(1

1− Ξ−[2ε(km)]Γ2B+

1

1− 2Ξ+[2ε(km)]Γ2B

)n

where we take the principal Cauchy value in the expressions for the functions Ξ± and n =∫dq

(2π)2N(q± km).

In the expressions for the parameters Eq. (3.75), these depends on the temperature and

external eld. The results obtained for YIG samples, with geometry as in Fig. (3.2), are

displayed in Fig. (3.12). The pseudo spin (associated valley degeneracy) has allowed us to

5.5

4.5

10-18

3.5

0.8 1.0 1.2 1.4 1.6 1.8 2.0

H[KOe]

1,2

γ [

meV

cm ]

0.8 1.0 1.2 1.4 1.6 1.8 2.0

H[KOe]

10-2

μ,ν

[

meV

]

3.0

2.0

1.0

μ

ν

γ

γ

2

1

(a) (b)

2

Figure 3.12: Numerical calculation of eective parameters calculated within the T-matrix orladder approximation. In the rst gure we display the normal self-energy, µ, and the anomalousself-energy, ν. In the second gure the eective interactions parameters γ1(blue line) and γ2(redline), where γ1 quanties the eective direct interaction of magnons and γ2 stands out as aneective anisotropy penalizing magnon distributions with an imbalance between the valleys.

write a Gross-Pitaevskii like action for two interacting condensates one lying at each valley.

We can extract valuable physical insight from analyzing the dynamical consequences of this

action. Obviously the U(1)×U(1) symmetry of the two condensates is explicitly broken in the

eective action by the terms proportional to ν, since this symmetry is associated to the number

63


of particle conservation. However, the eective theory for nonequilibrium magnons still keeps a

residual continuous symmetry transformation, namely

|Ψ〉 → exp(iφσz)|Ψ〉, (3.76)

that leaves the action invariant. This symmetry transformation is a rotation in opposite phases

for each component of the spinor. Moreover, selection of a specic stationary point corresponds

to a spontaneous breaking of such residual symmetry and the relative position of the pair wave

functions is xed, in the next chapter we will see the consequence of these broken symmetry

over the magnetization properties. Since, as can be easily shown, if γ2 > 0, the system will

4.0

3.0

1015

ρ[

c

m

] -2

2.0

0.8 1.0 1.2 1.4 1.6 1.8 2.0

H[KOe]

c

P > P

P < P

c

c

(a) (b)

No BEC

BEC

Figure 3.13: (a) The "free-energy" of the system as a function of the order parameter Φ1. Foreach complex value of Φ1 minimization over Φ2 is performed to nd the saddle point. (b) Thedensity of the condensed state (black line) as a function of applied magnetic eld when thenormal and anomalous self-energy satises the condition to the instability, µ < |ν|. The shadedarea represent the zone where the condensation is expected. For high magnetic eld, greaterthan 1.8[KOe], the instability condition is not satised and then there is not BEC.

remain in an in-plane pseudo-spin state and the two valley wave function can dier at most

by a phase. Stationary points of the system have a well dened value for the average of the

phase of the two components, hereafter to denoted as θ. This is due to the term proportional

to ν, involving gauge-breaking factors. On the other hand the phase dierence between the

64


components, denoted by φ, is left undetermined (see Fig. 3.13.a.) as long as

µ < |ν|+ P , (3.77)

the system will favor an specic, though arbitrary, phase dierence between the two condensates.

Therefore, a collective symmetry breaking state take place as long as the parameters of the

system fulll the inequality Eq. (3.77). Since the eective parameters of magnon gas, without

external pumping, satisfy the relation µ > |ν|, we can point out two important conclusions. The

rst one is referred to the impossibility of formation of condensate state, because of incoherent

scattering processes, in thermodynamical equilibrium conditions. Second, once the magnon gas

is removed from the thermodynamical equilibrium, the instability condition establishes a lower

bound for the formation of the condensate state and therefore a critical level for the amplitude

of the pumping, dened as Pc = µ− |ν|. It is worth noting that the validity of such condition

is linked to the period of the applied pumping pulse, i.e. for enough short pulses, the injected

magnons is small. Although this is evident, is necessary mentioning that we are assuming that

the parametric pumping is pulsed, which pulsed period is τp ∼ (10− 30)[ns].

1.510-2

1.0

0.8 1.0 1.2 1.4 1.6 1.8 2.0

H[KOe]

P [

meV

]

c

2.0

2.5

Figure 3.14: The critical pumping as a function of in-plane magnetic eld, in unity of energy.The dashed region shown the set of possible values for the amplitude of pumping eld, whichthe formation of Bose-Einstein condensation of magnons take place.

65


It is noted that the many-body T-matrix approximation implemented, based on the assump-

tion that due to the smallness of the gas parameter β = (ρa3s)

1/2, with ρ the density and as the

scattering length, it is very unlikely the scattering of three or more particles and only we need

to considerer all possible two-body scattering processes in the magnon gas. In fact for realistic

experimental conditions the gas parameter take values of order β ∼ 0.001 for the condensed

density, see Fig. (3.13), which is consistent with the main supposition for the use of many-body

T-matrix. In that sense such approach is enough. In the next chapter we will show in detail the

consequences of the bi-condensate and breaking symmetry term, presents in the eective action,

over the collective properties of the phase condensate. Moreover, we deduce a Gross-Pitaevskii

equation from the eective theory and apply this to unveil novel quantum coherent phenomena,

linking these characteristics with the magnetic properties of the magnon gas.

66

Chapter 4

Collective Dynamics of Magnons

Condensate

In the last chapter we have arrived to a low-energy eective theory for a nonequilibrium

magnon gas, which fundamentals interactions were renormalized taking account the many-

body eects. It was noted, within the perturbative scheme provided by the many-body T-

matrix approximation, the importance of the dipole-dipole interactions in the internal dynamics

of magnons, either in the renormalization of two-magnon interactions as in the anomalous

processes.

As our main result we presented the physical mechanism behind the formation of the conden-

sate magnons. This is crowned in a instability condition for the spontaneous broken symmetry

and consecutive emergence of magnon condensed. The condensate state is characterized by a

pseudo spinorial order parameter which collective dynamic is determined by the equation of

motion coming from a variational principle on the eective action. In this chapter we explore

the collective behavior of the magnon condensate, the minimum energy congurations and the

collective excitations. We emphasize the true role played by the pumping power in the build-

ing stage of the condensate state, through an qualitative analysis of the classical dynamical

equations.

4.1 Semiclassical interpretation of the condensed stage

Before to give a detailed description of the condensate stage achieved when the symme-

try breaking condition is fullled, a intuitive notion, from the eective action analysis, can be

extracted about of the condensed magnons and their meaning in terms of the magnetization.

Indeed this notion provides the rst link between the quantum mechanical properties of conden-

sate state and their relation with the nonlinear magnetization dynamics. In other words, the


mapping from the macroscopic quantum state characteristics to the magnetization structure,

as a result of the collective organization of the condensed magnons.

Subject to the condition that pumping is greater than a certain critical value, the magnon

gas will reach the condensation phase. The precise meaning of this symmetry breaking solution

is revealed once we go back from the bosonic degrees of freedom to the magnetization. This is

achieved recognizing the expectation value 〈bk〉 as the expectation value of Sx(k) + iSy(k), i.e.

〈Ψk〉 ∼ 〈bk〉 ∝ 〈Sx(k) + iSy(k)〉 (4.1)

in agreement with the Holstein-Primako transformation in Eq. (3.3). The essential feature

associated with magnon coherence is the emergence of the long-range order as a consequence of

the non-vanishing expectation value of the magnon creation an annihilation operators. Further-

more as was proved in the last chapter, the magnon interaction between valley is positive, i.e.

the strength interaction γ2 > 0, allowing us consider the situation with the two valleys equally

occupied by the coherent magnons whose eld operators expectation values are

|Ψ±k〉 =√ρ0 exp (iψ±). (4.2)

where ρ0 is the net density of magnons, ρ0 = (ν + P − µ) /2γ1. Let us denote the average

between the phases by θ = (ψ++ψ−)/2 and their dierence by φ = (ψ+ − ψ−) /2. Recapitulating

this basic ideas, we see in the Eq. (4.1) the single-wave function with a macroscopic occupation

in the ±km state which evidence (macroscopic)quantum coherence.

The magnetization eld deviation from saturation, M0z, can be easily determined from Eqs.

(4.1) and (4.2):

δ ~M(x) ∼ √ρ0 (cos θ, sin θ, 0) cos (km · x + φ) , (4.3)

where we have collected up to rst order in the Holstein-Primako magnons. Since this approach

can be said that the system of magnons condense into an spatial pattern that corresponds

to a spin density wave (SDW). From Eq. (4.3) it is straightforward to provide a physical

interpretation of the phases θ and φ.

The plane of polarization of the SDW is xed by θ, i.e. by the microscopic scattering

processes that broken the gauge symmetry U(1). While its spatial position is linked to φ that

parametrizes a sliding mode degree of freedom. Since θ is related to the average phase of

68


[μm]-0.8 0.0 0.8

x ||

[μm

]x

0.0

-0.8

0.8

(a) (b)

(c)| |

| |

0

0

Figure 4.1: (a) Spin density wave pattern with wavelength λm ∼ 0.8[µm] for typical valueof magnetic eld H ∼ 1[KOe]. (b) A representation of the spatial pattern described by themagnetization once the condensed phase is achieved. For xed phase dierence, φ, but dierentaveraged phase, θ, the pattern tilts around the magnetic eld. (c) Same as (b) varying thephase dierence, leading a sliding degree of freedom described by the phase dierence φ.

the magnon system, the symmetry related with the conservation of number of particles and

associated with changing it

|Ψ〉 → exp(iδθ1)|Ψ〉, (4.4)

is explicitly broken in the collective action by the terms proportional to eective parameter ν.

The specic value of θ is therefore xed by the gauge breaking energy term, see Fig. 3.13.b.

On the other hand changes in φ, implemented through

|Ψ〉 → exp(iφσz)|Ψ〉, (4.5)

leave the action unchanged. The symmetry breaking characterizing the magnon condensation

correspond precisely to the spatial symmetry breaking associated with the SDW.

The basic feature of this SDW is its wave length equal to wave length of the lowest en-

ergy magnons 2π/ |km|. In Brillouin light scattering experiments, only the absolute number of

magnons can be measured, making the pattern described above appear with an apparent wave

length equal to π/ |km|, see Fig. (4.2). Recent observations account for the prediction done,

69


where the spin density wave is identied for high pumping power. It is worth mentioning that

the pumping is continuous, because to realize the measurements with thw BLS technique is

necessary to keep the coherence in each time.

Fourier spectrum intensity

Spatial coordinate, Wave number, [cm ][μm]-1

1086420

BL

S in

tens

ity

0.5

1.0400mW

315mW

250mW

400mW k = 0.6 x 10 cm5 -1

5.0 x 10 4

1.0 x 10 5

1.5 x 10 5

(a) (b)

Figure 4.2: Experimental observation of the spin density wave for dierent continuous pumpingpower. (a) BLS signal from two interfering condensates with a wavelength ∼ 1[µm]. (b) Fourierspectra of the spatial proles indicating a large BLS signal for k ∼ 0.6 × 105[cm]−1. Themaximum is placed on the wavevector of the condensed magnons.

The above mentioning properties are an accurate description of the macroscopic behavior

of the condensate magnon state. However is illustrative to now what happens microscopically,

namely how is the collective motion of the set of spins in the lattice, YIG lm. The precessional

motion of each spin is realized in-phase and at the same frequency ωm, where the collective

oscillations are modulated by a factor cos(kmx|| + φ

), whose maximum and minimum values of

this factor reects the greater and lesser precession cone angle respectively.

In that sense we distinguish the long spatial scale behavior respect to the internal dynamics

of the spins, where the microscopic dynamics is irrelevant for us description at low-energy. In

fact not only the long-wavelength are important but also the time-scale proper of the low-energy

dynamics.

70

4.2 Two-components Gross-Pitaevskii equation

λm

Figure 4.3: Illustration of the Spin Density Wave from a microscopic point of view. In thecartoon the SDW is drawn the half-wavelength that correspond to a coherent oscillation of themagnetic moments in each site of the lattice with frequency ωm and whose precession angle isperiodically modulated. In the Fig. (4.1) the white and black colors, means the maximum andminimum precession angles respectively. The precession cone angle has been exaggerated forclarity.


