Magnetization process of the kagome magnets

Magnetization process of the S=1/2 distorted kagome magnets

RYUTARO OKUMA

JANUARY 21ST 2014

RYUTARO OKUMA JAN 1ST 2014 2

OutlineKagome magnetsMotivation for our work

ReviewMagnetization process of isotropic kagome antiferromagnet(KAF)Magnon picture from all up stateSpin wave HamiltonianLocalized magnon in KAFHexagram pattern at plateaus

Numerical resultsSummary

All of the contents are supposed to be at zero-temperature.


Kagome magnetsFrustration and low connectivityUnconventional state quantum spin liquid

Heisenberg Hamiltonian

Vesignieite J=77KJ=197K

Okamoto et al., arXiv:0901.2237 (2009).


Motivation for our work Distorted kagome magnetsCu1-Cu2 interaction J’ is ferromagnetic.(J<0)Cu2-Cu2 interaction J is antiferromagnetic.(J’>0)Tcw=(J’+2J)/3kB-13.4 K (By Curie Weiss law)

Other KAF’s Tcw: herbertsmithite:300K,vesinieite:77K

KCu3As2O7 (OH )3

J’

J Cu1

Cu2Some interesting phenomena might be caused by novel geometry and mixed FM-AFM interactions.

OKAMOTO et al., arXiv:1202.2913, 2012


Magnetization process of kagome antiferromagnet Calculation was carried out by density matrix renormalization group method.Numerical results are exact in the thermodynamic limitThere are several plateaus at fractional M/Msat.

Plateau implies the existence of