The spontaneous emergence of the magnon condensate is capitalized in a single-particle wave

function, the order parameter. All physical properties of the condensate stage are described by

these macroscopic wave function, and by macroscopic we mean that there are a huge density of

magnon occupying the double degenerate ground state. The main result of this chapter is to

provide a precise semiclassical picture of the collective dynamics of the magnon system, obtained

by the extremum of the eective action,

δSeffδ〈Ψr|

= 0. (4.6)

The condensate consists, roughly speaking, of two magnon condensates lying the vicinity of

the two points of minimum energy in momentum space, and magnetic interactions introduce

a coupling between them. Denoting the condensate wave functions around the two points of

minimum energy by Φ1 and Φ2 respectively, the dynamics that these two condensate follow can

71


be described by the following generalized pseudo-spin Gross-Pitaevskii equation:

i~(1 + iα)∂t|Ψ〉 = − ~2

2m||∇2|||Ψ〉 −

~2

2m⊥∇2⊥|Ψ〉+ µ|Ψ〉+ νσx|Ψ∗〉

+ γ1|Ψ|2|Ψ〉+ γ2〈Ψ|σz|Ψ〉σz|Ψ〉 (4.7)

where the pseudo-spinorial macroscopic wave function

|Ψ(r, t)〉 =

(Φ1(r, t)

Φ2(r, t)

), (4.8)

refers to valley degeneracy in momentum space, with each component representing the con-

densed magnons with momentum ±km, respectively and r = (x||,x⊥). While the eective

parameters α, m, µ, ν, γ1 and γ2 characterizes the dynamics. In the Eq. (4.7) the pump-

ing term is not considered since is assumed the pumping pulsed case, i.e. the condensation of

magnons take place when the pumping is switched o.

The only term in the equation that breaks the time reversal symmetry (associated with the

transformation |Ψ(t)〉 → σx|Ψ∗(−t)〉) is the term proportional to α. This term plays the role of

a damping constant much in the same way as the Gilbert damping term in magnetism.

The dominant dissipative process, namely the decay of magnon number, enters into the

stage through the imaginary part of the self energy Σ(1,1)(k, ε). For small energies, time reversal

invariance allows terms of this form only to rst order in the energy. Indeed if we expand in

Taylor's series,

~Σ(k, ε) = ~Σ(k, εm) +∂~Σ(k, ε)

∂ε

∣∣∣∣ε=εm

(ε− εm) +1

2!

∂2~Σ(k, ε)

∂ε2

∣∣∣∣ε=εm

(ε− εm)2 + · · · (4.9)

where we recognized the rst term as the eective parameter µ and identied the term propor-

tional to (ε− εm) as the dissipation, i.e.

α ≡ ∂Σ(k, ε)

∂ε

∣∣∣∣ε=εm

, (4.10)

since the time-representation of this term leads to the contribution i~∂εΣ(εm)∂t in the quadratic

part of the eective action Eq. (3.74). In agreement with the general expression for the self-

energy 3.70, we can evaluate within the approach mentioned above the Eq. (4.10) in terms of

72


the many-body T-matrix for the magnon gas,

∂Σ(k, ε)

∂ε

∣∣∣∣ε=εm

= 2n(∂εΓ

(+)(km,km,km,km; ε(km) + ε) + ∂εΓ(+)(−km,km,−km,km; ε(km) + ε)

)+ 2∂εI(ε, T,H)

[Γ(−)(km,km,km,km; 2ε(km))Γ(+)(km,km,km,km; 2ε(km))

+2Γ(+)(−km,km,−km,km; 2ε(km))Γ(−)(−km,km,−km,km; 2ε(km))]

(4.11)

where

I(ε, T,H) =

∫dqb

(2π)2

dqe(2π)2

N(km + qe)N(km + qb − qe)

ε(km + qe) + ε(km + qb − qe)− ε− ε(km + qb)− i0+

.

From the diagrammatic point of view, this term, Eq. (4.10), represent the rate of change of

the condensate density caused by the incoherent scattering processes, which contribute to the

decoherence mechanism. The contribution to the action, otherwise nonlocal in time, acquires a

105

Figure 4.4: Dissipation characteristic for the magnon gas in the condensate state. This eectiveparameter is calculated within the many-body T-matrix approach and displayed as a functionof the in-plane magnetic eld applied over YIG lms. The smallness of the dissipation is linkedwith the typical low-damping for spin waves in YIG thin lms.

73


low energy form that contributes to the equations of motion to conform Eq. (4.7) and whose

dependence with the in-plane magnetic eld is exhibited in the Fig. (4.4). Is worth mention

that this result is consistent with the propagation of spin waves, slightly attenuated on the YIG

material [25]. This analysis is in complete analogy with the microscopic description of the Gilbert

damping in the Landau-Lifshitz equations of motion for the classical magnetization [57, 58, 59]

The large discrepancy between the eective masses, along and perpendicular to the external

eld, make it more convenient to regard the magnon condensate as quasi one-dimensional,

with a rather weak dependence on the axis parallel to the magnetic eld. The parameter µ

plays the role of chemical potential shared between the two components of the condensate.

Meanwhile the coecient ν quanties the gauge invariance breaking that is associated with

magnetic interaction-driven processes that create or annihilate magnons in the system, see Fig.

(3.9), which direct eect over the non-conservation of magnon number follow,

dn

dt= =[ν]n(t)− α

1 + α2

((µ+ ν)n(t) + 2γ1n

2(t))

(4.12)

with n(t) = 〈Ψ(t)〉 where the rst term of the right-hand side, proportional to the imaginary

part of ν, is dominant respect to the second contribution. As was proven in the Chapter 3 these

eective parameters obeys the inequality µ > ν, Fig. (3.12), whereas that the emergence of the

magnon condensed state is subject to being satised the instability condition, i.e. µ < ν + P .When the pumping exceeds the critical value Pc ≡ µ − ν, this term acts eectively by phase-

locking the two condensates.

Finally, the interaction terms have a normal Gross-Pitaevskii-like contribution proportional

to γ1 and an eective, acting on the spinor degrees of freedom, easy plane ferromagnetic inter-

action term proportional to γ2. In fact, let us see more precisely the meaning of the eective

interaction between valley, γ2. From the Eq. (3.75) we see that

γ2

2= Γ(+) (km,km,km,km; εm)− Γ(+) (−km,km,−km,km; εm)

where the rst term on the right hand side represent just the eective magnons interaction

around of each valley, while the second contribution weights the interactions of them between

valley. In that sense, positivity of the net interaction γ2, means repulsive interplay for magnons

with momentum near to km and whose with momentum near to −km. Then once the occupation

of magnons in the minimum energy state take place, equal populations is energetically more

favorable.

74


net interactionγ > 02

Figure 4.5: Cartoon of the magnons population around of each valley when the net interactionγ2 > 0. In the background is sketched the dispersion relation for longitudinal wave-vector. Sincethe intra-valley interaction is greater respect to inter-valley interaction between magnons, thesame occupation of the ground-state is favored.

We emphasize that this form of the equation is essentially dierent than the phenomeno-

logical ones proposed in [60, 61], since Eq. (4.7) has a dierent form for the dissipation term,

an explicitly gauge symmetry breaking term and a spinor nature. An additional advantage of

the formalism described so far is that the eective parameters in Eq. (4.7) can be calculated in

terms of the system parameters.

The above process of condensation of magnons has been a matter of controversy due to

the presence of explicit gauge-symmetry breaking terms that are induced through dipolar in-

teractions of the spin wave excitations[62] or by their non-equilibrium nature[63]. We have

shown that neither of those criticisms is valid, since the system displays all the relevant features

expected of a legitimate BEC. Put it another way, the magnons as quasiparticles of magnetic

ordered materials, have nite lifetime and just is necessary, although obviously not enough, that

the magnon lifetime is much longer that the time they need to scatter with each other, and the

condensation is possible. It has been shown that this subtlety has no eect in the basic process

of condensation and that the magnon system shares several qualitative features with the usual

particle-based BEC's.

The parametric nature of the magnons excitation mechanism over YIG lms, in the build-up

of a huge population of magnons near the ground state and eventually undergo the spontaneous

coherence, should remain stealthily linked to the manifestation of magnon Bose-Einstein con-

densation. In fact, to explore this hypothesis, we motivate to observe the growth-up of the

occupation in the ±km state toward condensation density and evolution of the global phase,

75

4.3 Magnon BEC as a Hopf-Andronov Bifurcation

under parametric pumping. This is the point that will be treated in the next discussion.


The basic physical phenomena accounted for by Eq. (4.7), namely its dissipative nature,

its explicit symmetry breaking contributions and its spinorial structure, provide a consistent

description of the condensate dynamics valid when the condensate is fully formed. In this

section we use Eq. (4.7) to explore some features of the condensate process itself. Since we are

pushing the equation away from its domain of validity (i.e. far from the condensate state), the

conclusions of this section are expected to provide a qualitative and semiclassical picture of the

condensate formation. A fully quantum analysis of the process of condensation is outside the

scope of the present thesis. With this caveat in mind we proceed to obtain numerical solutions

of Eq. (4.7). The physics of such dynamics can be understood in terms of the Hopf-Andronov

bifurcation [65].

The question of the legitimacy of condensate of magnons, as a usual BEC's, is related to

the role played by the pumping power in the thermalization and nucleation mechanism, and

subsequent condensation stage. To unravel this trouble we may to explore in a simplied way

the Gross-Pitaevskii equation in the homogeneous case, analyzing the growth of the magnon

density in the bottom state, i.e. magnons with energy ~ωm and momentum km. Recently a

study of growing of the condensate [64] has been performed by means of the characterization

of the macroscopic state of the magnon system in terms of the thermomechanical statistics

based on the framework of a nonequilibrium statistical ensemble formalism. Assuming for

simplicity, without lost of generality, that the spinorial wave function doesn't depend on the

spatial coordinates the Gross-Pitaevskii equation with parametric pumping take the form

i~(1 + iα)∂tΨ(t) = µΨ(t) +(ν + P(t)e−iωpt

)σxΨ

∗(t) + γ1 |Ψ(t)|2 Ψ(t) + γ2 (Ψ∗σzΨ) Ψ(t)

(4.13)

where the pumping amplitude is considered as a pulse, therefore P(t) = PH[t] with H[t] the

Heaviside step function. We solve numerically the Eq. (4.13) for dierent pumping amplitude,

i.e. P/ (µ− ν) = 1.5, 5, 15 and a time pulse τ ∼ 10[ns]. The solutions presented in the Fig. (4.6)

shown a saturation in the condensate density where the time of thermalization is dependent of

76


the pumping power, and then over the density. The same applies to the rate of decay, once the

pumping pulse is switched-o, given as a result a shorter lifetime as increases the pumping.

0.1 0.2 0.3 0.4

2

6

10

14

Time@nsD

Mag

no

nd

ensi

ty,

Ρ@1

016D

50 100 150

2

6

10

14

Time@nsD

Mag

no

nd

ensi

ty,

Ρ@1

016D

Figure 4.6: Temporal evolution of the magnon density under parametric pumping above thethreshold value Pc = µ − ν. For dierent pumping values the saturation density is greaterwhen the pumping increase. (a) Once the pumping pulse is switched-o the rate of decaysdepend on the density and then the lifetime of the condensate state is smaller with increasingpumping power. (b) Consecutively the time of saturation to the condensed density follow thesame behavior that the lifetime of the magnon condensate.

The last point can be understood thinking in the following way: having in mind two main

things widely treated in the last chapter, the diluted character and the anomalous scattering

processes. Once the parametric pumping takes place, the increased density of magnons begins.

If the pumping is greater respect to critical value, Pc, the density will be higher and as a

consequence the scattering cross section increase. Thus those anomalous processes will be

most likely to be within the range of the interaction. In summary we conclude that both the

thermalization time to achieve the condensate stage and the rate decaying of these depends on

the pumping power, i.e. over the magnon density.

The last approach, referred to the appearance of the instability, can be interpreted classically

from a dynamical systems point of view. In fact we can establish a correspondence between

the spontaneous emergence of the macroscopic quantum magnon state and the Hopf-Andronov

bifurcation in the phase diagram for the magnon gas. Let us write the wave function as Ψ(t) =√ρ(t) exp (iΘ(t)) in the Eq. (4.13) and rewrite it just for the phase variable

Θ′′(t) = − sin Θ(t) (γ + cos Θ(t)) (4.14)

where the parameter γ = µ/νeff , with νeff ≡ ν + P . It is clear that the critical point γ = 1

distinguishes both phases, condensed and non-condensed phase. In this way we translate the

77


broken symmetry condition Eq. (3.77) in a constraint for the γ parameter. Is straightforward

to note that the equilibrium points of Eq. (4.14) may be viewed as extreme of the potential

energy dened as

Θ′′(t) = − ∂

∂ΘV [Θ] (4.15)

and V [Θ] = −γ cos Θ− 12

cos2 Θ. The equilibrium points are

Θeq =

nπ, cos−1[γ] γ < 1

nπ γ > 1, (4.16)

is worth mentioning that the equilibrium point Θ = nπ is xed for any value of γ. The next

step is to provide, through a perturbative scheme, a stability analysis of the trajectories around

of each one of such equilibrium points. Introducing a new variable ϕ = Θ′ we linearized the Eq.

(4.14) such that in a matrix form looks like(ϕ′

Θ′

)=

(0 γ − 1

γ − 1 0

)(ϕ

Θ

). (4.17)

solving the characteristic equation det[

(0 γ − 1

γ − 1 0

)] = 0 easily we nd the eigenvalues,

λ± = ±√γ − 1. (4.18)

Clearly when γ > 1 the equilibrium point Θ = nπ is unstable, since the trajectories near this

point depart. While for γ < 1 namely, when pumping exceeds the critical value, the eigenvalues

are purely imaginary and now the equilibrium point becomes is stable. This qualitative change,

in terms of the parameter of the system, correspond to a bifurcation so-called Hopf-Andronov

bifurcation[65], see Fig (4.9(a)). This region in the phase space is isolated, respect to the rest

of the phase space, by the limit cycle. A similar analysis done on the other equilibrium points,

results in unstable trajectories in a neighborhood to those points. In the portrait phase space

can be seen to those points as saddle dots since the trajectories converge in one direction but

not in the other.

Once we have characterized the dynamical system Eq. (4.14), characterized by the Hopf-

Andronov bifurcation in terms of the γ parameter, the magnon condensation mechanism can be

78


0 0.5 1 1.50.5Π

Π

1.5Π

Γ

Θep

Figure 4.7: Illustration of the Hopf-Andronov bifurcation in the parameter space. For γ > 1the equilibrium point is unstable, while than for γ < 1 that point becomes stable, which isrepresented by the red line. The attractor centered in Θ = π encloses the trajectories withinthe limit cycle between the pair of saddle points, represented by the black lines.

qualitatively viewed or interpreted in the following way. If we start to parametrically pumped

magnons, below the critical level, the magnon density can not be saturated and the systems fails

to condense. In the phase space the trajectories are driven towards the unstable equilibrium

point and due to the presence of dissipation the system falls into the initial state again.

However, if the pump is above the threshold Pc = µ−ν then γ < 1 and the magnons density

saturates in the value ρc = (P + ν − µ) /γ1 see Fig. (4.6). The trajectories are driven to the

attractor, if the pulse of pumping is large enough, and the magnon gas remains oscillating

within the basin of attraction until the pumping is switched-o. After these, the magnon

gas remains librating when it nally decays to the initial conguration. In such process we

identify two characteristic time scale associated to the oscillations in the condensate and non

condensate stages. The second is greater than the rst since both the thermalization and

relaxation processes depends on the density, as was discussed in the beginning of this section.

This point can be seen in the Fig. (4.8), noting that the trajectories within the attractor will

be kept librating after that the pumping is switched-o. In terms of the potential V [Θ], the

trajectories followed by the condensed magnon gas on the phase space are trapped when γ < 1,

where the region size is determined by the saddle dots Θ± = cos−1 γ.

In summary the bifurcation characteristic of the dynamical system Eq. (4.14), arising when

the γ parameter goes through a critical value, is associated with the spontaneous emergence of

the condensate magnon. Obviously, this analysis is not rigorous because the condensation has a

79


Figure 4.8: Portrait phase space for the global magnon phase trajectories in the γ < 1 regime,i.e. in the condensate stage. The trajectories are drawn for dierent initial conditions andγ = 0.5. The attractor, centered in Θ = π, encloses the trajectories within the limit cyclebetween the pair of saddle points. It should be noted a libration zone, whose solutions connecta complete oscillation period.

80

4.4 Topological excitations within the condensade phase

purely quantum nature, while the above description is classical. Nevertheless, is a useful insight

to understand the role played by the pumping power and the meaning of the broken symmetry

condition in our theory developed in the last chapter.

-2Π -Π 0 Π 2Π

-2

-1

0

1

Θ

V@Θ

D

Figure 4.9: Illustration of the potential energy associated with the trajectories followed by theglobal phase Θ(t) in phase space. The middle curve for γ = 1, splits the non-condensed andcondensed phases. The metastable state, for γ < 1, is characterized by a little valley(attractor)formed around of Θ = π. When the pumping power is large enough, the trajectories fall intothe attractor and the condensate stage begins.


The manifestation of a macroscopic coherent phase, as a consequence of spontaneous broken

symmetry of the system, give arise novel intricate phenomena where the kind of organization

is described by the Gross-Pitaevskii equation mentioned above. This collective excitations are

mapped over a new magnetic order achieved by the magnetization. In this section we are

going to explore topological collective excitations originated in the magnon condensate state,

where by topological we mean that the structure of the order parameter space is classied in

terms of the possibility to continuously shrink every closed loop, or n-sphere in more general

cases, to a point over the order parameter space. In other words, the order parameter of the

magnon condensate naturally dene a map from the real to the order parameter space, whose

classication is provided by the so-called Homotopy groups that reveals us the topological defects

81


present in the system. The most popular topological defect in condensed matter is the Votex

like solution and that we analyzed to continuation.

4.4.1 Vortex like structure

We have stated in the previous section that the broken residual-symmetry of the eective

action Eq. (3.74), manifests itself in a position xing in the Spin Density Wave. The simplest

scenery, displayed in the Fig. (4.1), correspond to the case when the coherent phase is constant.

A natural feature inherited from the breaking of the U(1) symmetry is the capability of the

system to sustain vortex like solutions. In fact the vortex solutions are described by a velocity

eld

v =~

√m⊥m||

∇φ, (4.19)

where φ is the phase of the condensed magnons. This approach comes from the equivalence

between the Gross-Pitaevskii equation and a quantum hydrodynamics representation. Resulting

these last in a continuity and Bernoulli equations for the condensate uid in terms of the density

n(r, t) = 〈Ψ(r, t)〉 and the velocity eld v. If we take the radial-dependent density where the

phase φ is two-dimensional polar phase the Eq. (4.19) gives the circulating velocity vortex

v =(~/√m⊥m||

)φ and the circulation associated obeys

κ =

∮dl · v = 2π

~√m⊥m||

n, n ∈ Z (4.20)

is quantized due to the single-valued of the wave function of the condensate. By the Stokes's

theorem we know that∫dr · ∇ × v = 2π~/√m⊥m||, implying a singular localized vorticity at

the center of the vortex core which explicitly reads

∇× v = 2π~

√m⊥m||

δ(2)(r)z, (4.21)

where δ(2)(r) = (2πr)−1 δ(r) and the circulating ow becomes irrotational except in the origin.

It is worth noting that, unlike the minimium energy conguration for the Spin Density Wave,

Eq. (4.3), is a essential requirement consider the position-dependent phase to expect vortex

solution, or topological defects in a broad sense.

82


From a formal point of view the vortex structures can be classied using the rst homotopy

groups[66]. The rst homotopy group of the order parameter M, denoted by π1(M), is the

group of all maps from the circle S1 toM. Obviously if the surfaceM is smooth, each closed

curve can be continuously shrunk to a point. However, this statement can not be successful

when M has a singularity, see Fig. (4.10). To understand with most accurately let us see

this concept applied in our system. First, to nd all possible topological excitations of the

magnon condensate, we need to know the full symmetry of the macroscopic wave function,

i.e. each component have the form Ψ±(r) ≡√n(r)e±iφ(r), where

√n(r) is the total density

of the condensed magnons with momentum ±km and φ(r) the macroscopic phase, assuming

both valleys equally occupied. As a result the order parameter space for Ψ±(r) take the form

M = U(1) ' S1 and the rst homotopy group obeys

π1(M) = Z, (4.22)

since the order parameter space can be n-times encircles by the loop, with n the winding number.

We conclude that a condensed magnon can have only vortices structures with winding number

that are an arbitrary integer when the order parameter parameterized by the phase.

Figure 4.10: Sketch of the deformation for a closed loop over a surface M. (a) Each closedcurve on a smooth surface can be continuously shrunk to a point. (b) While a loop α on asurface with a singularity, represented by a little sphere, cannot be shrunk to a point due to theexistence of the hole. The classication scheme for all the loops encircles the hole is providedby the homotopy group.

We now will pursue the characterization of the vortex prole and its eects on the SDW

described in section (5.3). Due to the dierent longitudinal and transverse masses, the vortex

is anisotropic[67], with an elliptic cross section of aspect ratio γ =√m||/m⊥ ∼ 5, for in-

plane magnetic eld ∼ 1[KOe]. We can change variables in Eq. (4.7), x|| = 1/√γ (x||/ζ) and

83


x⊥ =√γ (x⊥/ζ) where

ζ2 =~2

2√m||m⊥|P + ν − µ|

(4.23)

with P the pumping eld. With those new variables we look for the specic physical properties

of those solutions through the ansatz

Ψ (r, t) =√ρ0R(r)

(ei`ϕ

e−i`ϕ

), (4.24)

where (r, ϕ) are polar space coordinates centered in an arbitrary point and ρ0 = |µ− ν|/γ1. It

is straightforward to verify that the vortex prole is determined by the equation:

R′′(r) +1

rR′(r) +

(1− `2

r2

)R(r) = R3(r) (4.25)

This equation is just the one that characterizes the vortex proles in the single component

Gross-Pitaevskii equation in standard BECs[34]. This vortex solution correspond to a congu-

ρ/ρ

η/ζ[μm]

0

H[KOe]

1.0

0.0

0.5

0.0 1.0 2.01.50.5 0.8 1.0 1.2 1.4 1.6 1.8 2.0

5.0

1.5

3.0

ζ 10

[μ

m]

-1

Figure 4.11: (a) Normalized density prole of the vortex for a healing length ζ ∼ 0.8[µm] andmagnetic eld ∼ 1.8[KOe]. (b) Healing length of the vortex structure as a function of in-planemagnetic eld. Both calculation were performed for typical experimental values of YIG samples.

ration with a constant density ρ0 in the bulk and the prole density satisfying the Eq. (4.25),

where the density is nullied in the origin. The single-valuedness of |Ψ〉 requires that ` be aninteger (` the winding number). The prole density of the vortex is dependent of a characteristic

length, determined by the eective parameters, Eq. (4.23). In the Fig. (4.12) is displayed the

84


dependence of the healing length as a function of the in-plance magnetic eld together with

the density prole for a specic value of the healing length, ζ = 0.8[µm]. The healing length

characterize the size of the vortex core and for typical in-plane magnetic eld utilized in the

experiments over YIG thin lm, we predict a size of the vortex core of order ζ ∼ 2π/km in the

longitudinal direction.

5.0

0.0

2.5

4.2

1.7

X [

μm]

X [μm]

8.00.0 4.0 6.02.0

3.2

-3.2

0.0

1.6

-1.6

-0.8 0.0 0.8

X [μm]

X [

μm]

(a) (b)

Figure 4.12: (a) Vortex structure with an elliptic cross section of aspect ratio γ =√m||/m⊥ ∼ 5.

Such structure emerge as a dislocation over the spin density wave, and with a Burgers vectorproportional to the winding number of the vortex. The presented gure correspond to the squareof magnetization module |δ ~M |2. (b) Experimental evidence of a vortex pair in the magnon BEC,courtesy of S. Demokritov[68]. It is clearly seen the dislocation on the spin density wave. Thecolors represent the BLS intensity for continuous pumping.

This topological defect can also be understood, just like the homogeneous solution, in terms

of the expectation values of the magnetization eld. It is straightforward to see that the vortex

solution corresponds to an edge dislocation in the SDW with Burgers vector proportional to the

winding number of the vortex. In Fig. (4.12) the SDW pattern is displayed with a vortex with

` = 1 and the original spatial variables restored, where a recent evidence have been observed[68].

The connection between the vortex structure and the dislocation upon the magnetization wave

pattern can be seen in the following way. We know that the macroscopic phase is position-

dependent and then if we draw a semicircle contour we accumulate a π phase, as a consequence

the spin in that position will be oscillating in counterclockwise respect to the original precession.

85


In that sense the maximum is converted in a minimum on the spin density wave.

86

Chapter 5

Josephson eect in the condensate of

magnons

Along with the uncontroversial evidence of macroscopic occupation of the lowest lying state,

several questions arose regarding the appropriateness of the concept of BEC to refer to collective

magnon behavior [?]. One of the most striking phenomena in the nature of macroscopic BECs

is the emergence of a macroscopic wavefunction that displays phase coherence over macroscopic

length-scales. Finding eects associated with such coherence is important in clarifying the true

nature of the magnon cloud. In the present work we start from a phenomenological stand-

point and proceed to explore the physical nature of the dynamics of the condensate. To exploit

the occurrence of the quantum coherence is necessary to analyze and perform the macroscopic

interference eect between magnon condensates. The interference phenomena of such states is

referred as the Josephson eect. The quantum collective dynamics between the coherent many-

body systems will be described by macroscopic observables either by the population imbalance

and the relative phase between them. Throughout this chapter will distinguish between internal

and external Josephson eect due to the spinorial characteristic of the condensate.

Discovered and observed early in superconductivity[69, 70, 71], the Josephson eect have

been demonstrated too in superuid helium 3He[72] and 4He[73], and in Bose-Einstein conden-

sates of alkali atomic gases in double well traps[74]. In the last case, the Josephson dynamics

between weakly coupled BEC's manifests itself in several novel phenomena, mainly due to the

nonlinearity arising from the self-interaction among bosons. In this chapter we propose an

magnon Josephson's eect to report the existence of a long-range phase coherence [75]. This

provide a irrefutable evidence of the spontaneous quantum coherence, at macroscopic scales,

where a macroscopic number of magnons suddenly occupy the same quantum state.

The realization of the magnon josephson junction (MJJ) will consist principally of two

stages. The rst is based in the usual way for modeling the splitting of condensed clouds in

alkali atomic gases, i.e. introducing a potential well inside the trap that splits the single trapped

condensate into two parts. The partitioning leads to two weakly coupled condensates, where the

tunneling can be tuned modifying the parameters of the system. The second point is related

to the spin-wave tunneling eect on ferromagnetic thin lms. In that situation a magnetic

eld inhomogeneity is induced over the thin lm by a conductor placed transversely this. The

magnon condensate created on the lm will be divided in two parts when, by the wire conductor,

across a dc current in such direction that the locally increases the magnetization saturation.

Within this scenery the dynamics of condensed magnons, in a double-well potential, will

be described by the Gross-Pitaevskii equation where to carry out a approximation on the full

macroscopic wave function, to represent the dynamics by the essential degrees of freedom. Such

approach is based on a widely explored soluble two-mode approximation to the many-body

Hamiltonian. That approximation assume a two-state model due to the weak coupling between

them and resulting in a eective semiclassical description.

H0H j

[z] x

z

j y

dc conductor

5 μmYIG film

Microstripe resonator

Figure 5.1: Sketch of the experimental setup for the magnon Josephson's eect realization. Thespatial fragmentation of the cloud magnon condensate over the YIG thin lms is allowed bymeans of a wire conductor, for which crosses a dc current J and producing an local inhomo-geneity, H(z), in the magnetization. The current direction determines the sign of the potentialbarrier which magnons feel.

A traditional starting point to the realization and implementation of the Bose Josephson

junction in alkali gases consists of split the condensed clouds, applying a Gaussian shaped laser

sheet at the center of the trap that cuts a single trapped condensate into two parts. At that

scenario the weak link between the two condensates can be tailored by tuning the width or the

88

height of the laser sheet. On the other hand, recent experiments about nonlinear magnetization

dynamics on ferromagnetic thin lms have shown the eect of spin-wave tunneling. By a

conductor, placed across the lm, carries a current which is used to create a local inhomogeneous

eld. From a microscopically point of view, the magnons feel the presence of a potential, either

enhanced their kinetic energy or tunneling to the other side of the barrier, depending on the

sign of the inhomogeneity.

Inspired by these ideas we propose the Magnon Josephson Junction(MJJ) on ferromagnetic

YIG thin lms for weakly linked magnon condensates. The splitting of the cloud condensate is

implemented by applying the DC current, where the tunneling between both states is adjusted

by varying the current and geometric parameters of the setup. In this section we take out

the realization of MJJ and will build, from a phenomenological point of view, the semiclassical

equations of motion for the collective dynamics between the condensate states.

Once the population of magnons, created through a parametric pumping in a YIG thin

lm, surpasses a critical level, the systems develops a long range order which manifests in a

macroscopic quantum coherence where both the density and phase coherent characterizes the

physical properties. As mentioned above and established in the Chapter 3, the condensed

magnons with momentum and energy (±km, ~ωm) contribute to the magnetization, emerging as

a spin density wave with wavelength 2π/km. When the macroscopic quantum state is partitioned

in two condensate clouds, the dynamics between them is completely captured by the macroscopic

observables either, the population imbalance η and the relative phase φ, dened as follow

η ≡ 1

nT(nL − nR)

φ ≡ φR − φL.

On the other hand and within a experimental context, the realization of the fractionalization

of cloud condensed naturally arises from the studies of spin wave tunneling in a nonuniform

magnetized thin lms[76][77], as already mentioned. In that sense let us introduce a local

inhomogeneity H(z) in the magnetization, produced by means of a current that goes through a

conductor and placed transversely to the YIG sample. In the Fig. (5.1) the experimental setup

in mind is schematized, where the spatial separation of the magnon cloud condensate can be

made.

89

5.1 Microscopic fundamentals for the magnon Josephson effect

5.1 Microscopic fundamentals for the magnon Josephson

eect

In this section we will give the theoretical support to deduce the magnon Josephson equa-

tions, from a semiclassical perspective characterized by the Gross-Pitaevskii equation for the

condensed magnons. As was shown in the previous chapters, where we provide a precise scheme

for the collective dynamics of the magnons system, the condensate state of magnons has two

components that belong to the vicinity of energy minimum due to the double valley degeneracy,

±km in the spectrum, Fig. (5.1). The macroscopic wave function for the condensed magnons

is denoted by |Ψ〉 = (Φk,Φ−k)t, and which dynamics follow the pseudo-spin GP equation,

i~(1 + iα)∂t|Ψ〉 = − ~2

2m||∇2|||Ψ〉 −

~2

2m⊥∇2⊥|Ψ〉+µ|Ψ〉+ Vext(r)|Ψ〉+ νσx|Ψ∗〉

+γ1|Ψ|2|Ψ〉+ γ2〈Ψ|σz|Ψ〉σz|Ψ〉 (5.1)

whose parameters α, µ, ν, γ1 and γ2 determine the dynamics of the condensate phase and are

calculated from a microscopic theory developed in the chapter 3. Those eective parameters

depends both of the temperature and the in-plane magnetic eld applied.

In agreement with the experimental realization Fig. (5.1), both the splitting of the magnon

condensate and their consecutive Josephson oscillations, the inhomogeneity in the magnetic eld

is represented by a potential barrier Vext(r). That external potential introduce a energetic gap

than the magnons in the cloud condensate must overcome and then is favored an occupation

on both sides of barrier. From a classical point of view, the above scheme result in a interplay

among the spin density waves where the dispersion relation have a spatial dependence. As a

result of the above, the spectrum have a forbidden region for the spin waves propagation and a

locking of oscillation modes of the spins at each site. Obviously the phenomenon of Josephson'

eect in consideration is intrinsically quantum and a suitable analysis will focus on a microscopic

description. In resume the localization of both condensate states is allowed by the applying of

a dc current, which creates a potential well for the magnons and therefore the amplitude of

tunneling will be controlled changing the current.

90


5.1.1 The Nonlinear Two-Mode Approximation

The previous phenomenological motivation, given in the last section, is essentially the main

point in order to study the dynamical oscillations of the two weakly linked magnon condensates.

Indeed the spatial localization of both condensed magnon clouds, obtained by applying a dc

current transversally to the in-plane magnetic eld, can be formally summarized as a eective

description in terms of two states, the right and left states, to characterize the relevant physical

observables in the dynamical behavior between them.

To be more specic we write the full macroscopic wave function as the addition of a two

spatially splitted time-dependent states,

Ψ(x, t) = ψL(t)ΦL(x) + ψR(t)ΦR(x), (5.2)

where we have assumed with this variational anzats two things: each wave function are separable

variables and both states are weakly interacting. This approximation, so-called the two-mode

approximation[78][79], in the Gross-Pitaevskii equation has proven to be a successful description

to predict the existence of a Josephson tunneling phenomena in clouds of bosonic system conned

in a double-well potential. The left and right modes

ΦL,R(x) =1√2

(Φ1 ± Φ2) , (5.3)

corresponding to the symmetric and antisymmetric functions, are constructed from the ground-

state Φ1, and the rst excited state Φ2 satisfying the stationary Gross-Pitaevskii equation

0 = − ~2

2m||∂2|||Ψ〉 −

~2

2m⊥∂2⊥|Ψ〉+ µ|Ψ〉

+Vext(r)|Ψ〉+νσx|Ψ∗〉+ γ1|Ψ|2|Ψ〉+ γ2〈Ψ|σz|Ψ〉σz|Ψ〉 (5.4)

where the spatial part of the wave function satisfy a orthogonality relation∫dxΦiΦj = δij, i, j = 1, 2. (5.5)

Assuming a uniformity in the perpendicular direction, the wavefunction depend just along

the longitudinal coordinate and the Eq. (5.4) can be reduced to the one-dimensional nonlinear

Shroedinger equation with a external potential given by Eq. (5.6). This equation can be even

91


more approximated for a square potential where the solutions are very well known. In fact

the functions Φ(1,2) are expanded in terms of symmetric and antisymmetric elliptical bessel

functions.

The potential barrier produced by the current crossing the wire conductor have the simple

form,

Vj(x||) =γ~Hj√δ2 + x||2

. (5.6)

where Hj is the inhomogeneity produced by the dc current and δ the separation between the

wire and the YIG lm. Putting the Eq. (5.2) in the full Gross-Pitaevskii equation Eq. (4.7),

using the mentioning above statements we nd the Josephson equations for the two dynamical

modes to obeys

Ψ (x,t)R

YIG film

Potential Barrier

Hj[z]

Ψ (x,t)L

Figure 5.2: Cartoon of the two-mode approximation over the full macroscopic wave function ofthe condensate state. The local inhomogeneity in the magnetization produced by a dc currentthrough a wire conductor, corresponds to a potential barrier to the magnons.

i~ (1 + iα) ∂tψL(t) =[EL + UL

(γ1 |ψL|2 + γ2 (ψ∗LσzψL)σz

)]ψL(t) + νσxψ

∗L(t) + KψR(t)

i~ (1 + iα) ∂tψR(t) =[ER + UR

(γ1 |ψR|2 + γ2 (ψ∗RσzψR)σz

)]ψR(t) + νσxψ

∗R(t) + KψL(t) (5.7)

92


where the spatial dependence was integrated utilizing the orthogonality condition Eq. (5.5),

assuming that the spatial conguration for the components ±km are the same. These systems

of non-linear equations represents the dynamics between two magnon condensate states with

a coupling factor, proportional to their overlapping, with the higher order contribution were

neglected.

The information about of the spatial dependence is contained in the coecients Ei, Ui andK. The meaning of such parameters are the following: the coecient Ei represent the zero pointenergy in each region, Uin±km

i are proportional to the self-interaction energies, while K describe

the amplitude of the tunneling between both condensates. Those coecients can be written

in terms of Φ1,2(r) wave-function overlaps and the eective parameters which characterize the

condensed phase determined by the experimental realization displayed in the Fig. (5.1). The

expressions for each one of this coecients, where the weakly linked approximations was used,

correspond to

Ei =

∫drΦi(r)

[− ~2

2m∇2 + µ+ Vj(r)

]Φi(r) (5.8)

Ui =

∫dr |Φi|4 (r) (5.9)

K =

∫dr

(~2

2m∇ΦL(r)∇ΦR(r) + V (r)ΦL(r)ΦR(r)

)(5.10)

From this motivation the wave functions ψi, i = L,R can be written in occupation density-

phase representation, i.e.

ψi =

√nkmi (t)eiφi(t)√

nkmi (t)e−iφi(t)

(5.11)

where we are limiting ourselves to consider the in-phase oscillations with the total number of

magnons related to |ψL|2 + |ψR|2 =(nkm

L + n−kmL

)+(nkm

R + n−kmR

)≡ nT.

As has been widely discussed in the last chapter, the symmetry breaking term proportional

to ν in the Eq. (5.7) implies non-conservation of magnon population. In that sense the total

occupation density of magnons, |ψL|2 + |ψR|2 = nT(t), is time-dependent, in fact we can see

explicitly such temporal dependence in the occupation density in the Eq. (4.12). However can

be shown that the internal time scale involved in such process is greater than the Josephson

93

5.2 The Magnon Josephson Junctions equations

-3 -2 -1 0 1 2 3

5

10

15

20

x°@ΜmD

VV

0

Figure 5.3: Prole of the potential barrier produced by a wire conductor. The local inhomo-geneity of the magnetization is determined by the current J and the diameter of conductor.Both curves are calculated for HI = 200[Oe] and HI = 500[Oe], and δ = 50[nm] in a meanregion of L = 6[µm].

oscillations between the system of two states, Fig. (5.2), and as a approximation we can assume

that the total density system of two clouds magnon condensate remains constant.


In the last section we have established several properties, characteristics of the magnon

condensate system fragmented by a potential well. For now we will restrict ourselves to the

case where the internal oscillations are frozen and just we considered, as a relevant variables,

the oscillations between both left and right states, i.e. the external magnon Josephson eect.

Considering the expression for the wave functions Eq. (5.11), the two-mode dynamical equa-

tion Eq. (5.7) can be written in terms of macroscopic observables as the population imbalance

94


and relative phase, dened as

φ(t) ≡ φR(t)− φL(t)

η(t) ≡ 1

nT(nL(t)− nR(t)) ,

leading to the formal expression for the Magnon Josephson Junctions(MJJ) equations, phe-

nomenologically deduced in the last section, given by

η = −αΓη −√

1− η2 sinφ (5.12)

φ = Λη +1√

1− η2(η cosφ− α sinφ) (5.13)

where the time is rescaled to a dimensionless characteristic time tc = ~/2K and the phenomeno-

logical parameters acquired the following form

Λ =1

2Kγ1Uρc

Γ = (2E + 2γ1Uρc + µ− ν) /2K.

The total conserved energy, i.e. without dissipation, can be written as

H[φ, η] =Λ

2η2 −

√1− η2 cosφ (5.14)

and which give rise to the equations Eq. (5.12) for α = 0. The dimensionless parameters Λ,

Γ and the dissipation determine the dynamic regimes of the magnon condensate tunneling. Is

evident that if there is not dissipation, only the Λ parameter determine the type of oscillation

in the Eq. (5.12) being, in turn, characterized by the in-plane magnetic eld and the size of the

inhomogeneity. The phase diagram for Λ parameter is displayed in the Fig. (5.4) as a function

of in-plane magnetic el H0 and the local inhomogeneity in the magnetization Hj for typical

experimental values in YIG thin lm.

From a formal point of view, the problem of macroscopic tunneling between two states ψL(t)

and ψR(t) is identical to the problem of a single electron in a polarizable medium, forming a

polaron[80]. In fact, the common point is the nonlinear Schroedinger equation.From a standard

analysis the magnon Josephson equation Eq. (5.12), without dissipation, can be reduced to

95


elliptic integrals which are resolved in terms of Jacobian elliptic functions. Combining the Eq.

(5.12) and Eq. (5.14), we obtain

η2 +

[Λη2

2−H0

]2

= 1− η2. (5.15)

Integrating the Eq. (5.15) we have a formal solution for η(t) in terms of quadratures

Λt

2=

∫ η0

η(t)

dη√(α2

1 + η2) (α22 − η2)

(5.16)

where the coecient obeys

α21 =

2

Λ2

[ζ2 − (H0Λ− 1)

], α2

2 =2

Λ2

[ζ2

2+ (H0Λ− 1)

](5.17)

ζ2(Λ) =2√

Λ2 + 1− 2H0Λ. (5.18)

The solution for Eq. (5.16) can be written in terms of the cn and dn Jacobian elliptic

functions.

η(t) =

α2cn [(α2Λ/k) (t− t0), k] for 0 < k < 1

α2dn [(α2Λ) (t− t0), 1/k] for k > 1(5.19)

with the elliptic modulus as

k2 =1

2

(CΛ

ζ(Λ)

)2

=1

2

[1 +

(H0Λ− 1)√Λ2 + 1− 2H0Λ

], (5.20)

t0 = 2

[Λ√α2

1 + α22F (arccos [η0/α2])

]−1

, (5.21)

where F [φ, k] =∫ φ

0dφ(1− k2 sin2 φ

)−1/2correspond to the incomplete elliptic integral of

the rst kind. The Jacobian elliptic functions Eq. (5.19) are periodic in the rst argument

with period 4F (π/2, k) and 2F (π/2, k), respectively. It should be noted than that character of

those solutions changes qualitatively in the k = 1 value. In the next subsections we presents

the physical meaning of this analysis and characterize the two oscillation regimes separated

by the condition k = 1, which correspond to solutions whose period becomes innite and the

96


population imbalance is given by a non oscillatory hyperbolic secant. Physically these solutions

can be classied in terms of the Λ parameter, but the transition point between them is subject

to initial conditions, and then to the initial preparation of the condensate system.

ac Josephson's Oscillations

MQST State

H [

KO

e]

1.0

0.8

0.6

0.4

H [KOe]

0.8 1.0 1.2 1.4 1.6 1.81.0

0.8

0.6

0.4

0.2

0.0

0

j

0.2

ΛΛmax

Figure 5.4: Phase diagram displaying the behavior of Λ parameter which delimits two, qualita-tively dierent, Josephson's oscillation regimes. The Λ-parameter is determined as a functionof the inhomogeneity in the magnetization, Hj, produced by the dc current, and the in-planemagnetic eld applied H0. In the diagram is drawn a Λ-contour for a xed value of externalmagnetic eld applied over the sample of YIG, H0 = 1[KOe]. The curve determine values for Hj,i.e. for the dc current, for which the Josephson's oscillations are in the ac Josephson oscillationor macroscopic quantum self trapping state.

5.2.1 AC magnon Josephson oscillations

As mentioned before, these are classied in terms of the Λ parameter. Here we restrict

ourselves to the case only for k < 1, with special emphasis in small oscillations where the

nonlinear contributions can be disregarded.

The solutions of Magnon Josephson equations, without dissipation, have been widely studied

97


in Bosonic Josephson Junctions(BJJ) for alkali gases which solution can be classied in terms

of Jacobian elliptic functions, as shown in the previous section. Nevertheless, the subtle con-

sideration of dissipation not only increases the complexity but also contains, through of Γ, the

symmetry breaking parameter ν. In other words, the anomalous scattering process that doesn't

conserve the magnon density, enhanced the damping of the Josephson's oscillations.

Pop

ulat

ion

Imba

lanc

eP

opul

atio

n Im

bala

nce

Rel

ativ

e P

hase

Time , ns Time , ns

0.1

-0.1

0.0

0.2

-0.2

0.0

0.2

-0.2

0.0

π

πR

elat

ive

Pha

se

0.0

π

-π

0 20 40 0 20 40

Time , ns Time , ns

0 15 30 0 20 40

Figure 5.5: Dynamical behavior of both relative phase φ and population imbalance η between theclouds magnons condensate. The solutions are calculated for the typical experimental conditionsover the YIG thin lm, a in-plane magnetic eld H0 = 1[KOe] and Λ(Ha = 250[Oe]) = 40 wherethe dissipation α = 10−5. The initial conditions, φ(0) = 0.5π-η(0) = 0.1 and φ(0) = 0.9π-η(0) = 0.05 determine the small amplitude and long wave oscillations respectively.

Let us start with the most elemental case, i.e. the Josephson's oscillation of small ampli-

tude. In fact, is evidenced in this straightforward example the quantum nature of the magnon

condensation phenomena discussed so far. The oscillatory behavior of the relative phase reect

the macroscopic interference of the magnon condensate states. In other words, the oscillatory

nature both of the magnon current and relative phase, is reminiscent to the elementary behavior

in superconducting junctions.

98


The solutions for magnon current and relative phase are calculated for several values of

the potential well, i.e. for dierent intensities of the dc current applied. In the Fig. (5.5) we

illustrate experimentally accessible solutions for those macroscopic observables of the system,

when the inhomogeneity in the magnetization take values in the range H0 ∈ [250 − 500][Oe].

For small amplitude of oscillations, both the magnon current as relative phase can be written

as

η(t) = η0 cosωact (5.22)

φ(t) = C sinωact+ φ0 (5.23)

where (η0, φ0) are the initial conditions, ωac =√

1 + Λ the frequency of oscillations and C

the integration constant. In fact in a realistic scenery the frequency is about ωJ ∼ 0.6[GHz]

for H0 ∈ [250 − 500][Oe]. It is worth noting that the critical magnon current is related to

the amplitude ηc = η0 which is determined by the initial population imbalance between the

condensate states, while the frequency of oscillation ωac, is determined by the dc current.

The solutions displayed in the Fig. (5.5) are calculated taking into account the dissipation

and considering small and long-amplitude oscillations, where we considered a in-plane magnetic

eld H0 = 1[KOe] and dissipation coecient α = 10−4. The small amplitude oscillation is

determined by the values of parameters Λ = 20(Hj = 250[Oe]) and initial conditions η0 =

0.1, φ0 = 0.1, while for long-amplitude oscillations we use Λ = 70(Hj = 500[Oe]) and initial

conditions, η0 = 0.1, φ0 = π/2. It is worth point than that solutions, near of critical value,

Λc, which separates the two regimes presents in the Eq. (5.12), the period of oscillation is

sensitive to initial conditions since the nonlinear contributions begin to be relevant. Aa feature

which, however, distinguishes this regime of oscillation is than that mean value of the magnon

current 〈η(t)〉 = 0, i.e. the system of two quantum states oscillate around an equilibrium value

set in ηeq = 0. Moreover, it follows from Fig. (5.5) that the long-amplitude oscillations are

quickly damped, respect to the small amplitude oscillations, due to the nonlinearity of system.

In this sense, the dissipation prevents the spread of nonlinearity more eciently than the small

amplitude solutions, driving rapidly the nonlinear solution to the linear regime.

The solutions presented here correspond to the ac Josephson eect for the condensed magnons.

The frequency of oscillations of the magnon current is directly related to the macroscopic rela-

tive phase of the pair condensate, which are within the typical experimental resolution range.

Therefore we provide a precise scheme for the observation of ac Josephson's oscillations, over

the YIG thin lms, as a prove of the macroscopic quantum coherence of magnon condensate.

99


It is worth note that there is nothing original in these results, since just we are conrming that

the condensate of magnons satises the Josephson's relation Eq. (5.12), where the dissipation,

proper of the magnon gas, play a important role. However, novel phenomena will appear when

the internal degrees of freedom are released. The connection with the magnetization dynamics,

i.e. the behavior of the spin density wave during the Josephson's oscillations, will be established

in the next chapter.

5.2.2 Macroscopic quantum self-trapping of magnons

The magnon ac Josephson's oscillations discussed above, corresponds to an oscillating stream

of magnons through of potential barrier. The frequency and amplitude of the oscillations, within

the linear regime, are dictated by the relative phase between the condensate pair and by the

initial condensate density, respectively. Although these analysis doesn't apply when the self-

interaction Λ parameter increase below the critical value, since the nonlinear eects become

relevant, that oscillations are characteristic of a mean occupation density value in the time,

〈η(t)〉 = 0. Namely, the magnon current oscillates around of a null value. However, this

scenery change drastically when the self-interaction surpasses a critical value Λ > Λc, where the

Josephon's oscillations follow a qualitatively dierent behavior. In this stage the evolution of

magnon population imbalance is characteristic of a nonzero time-average 〈η(t)〉 6= 0, i.e. there

is a self-trapping of magnons at one side of the wire conductor, where the dynamics evolution

is strongly inuenced by the initial conditions.

This intricate nonlinear phenomena, discovered by [79] in the context of BEC's of alkali

gases, is so called Macroscopic Quantum Self-Trapping(MQST), which quantum nature involves

the coherence of a macroscopic number of bosons in the two condensates. Here we show that

the magnons condensate manifest themselves in a macroscopic quantum self-trapping state, for

certain values of the self-interaction and where the dissipation play a important role in the

dynamics of such state.

The phase diagram displayed in the Fig. (5.4), indicating the dierent oscillation regimes,

shows the zone where the self-trapping phenomena of magnons occurs. That region is determined

when the self-interaction parameter Λ exceed the critical value Λc, which is in turn specied by

the initial conditions η0 and φ0. In fact the MQST state will appear when the system satisfy,

100


in the dissipationless case, the condition

H0 =Λ

2η2

0 −√

1− η20 cosφ0 > 1. (5.24)

which means that trajectories that connect one complete period in the phase-space portrait. In

the dynamical-system language that phenomena is denominated libration. Subject to the initial

preparation of the system, i.e. controlling the condensate density by means of the pumping

power, the critical parameter for MQST obeys

Λc =1 +

√1− η2

0 cosφ0

η20/2

. (5.25)

Once again the self-interaction satisfy Λ > Λc, the Josephson's oscillations are driven to the

completely nonlinear regime, which are determined by the Eq. (5.19) for k > 1. In expression

that dene the elliptic modulus, Eq. (5.20), we note the dependence in the initial energy of

system H0. The frequency of the Josephson's oscillations can be obtained from the solutions

of Eq. (5.12), doing α = 0, following the analysis of the above section. In fact the frequency

obeys the following form,

ΩJ = Ω

πα2Λ

2kF [π/2,k]0 < k < 1

πα2ΛF [π/2,1/k]

k > 1(5.26)

where Ω =√

2((H0Λ− 1) +

√Λ2 + 1− 2H0Λ

)and the K-function dened by Eq. (5.20). This

calculation is a good approximation, while the self-trapping magnons state remains. However

in real situations, where the dissipation is considered, the Josephson's oscillation will follow a

much more complicated behavior. Indeed, from the Eq. (5.26), the Josephson's frequency for

long-amplitude and MQST oscillations are obtained by k . 1 and k > 1, respectively. Is clear

the sensitivity of the frequency on the initial conditions through of k-dependence, due to the

strong nonlinearity of the magnon Josephson equations, unlike in small amplitude oscillations

case.

Considering the presence of damping, can be seen that the condensed magnons remaining in

the MQST state for some time until decay into a ac long-amplitude oscillation, see Fig. (5.6).

Indeed the dissipation quickly destroys such state giving way to the ac Josephson's oscillations

studied before, i.e. for Λ > Λc xed but increasing the initial energy H0, the MQST solutions

are more quickly damped, and then the lifetime of the magnons self-trapped, is smaller.

As we have learned, the appearance of MQST is subject to the initial preparation of the

101


Pop

ulat

ion

Imba

lanc

eP

opul

atio

n Im

bala

nce

Rel

ativ

e P

hase

Time , ns Time , ns

0.4

-0.4

0.0

0

0.3

0.0

0.15

Rel

ativ

e P

hase

0 20 40 0 20 40

Time , ns Time , ns

0 20 40 0 20 40

6π

12π

0

6π

12π

Figure 5.6: Dynamical behavior of both relative phase φ and population imbalance η between theclouds magnons condensate. The solutions are calculated for the typical experimental conditionsover the YIG thin lm, a in-plane magnetic eld H0 = 1000[Oe] and Λ(Ha = [Oe]) = 20 andΛ(Ha = [Oe]) = 50 where the dissipation α = 10−5. The initial conditions for both solutionsare φ0 = π, η0 = 0.1.

102


condensed states. In the Fig. (5.6) the solutions are exposed to a initial relative phase φ0 = π

with a imbalance population in the range η0 = [0.1 − 0.5] and the critical parameter Λc =,

determined by the phase diagram, Fig. (5.4). With all of these considerations is possible pass

across from the ac oscillations until MQST state changing the applied dc current. The eect

over the spin waves degrees of freedom will be shown in depth in the next section.

The dissipation in the condensate magnons system play a signicant role contributing to

the decoherence of condensate gas, by means of incoherent scattering and anomalous processes,

determining their life time. This, obviously, leads to a damping on the oscillation amplitude

between the weakly coupled states ψL and ψR. However, the damping eect on the oscillations

depend if the self-interaction parameter is greater or lower to the critical value. That point

can be visualized in the breakdown of the MQST state, establishing a characteristic life-time

mean for this state, and for later give rise to the usual ac Josephson's oscillations. From a

microscopic point of view, both the precessional angle of the spins, in each site of the lattice,

as their longitudinal projection, i.e. δMz, are damped to a stable conguration, in complete

analogy with the Gilbert damping in the magnetization dynamics.

2Π 4Π

-0.3

-0.1

0.1

0.3

Relative Phase Φ

Po

pu

lati

on

Imb

alan

ceΗ

Figure 5.7: Phase-space portrait of the dynamical variables η and φ for Λ > Λc values. Thetrajectories followed by the oscillations in the MQST regimes correspond to librations, in thenonlinear pendulum language, since connect a complete period in the phase-space. The solutionsare calculated in a period of τ ∼ 200[ns]

With our knowledge acquired in the last chapter, i.e. the relationship between the condensate

of magnons and Hopf-Andronov bifurcation, the MQST state can be conceived in the similar

terms. The MQST can be understood by mapping the system into an equivalent classical

103

5.3 Dynamical behavior of the spin density wave

nonrigid pendulum system, where the pendulum length depends on the angular momentum.

The angular momentum in the pendulum maps to the population imbalance in the weakly

linked magnon condensates, and likewise angular displacement maps to the relative phase.

Then the Josephson's eect is analogous to small oscillations of the pendulum around of the

equilibrium. Meanwhile the MQST correspond to the pendulum to make complete revolutions,

i.e. librations. In that case the angular momentum is nonzero and the angular displacement is

monotonically increases in time. In our language, that is associated with the nonzero average

population imbalance. For the solutions presented in the Fig. (5.6) the phase-space portrait

is displayed for the trajectories dened by the dynamical variables η and φ, see Fig. (5.7). In

that gure the libration of the magnon condensate pair and the dissipation eect, extensively

discussed, is clearly evidenced.


In the above section we have showed the dynamical behavior of the population imbalance and

the relative phase between the weakly coupled magnon condensates. It was emphasized the role

of dissipation α and the broken symmetry term ν, important properties that characterize the

magnon condensate upon the ferromagnetic system. In the Josephson eect previously studied,

we studied two qualitatively dierent regimes of magnon's current which classical counterpart

can be seen in the spin density wave dynamics. As already established, theoretically in the

last chapters and experimentally[81], the manifestation of the condensate in real space is as a

spin density wave with wavelength 2π/km, as a result of the bi-condensed property, and which

position is xed by the spontaneous symmetry breaking.

The essential feature associated with magnon coherence is the non-vanishing expectation

value of the magnon creation an annihilation operators. Let us consider the situation with the

two valleys equally occupied by the coherent magnons, since such state is energetically favorable

because γ2 > 0, whose eld operators expectation values are

|Ψ±k〉 =√n exp (iθ ± iφ). (5.27)

where θ is the phase related to the explicit U(1) symmetry breaking due to the non-conservative

of magnon population and φ to the spontaneous symmetry breaking associated with the position

104


of the SDW. The magnetization dynamics eld deviation from saturation is determined by

δ ~M(x) ∼√n(t) (cos θ, sin θ, 0) cos

(kmx|| + φ(t)

), (5.28)

where unlike the minimum energy SDW conguration, Fig. (4.1), the density and the phase

are time-dependent due to the phase-dependence of the coupling energy between both magnon

condensates, in the regime of Josephson oscillations. Let us provide at continuation an accurate

interpretation of the meaning of equation Eq. (5.28).

Spin density wave dynamics in the ac Josephson's oscillations regime

Before to carry out the mapping to the magnetization degrees of freedom, it must be kept

in mind that when the spontaneous symmetry breaking of the residual symmetry happens,

the magnon gas choose an arbitrary phase, the position of the spin density wave is xed but

undetermined. However, in the weakly linked magnon condensates scenery the relative phase

is completely determined and, as a consequence, the relative position between the spin density

waves too. Due to the Eq. (5.28) the size of the magnetization eld is related to the phase-

dependent current of magnons.

H0 MRML

Pot

enti

al w

ell

12 μm

20

40

60

80

100

Tim

e, n

s

Figure 5.8: Dynamical evolution for the spin density wave at each side of the barrier. Thesolution presented correspond to a inhomogeneity Hj = 250[Oe] and initial conditions φ(0) =π/2, η(0) = 0.2 for a in-plane magnetic eld H0 = 1[KOe]. The evolution is ∼ 100[ns] and thedisplacement take place in the order of ∼ π/km.

The evolution of the cloud condensate at each side of the potential well is illustrated in Fig.

105


(5.8) for the ac Josephson oscillations regime. The solution presented correspond to long-wave

oscillations with period ∼ 30[ns], for a inhomogeneity Hj = 250[Oe] and a in-plane magnetic

eld H0 = 1[KOe]. Within the Fig. (5.8) can be observed the typical displacement of a SDW for

ac Josephson's oscillations, where the amplitude of oscillation is directly related to the critical

magnon current ηc.

The resulting magnetization correspond to a oscillatory displacement of the spin density

waves in opposite phase where the coupling is mainly dominated by the long-range dipolar

interaction. In the sense of the discussion given in the subsection 4.1, the precessional cone

angle of each magnetic moment is periodically varied.

Nonlinear spin density wave dynamics in the MQST of magnons regime

As well as the magnetization in the ac Josephson oscillations regime, the behavior of spin

density wave in the macroscopic quantum self-trapping oscillation state is determined and dis-

played in the Fig. (5.9)(d).

The complete Fig. (5.9) represent the transition towards the macroscopic quantum self-

trapping regime from the ac Josephson oscillations, Fig. (5.8). That transition is achieved

changing the local magnetizationHj. Indeed when the current is decreased until the Λ parameter

surpass the critical value Λc, the MQST state is allowed. The solution for each frame is calculated

for the prepared state in the initial conditions φ(0) = π, η(0) = 0.1 and H0 = 1[KOe] at dierent

values of Hj.

Is worth mentioning two important characteristics of the magnon condensate system; rst,

as was mentioned, the MQST eect manifests itself as a localization of a large number of

condensed particles in a specic region in space. However, on the other hand such agglomeration

of condensed magnons result in a unidirectional movement of the spin density wave, due to

the monotonically increasing of relative phase, unlike the normal ac oscillations Fig. (5.6-5.8)

where the sliding modes are oscillating. The propagation direction of the SDW is opposite to

the localization region of the condensed magnons. Second, the movement of the SDW remains

until the dissipation destroys the macroscopic quantum self-trapping state and falls into the ac

oscillation regime again.

106


Figure 5.9: Behavior of the Spin Density Wave in the macroscopic quantum self trappingoscillation regime. The displayed solutions are for initial conditions φ(0) = π, η(0) = 0.1and a in plane magnetic eld H0 = 1[KOe] to dierent values of the inhomogeneity in themagnetization values where each frame represent the transition to the MQST state, by changingthe dc current.

107

5.4 Internal Josephson's oscillations and asymmetric current of magnons

5.4 Internal Josephson's oscillations and asymmetric cur-

rent of magnons

The external Josephson's eect studied in the last sections, and by external we mean that

we do not consider oscillations between valley of each condensate, is described by the magnon

Josephson equations and, as a result, obeys the same behavior as bosonic Josephson junctions

for alkali gases. The characteristic dissipation and non-conservation of spin in the magnon gas

contribute to the damping of the Josephson oscillation for each regime, i.e. ac oscillations or

MQST state. However, the natural question that arise is what happen with the behavior of

Left state

k ,m -φ- k ,m φ

Right state k ,m -φ- k ,m φ

k ,m -φ- k ,m φ

k ,m -φ- k ,m φ

Figure 5.10: Magnon asymmetric-current Josephson eect between two cloud condensates. Therst cartoon illustrates the external Josephson oscillations where the internal oscillations aresuppressed. The red line represent the dynamical oscillations where the condensed componentshave the same occupation nkm

a = n−kma , for both states a = (L,R). The second graph shown the

case when the internal oscillations are allowed. In that situation appears a nonzero contribution,represented by the blue arrow, for the magnon current from the nkm

L → n−kmR states, due to the

combination between internal and external magnon currents.

spin density wave when the internal oscillations are unfrozen. This idea suggests to consider

the mixing of internal and external degrees of freedom, i.e. the coupling between the inter-

nal oscillations and the external dynamics with others macroscopic states. We refer to this

phenomena as the internal Josephson's eect, where the magnon oscillation linking the valleys

±km are allowed. As a consequence of the internal Josephson eect in a system partitioned, like

two clouds condensed of magnons, the system can experience asymmetrical current, where by

108


asymmetric-current we mean a magnon's ow connecting dierent valleys, at each side of the

barrier. The asymmetric Josephson eect have been studied in the context of ultracold Fermi

gases[82], which the asymmetric Josephson current appears between two spin components of

a Cooper pair in a Josephson junction of two superconductors. Here we perform the analog

to the magnons current between two magnon condensates [83]. The Fig. (5.10) depicts the

magnon asymmetric-current in the Josephson eect. In this illustration is appreciated the ex-

ternal oscillations, arrowed by red lines, where the occupation imbalance between valley, for

each condensate, are suppressed. While in the other gure is shows that the internal oscilla-

tions are allowed, whose dynamics result in complex collective behavior mixing several degrees

of freedom associated to the relative observables between valley and condensates.

In order to explore the behavior above mentioned, i.e. the interplay between internal and

external Josephson-modes, we write the equations Eq. (5.7) in occupation-phase representation

again. Unlike the case studied in the above sections, the wave function Eq. (5.11) has unequals

occupations for each condensed component. In this respect the wave function can be written as

ψa(t) =

(√nkma eiφa(t) 0

0√n−kma e−iφa(t)

)(5.29)

where a = (L,R) indexes both states and both components have dierent densities of occupation.

Let us introduce the following denitions for the relative phase between the condensates and

population imbalance among valley and the magnon condensates,

δn+ = nkmL − n

kmR

δnL,R = nkmL,R − n

−kmL,R

φ = φR − φL, (5.30)

the δnL,R quanties the distribution of magnons between the valleys and. Starting from the

expression for the wave function Eq. (5.29) and using the introduced denitions Eq. (5.30), the

respective combinations gives rise dierent equations of motion for the population imbalance

and phase dierence between both condensates (L and R) and their respective components (km

and −km). In fact with some manipulations, in the same way as in Section 5.3, is possible to

lead to the following set of equations

109


∂tφ =1

K(Λ1δn+ + Λ2 (δnR − δnL)) + 2K

(α (1 + ηL + ηR) sinφ+ 2η+ cosφ(

(1 + ηL + ηR)2 − 4η2+

)1/2

)

− ν

(((1 + ηL − ηR)2 − 4 (ηL − η+)2)1/2 −

((1− ηL + ηR)2 − 4 (ηR − η+)2)1/2(

(1 + ηL + ηR)2 − 4η2+

)1/2

)(5.31)

∂

∂tδn+ =

α

2K[Γ1δn+ + Γ2δn+ (δnL + δnR) + Γ3 (δnL + δnR + nT ) (δnL − δnR)]

+1

2K

(1

4(nT + δnL + δnR)2 − δn2

+

)1/2

sinφ

+ αν

4K

((1

4(δn+ + δn− + nT )2 − δn2

L

)1/2

−(

1

4(nT − δn+ − δn−)2 − δn2

R

)1/2)

(5.32)

∂

∂tδnL = −α

[∆0 + ∆1 (1 + 2η+ + ηR) ηL + ∆3η

2L

]+

(1

4(1 + ηL + ηR)2 − η2

+

)1/2

(sinφ− α cosφ)

+

(1

4(1− ηL − ηR)2 − (η+ − (ηL − ηR))2

)1/2

(sinφ+ α cosφ) (5.33)

∂tηR = −α[(2ER/nT + URγ1 (1− 2η+ − ηL)) ηR + UR (2γ2 + γ1) η2

R

]/K

−(

1

4(1 + ηR + ηL)2 − η2

+

)1/2

(sinφ+ α cosφ)

+

(1

4(1− ηR − ηL)2 − (η+ − (ηL − ηR))2

)1/2

(α cosφ− sinφ) (5.34)

where each coecient is a function of the physical parameters for the condensate magnon gas

and determined from the microscopic theory in the Chapter 3, and eective parameters which

characterize the weakly coupled magnon condensates in the potential well for the Josephson's

oscillations. The set of equations Eq. (5.31-5.34) are the generalized Josephson's equations

where the internal degrees of freedom are taken into consideration. These equations have a

110


complex structure and just we analyze some limited cases.

That system of non-linear dierential equations represents the dynamic behavior of the

fractional population imbalance and relative phase between condensates. Those collective in-

teractions, among the condensed magnons with momentum ±km in the L and R states, gives as

a result a magnon current connecting each valley of the system of two states. To illustrate that

idea let us denominate each component of the condensate by |km, φ〉 and |−km,−φ〉, see the Fig.(5.10). So, in that sense, the magnon current will appear connecting the

∣∣kLm, φL⟩→ ∣∣kRm, φR⟩,∣∣kLm, φL⟩ → ∣∣−kLm,−φL⟩and

∣∣kLm, φL⟩ → ∣∣−kRm,−φR⟩states. From a microscopic perspective

such kind of oscillations, the two last processes, envelops a non-conservative scattering processes

and that will become evidenced in the following.

It should be pointed that all these properties arise essentially due to the spinorial charac-

teristic of the magnon condensate and the interplay between these give rise novel phenomena.

In the next subsections will be presented some outstanding results that reect and deepen the

ideas previously introduced.

5.4.1 Internal Josephson eect

To be self-consistent with these discussion let us start from the most simply case, namely

considering just one magnon condensate, for the Josephson-like oscillations. By Josephson-like

we mean a coherent tunneling of magnons from one valley to the other. It is worth mentioning

that in this situation, the coherent interplay between the magnons on each valley result in a

phase-locking among components which manifest itself in the spin density wave discussed before.

Any internal oscillation, reected in a dynamical behavior of the phase between component, will

not be a test of quantum coherence. The latter is because both components have opposite phase

and these were arbitrarily dened by the spontaneous symmetry breaking mechanism.

Internal Josephson-like oscillations

Spontaneous coherence over macroscopic length scales is readily associated with the in-

teresting phenomenology of interference and Josephson eects. In this section we study the

possibility of internal Josephson oscillations between the magnons lying in the dierent val-

leys. The internal Josephson eect has been studied on other spinorial condensates [84].

If we separate the phase dierence, φ, between the valleys from Eq. (4.7), doing |Ψ〉 =

111


(eφ(t)√nk(t), e−φ(t)

√n−k(t))t where n±k(t) are the density in each valleys, we can easily nd

a Josephson-like relationship. For a small imbalance of magnons density between valleys,

δn = 〈Ψ|σz|Ψ〉/〈Ψ|Ψ〉, the phase dierence displays a damped behavior due to the dissipa-

tion coecient α. This coecient couples the equations for φ and δn

~φ = − (µ+ (γ1 + γ2)ρ0) δn− ~αδn,

δn = εf(sinφ)− αφ (5.35)

that can be derived from Eq. (4.7). Due to the translational invariance of the system, the case

Figure 5.11: (a) Illustration of internal Josephson oscillations between valley ±k0, the unbalancebetween the valleys drives phase oscillations. Damping works restoring the balance betweenvalleys. (b) Numerical solutions of Eq. (5.35) for α = 0.1, with initial conditions p(0) = 1(complete polarization) and ϕ(0) = 0. Such mode characterized by the imbalance population,p(t), and the dierence phase ϕ(t) between both components of the condensed cloud.

we have studied is described by ε = 0 In this case, Eq. (5.35) have an interesting implication

concerning the damping of the sliding modes. The equations can be rearranged in the form

δn = −δn/τn, where τn = ~/2α (µ+ (γ1 + γ2)ρ0) , and correspond to a simple dependence of

the damping rate of sliding modes on the net magnon density of the system. The sliding modes

will become gapped if the system has spatial inhomogeneities, this is represented in our Eq.

(5.35) by a parameter ε 6= 0, the function f is to be determined from Eq. (4.7), for a specic

form of the inhomogeneities.

112


There is a profound relationship between Eq. (5.35), and the semiclassical equations of

motion for the collective dynamics of a Josephson junction[85] and a single-domain easy plane

ferromagnet in an in-plane eld [86]. This phase relationship over the relative phase of the

condensate indicates that Josephson-related eects should be displayed by the system. Those

eects, that link the dynamics of magnon population and the spatial conguration of the mag-

netic patterns, will be studied in detail in a forthcoming work.

5.4.2 Magnon asymmetric current

As was discussed at the beginning of this section, the internal population imbalance is cou-

pled with the Josephson's modes. Here we analyze that oscillations regime where for simplicity,

but without loss of generality, is assumed a dissipationless case. The set of magnon Josephson's

equation can be reduced to

∂

∂tφ =Λ1δn+ + Λ2δnL +

4δn+ cosφ

(1− 4δn2+)

1/2

− ν

(((1 + 2δnL)2 − 4 (δnL − δn+)2)1/2 −

((1− 2δnL)2 − 4 (δnL + δn+)2)1/2

(1− 4δn2+)

1/2

)(5.36)

∂

∂tδn+ =

1

2K

(1

4(1 + δnL + δnR)2 − δn2

+

)1/2

sinφ (5.37)

∂

∂tδnL =

1

2

[(1− 4δn2

+

)1/2+(1− 2 (δn+ − 2δnL)2)1/2

]sinφ. (5.38)

We note that the dissipationless condition implies a counterclockwise oscillations between

the internal magnon currents, i.e. δnL = −δnR. From that restriction the magnon current from∣∣−kLm,−φL⟩→∣∣−kRm,−φR

⟩is determined by δn− = δn+ − 2δnL. A numerical integration of

those equations are displayed in the Fig. (5.12), for several initial conditions. In that gure the

population imbalance, either δn+ and δnL are identied by the blue and black lines, respectively,

where we note the self-trapping of magnons between the L and R states for the population

imbalance δn+ as well as for the internal magnon current δnL. The other population variables

are arrowed in the plot, however are determined by δn+ and δnL. The nonlinear nature of these

oscillations are illustrated in the phase-space portrait. From a microscopic point of view we can

imagine that situation as follow: given a little initial imbalance in the densities δn+(0) 6= 0,

113


appears a internal current of magnons δnL connecting the valleys ±kLm. As a consequence of that

the time-average of internal magnon current 〈δnL〉 6= 0, there are a constant ow of magnons

from one valley to the other, which result in a external magnon current with 〈δn+〉 6= 0.

Pop

ulat

ion

Im

bal

ance

Time, ns

Pop

ula

tion

Im

bal

ance

Relative phase

0.5

-0.5

0.0

0 0.1π0.05π

0.10

-0.10

0.0

0.10

-0.20

0.0

-0.10

0 2 4

0 2 4

Figure 5.12: Dynamical evolution of population imbalances in a system consisting of two magnoncondensates, when the internal oscillations are allowed. (a) The population imbalance dynamicsfor δn+ and δnL for dierent initial conditions, manifesting magnon self-trapping. (b) Trajecto-ries in the phase-space portrait for the dynamical variables displaying librating behavior in (a)and represented by the blue and black curves respectively.

It is important to note that albeit the self-interaction play a essential role in the formation

of a MQST state, for example Fig. (5.9), the internal magnons ow contribute to the formation

of these state. Indeed that properties of the internal currents is evidenced in the case reported

in Fig. (5.13). In that gure a linear regime for the external magnon Josephson's currents are

presented, where for small amplitude of the oscillations δn+(0) 1 one has even 〈δn+(t)〉 6= 0,

while that for the internal currents we have 〈δnL(t)〉 = 0. In other words, is possible to create

a macroscopic quantum self-trapping for magnons with a internal magnon current, even in the

linear regime for δn+. Obviously the physical mechanism behind the MQST state, in that

case, is dierent respect to the presented case in Section 5.3 since the self-interaction between

the L and R states is not relevant, but the stream of magnons which connect each valley, i.e.

the condensate components. In the Fig. (5.13) we presents two cases for the same initial

conditions, but dierent values for the physical parameters. The most curious case is the rst,

since the regime of oscillation is completely linear, which is enough to clear that the mechanism

114


of formation of MQST state is the synchronized ow of magnons, either connecting L and R

states, and dierent valleys.

Pop

ulat

ion

Im

bal

ance

Pop

ula

tion

Im

bal

ance

Time, ns

0.015

0.010

-0.005

0.005

0.000

0.020

0.015

0.000

0.010

0.005

0 25 50

Time, ns

0 25 50

δn δn + L δn + δn L

Figure 5.13: Small amplitude oscillations for the imbalance populations δn+ and δnL. In therst case the δn+(t) solution presents self-trapping while 〈δnL(t)〉 = 0, unlike in the second plotwhere we have 〈δnL(t)〉 6= 0.

Before closing it is worth commenting on the experimental signals that might be expected

from the eects discussed in this work. Magnetization oscillations can be measured in several

ways. The basic mechanism used so far in the context of magnon condensates, is the Brillouin

light scattering technique (BLS) [26]. Such technique probes the magnons system by studying

their eect on microwave radiation reected by the sample. In this way it might be expected

that the oscillations in magnon density between the two magnonic clouds might be detected.

Since our predictions involve oscillation periods on the order of 5− 20[ns]. Such oscillations are

however shorter than the characteristic resolution of the BLS measurement. As an alternative

the magnon dynamics can be mapped into spin currents pumped into a metallic sample in

contact with the system [87]. In fact a time-dependent magnetization in the ferromagnetic

system induce a spin-current on the adjacent metal

Is = A↑↓rdm

dt+A↑↓i m× dm

dt

, where the A↑↓i coecients characterize the interface. Such currents have been measured by

means of the inverse spin Hall eect in Pt [88], that converts them in charge currents. In the

present case, is easy to show that the presence of magnon-condensate implies a constant current.

Oscillations in such current can be detected and interpreted as signatures of the underlying

115


oscillations, see Fig (5.14).

VISHE ∼ −(πωmkmA↑↓i)n(t) (5.39)

Time, ns

0 20 40

MQST

AC

VIS

HE

k ωA

(

) ↑ ↓x

(a) (b)

Figure 5.14: (a) Schematic illustration of spin pumping setup by Josephson oscillations in theferromagnet. (b) Voltage signal, VISHE, induced in the Pt metal, proportional to the densityimbalance, in the AC and MQST oscillations.

In conclusion we have presented a phenomenological theory, that focusing only on the low-

energy and momentum projections of the magnon spectrum, accounts for the collective dynamics

of a Bose-Einstein condensate of magnons. Such theory has allowed us to provide a simple

understanding of the mechanisms behind the magnon condensation and to establish a clear

understanding of the meaning of collective wave function used to described it. In terms of such

description we discuss the nature of quantum interference between magnon clouds. Starting

with the discussion of the internal Josephson oscillations, that correspond to oscillations between

the ±k0 components of the condensed cloud. We have highlighted the close relation between

such eects and the well known Josephson eects. Using those ideas we presented a detailed

calculation of the Josephson oscillations between two magnon clouds, spatially separated in a

magnonic Josephson junction. Among the results we remark the clear and distinctive oscillations

that characterized common Josephson oscillations and also a regime that correspond to the so-

called macroscopic quantum self-trapping., that locks the oscillations favoring one side of the

junction over the other.

116

Chapter 6

Conclusions and Outlook

This thesis has focused on a type of behavior followed by a magnon gas parametrically

excited on a ferromagnetic media, which is interesting from the basic science viewpoint as well

as its implications in the magnonics eld, either highly coherent storage and transmission of

information in magnetic devices.

In this thesis we have presented the derivation of a eective quantum eld theory accuracy

for the description of a highly dense non-equilibrium magnon gas in a neighborhood of the

bottom state in the energy spectrum. Precisely, as evidenced at the beginning, the externally

excited magnons is a genuine out of equilibrium system and the eects of interaction with the

rest of gas for the thermalization and eventually condensation, must be taken into account.

Moreover, if we intended the description of a Bose-Einstein condensed phase, a suitable analysis

for the many-body eects is required, which can not be explained on the basis of two-body

physics, in the eective theory. The way followed in this thesis to include these fundamental

situations is allowed via a functional formulation of the Schwinger-Keldysh nonequilibrium the-

ory for many-body systems, where the many-body eect is treated by using the many-body

T-matrix approximation. From the eective low-energy theory we have been able to probe and

quantitatively explain the appearance of long-range order dictated by the spontaneous quantum

coherence. Such eorts were crowned by the characterization of the condensate stage through of

a semiclassical equation, the Gross-Pitaevskii equation, which exhibits the collective behavior of

the magnon condensate. In addition to investigate the macroscopic quantum coherence, the last

part of this thesis has been addressed in order to show the macroscopic interference phenomena

investigating in detail the Josephson eect for magnons in a ferromagnet.

In a more concrete sense, through of application of those eective low-energy theory, we have

provides a instability condition for the spontaneous emergence of macroscopic quantum coher-

ence of the magnon gas, i.e. the magnon BEC transition. The condensate state consists in a

pseudo-spinor with two component condensates, representing each components to the condensed

magnons with energy εm and momentum ±km respectively, whose coherent phases are opposite

and are associated to the symmetry breaking of the residual symmetry. The semiclassical inter-

pretation of the condensate state, which is a excellent agreement with the recent observations,

correspond to a spin density wave state, which amplitude is linked to the density of the conden-

sate state and which position, the phase coherent, is the result of the phase locking between the

component condensates. We have shown several properties of the Bose-Einstein condensed of

magnons stage that derive from the Gross-Pitaevskii equation: it contains the minimum energy

state as a trivial solution with the density determined by the eective parameters of the the-

ory. As a consequence of Holstein-Primako mapping a link is established between the bosonic

degrees of freedom of condensate state and their magnetization variables counterpart. Finally,

we report the existence of vortex structures for the magnon condensed that nicely emerge as a

dislocation in the spin density wave pattern. Nevertheless, challenges at this address are still

many, such as the inuence of the denite chirality in the spin wave propagation through of the

vortex state, as a source of transversal magnon current, or the existence of another topological

structure like skyrmions.

As discussed, we have presented a description for the condensate magnons which is closely

related to the phenomenology of particle condensates, like alkali gases, whose relationship is

evidenced in the Gross-Pitaevskii equation. We adopt the semiclassical approach. That discus-

sion was mainly concerned to investigate the macroscopic interference properties of condensate

magnons. An a important conclusion about this work is that, for certain values of the param-

eters, like the dc current or some other geometrical parameter, there exist a Josephson eect

for condensed magnons which in their more straightforward form, is analogous to previously

reported. We predict magnon Josephson oscillations within a regime experimentally accessible

and expect novel phenomena related to nonlinearity and spinorial characteristic of the magnon

condensate.

The theory presented in this thesis was focused to explain the physics near the BEC tran-

sition, however it can be generalized to a magnon gas widely distributed in the spectrum, i.e.

the formalism provides us a straightforward manner to include scattering processes relevant to

the microscopic physics in each region of spectrum.

This thesis can be intended as a building block in the magnonic subject where applications

can be performed to investigate either quantum interference, storage or transmission of spin

waves on magnetic systems. In this work provides us an accurate framework to study novel

phenomena in the physics of magnons on ferromagnetic lms. Just to name a few potential

118

applications:

1. Transmission of spin waves through periodic structures on ferromagnetic lms. Several

interesting phenomena can be detached like magnon band conductions or propagation on

periodic structures without spatial inversion symmetry.

2. Spin waves quantum logic gates. In lattices of magnon BEC's the quantum interference

can be exploited to perform an array of magnonic Josephson juncions.

3. Second sound in the magnon condensate. The uctuations in temperature, produced by

density perturbations, could be used to induce new spin waves in the system by Spin

Seebeck eect.

119

Appendix A

Eective parameters within the

many-body T-matrix approximation

In the third chapter of this thesis we described how, starting from a microscopic model for

the magnon dynamics, it was possible to obtain an eective theory that describes the low energy

physics on the condensate. In this appendix we will give a detailed discussion and principals

highlights of the procedure to get expressions for the eective coecients in the theory (3.74).

A.0.3 Eective interaction between Magnons; γ1 and γ2 parameters.

In this section we are going to give some details on the determination of the eective param-

eters γ1 and γ2. γ1 quanties the eective direct interaction of magnons while γ2 stands out as

an eective anisotropy penalizing magnon distributions with an imbalance between the valleys.

The values for those parameters are given by

γ1

2=[Γ(+−+−) + Γ(−+ +−)

](A.1)

while γ2/2 = Γ(+ + ++) − γ1/2, where Γ(± ± ±±) correspond to solutions of the Bethe-

Salpeter equation in the values Γ(+)(±km,±km,±km,±km; ε = 0). The equation includes the

renormalization of the bare interactions between two magnons due to the interactions with other

magnons present in the system. The many body T-matrix Γ, that describes the scattering of

two particles from momenta ka and kb to the momenta kc and kd in the medium is,

Γ+(ka,kb,kc,kd; ε) =Γ2B(ka,kb,kc,kd; ε)

+

∫dke

(2π)2

dkh(2π)2

Γ2B(ke,kh,kc,kd; ε)[N(ke) +N(kh)]

ε− ε(ke)− ε(kh) + i0Γ+(ka,kb,kh,ke; ε)

(A.2)

where N(k) = iG<0 (k; t, t) is the Bose distribution for particles with momenta ~k. Here we

have supposed that the many body T-matrix depends only on the dierence τ − τ ′ and the dia-

grammatic representation or ladder approximation is displayed in the Fig.(3.10). This equation

has been written using the two body T-matrix as a building block. This matrix represents the

scattering amplitude due to the interaction of otherwise non-interacting magnons. The latter

is well known to satisfy the Lippmann-Schwinger equation for the two-body T-matrix, Γ2B,

Γ2B(ka,kb,kc,kd; ε) =Γ0(ka,kb,kc,kd; ε)

+

∫dke

(2π)2

dkh(2π)2

Γ0(ke,kh,kc,kd; ε)

ε− ε(ke)− ε(kh) + i0+Γ2B(ka,kb,kh,ke; ε) (A.3)

Before we solve the Bethe-Salpeter equation it is necessary that the following physical con-

sideration is made. The bare interaction of 4-magnons Γ0(ka,kb,kc,kd; ε) consist in two kind

of processes of dierent origins. The rst scattering process comes directly from the exchange

interactions, and is proportional to V . The second scattering process arises from dipolar con-

tributions and are proportional to U2. In agreement with phenomenological considerations the

net bare interaction Γ0(ka,kb,kc,kd; ε) is renormalized by the two-body t-matrix and is giving

by

121

Γ2B(+ + ++; ε) =Γ0

1− Ξ−0 [ε]Γ0

Γ2B(+−−+ ε) =Γ2B(+−+−; ε) =Γ0

1− 2Ξ+0 [ε]Γ0

. (A.4)

Now we procedure to solve that equation considering to principal things: the dominant contri-

bution comes from the momenta in a neighborhood of the both valleys ±km and we separate

the integrals around of each valleys dened by the momentum ±km, see Fig.(3.2). In this way

we have obtained expressions for each component of the many-body T-matrix, assuming to Γ2B

constant, as a good approximation,

Γ+(+ + ++; ε) =Γ2B

1− Ξ−1 [ε]Γ2B

Γ+(+−+−; ε) =Γ+(+−−+; ε)

=Γ2B

1− 2Ξ+1 [ε]Γ2B

(A.5)

where we have dened

Ξ±ν [ε] = P∫

dq

(2π)2

[[N(q + km) +N(q± km)]ν

ε− ε(q + km)− ε(q± km) + i0+

](A.6)

A.0.4 Determination of the Self-Energy

In this section we analize the self-energy in the low-energy limit utilizing the principal results

of above section. First, to separate the dynamics over each valley we procedure to separate the

momentum space integrals around of each valleys dened by the momentum ±km, see Fig. (3.2),

and we express φAΓBAφB, from the action Eq.(3.69), in terms of pseudo spin eld Ψ = (Φ1,Φ2)t

to obtain a characterization of the both magnons gases in the condensed stage.

Introducing the self-energy ΓAB by the Dyson equation GAB = GAB0 + GAC 0ΓCDGDB , where G0

122

is the free propagator and G represent the dressed propagator by the interactions due to the

sorrounding media, we can write the retarded component, Σ+ in theKeldysh contour, in Fourier's

space as

~Σ+(ka,kd; ε) =

∫dkb

(2π)2V+(ka,kb,kb,kd; ε(kb) + ε)N(kb)

+ i

∫dkb

(2π)2

∫dε′

2π~V<(ka,kb,kb,kd; ε

′)~

ε′ − (ε(kb) + ε)− i0−(A.7)

where we have dened

V(<,+)(ka,kb,kb,kd; ε) = Γ(<,+)(ka,kb,kb,kd; ε) + Γ(<,+)(ka,kb,kd,kb; ε) (A.8)

Now we can consider that in the low-energy limit the dominant contribution comes from

the momenta in a neighborhood of the both valleys ±km, smaller than the thermal magnons

2π/Λ ∼ 107[cm−1] (Λ the thermal de Broglie wavelenght), due to the presence of the distribution

function N(k). In such regime the second term in the right-hand side of (A.7) is neglegible and

then the self-energy is well approximated by the straightforward relation

~Σ+(+; εm) = 2[Γ(+,+,+,+; 2εm) + Γ(+,−,+,−; 2εm)

]n (A.9)

where n =∫

dq(2π)2

N(q±km). Using the result (A.5), the expression for µ = <[~Σ+(km; ε(km))],

looks like

µ = Γ2B

(1

1− Ξ−[2ε(km)]Γ2B+

1

1− 2Ξ+[2ε(km)]Γ2B

)n (A.10)

where we take the principal Cauchy value in the exppresions for the funtions Ξ±.

123

A.0.5 Determination of the Anomalous Self-Energy

How it was above mentioned the magnons gas is not conservative in the number of particle,

and hence the U(1) symmetry is broken. Such characteristic property of this quasiparticle

systems is captured, in the diagrammatic description, for those scattering anomalous. For

energies of scattering processes∼ εm and by momentum conservation, those anomalous processes

are well approximated by the diagram in Fig. (3.9).

To quantify that contribution we write explicity φAφBΓAB, from the action Eq. (3.69),

in terms of pseudo spin eld Ψ = (Φ1,Φ2)t and we search the expression that obeys in the

low-energy limit. In such limit φAφBΓAB = ν∫dt∫dxΨ†(x, t)σxΨ

∗(x, t) where the eective

parameter ν = Γ+(0,2)(km,−km; εm) quantify the energy involved in the anomalous processes un

the magnon gas, and the retarded anomalous vertex function Γ+(0,2) satisfy in the Fourier's space

Γ+(0,2)(k,−k; ε) = iU2

∫dk′

(2π)2

∫dε12π~

(G+(k′; ε1)G<(k′ − k; ε1 + ε)

+ G<(k′; ε1)G−(k′ − k; ε1 + ε)). (A.11)

Before to renormalize the interactions a simple calculation shows the behavior of ν at low

temperatures which is displayed in the Fig. (A.1). The anomalous vertex is determined as a

function of temperature, considering bare interactions, which vanishes at zero temperatures for

every in-plane magnetic el applied.

The Green's functions G in Eq.(A.11) are dressed propagators by the interactions with the

surrounding media of magnons. Where by means of Dyson's equation GAB = GAB0 + GAC 0ΣCDGDB

we are able to give exact expressions for G+ and G< decomposing G into the dierents branch

124

H=1500[Oe]H=1000[Oe]H=800[Oe]

Figure A.1: Behavior of anomalous vertex in the magnon dynamics as a function of temperatureand in-plane magnetic eld, considering bare interactions. The anomalous vertex vanishes atzero temperatures for every in-plane magnetic el applied, either H = 800[Oe], H = 1000[Oe]and H = 1500[Oe].

of the Keldysh contour,

G±(k; ε) =~(ε− ε(k)± i0+ + ~Σ±(k; ε)

)−1(A.12)

G<(k; ε) =G+(k; ε)Σ<(k; ε)G−(k; ε), (A.13)

where the last equation equation is the so called the Quantum Boltzmann equation. Replacing

the last expressions in (A.11) we arrive to a exact expression that obey the retarded anomalous

self-energy

Γ(0,2)(k,−k; ε) =− U2

~

∫dε12π~

∫dk′

(2π)2

∫dε′

2π~

∫dk′′

(2π)2

[G+(k′; ε1)V<(k′,k′′,k′′,k′; ε′)G>0 (k′′; ε′ + ε1)

× G−(k′; ε1)G(−)(k′ − k; ε1 + ε) + G(+)(k′; ε1)G+(k′ − k; ε1 + ε)

× V<(k′ − k,k′′,k′′,k′ − k; ε′)G>0 (k′′; ε′ + ε+ ε1)G−(k′ − k; ε1 + ε)]

(A.14)

125

where V< is the lesser component of the T-matrix. A good approximation obtained for

Γ0,2(km,−km; εm) with the considerations above mentioned is displayed for ν in the Fig.(3.12).

126

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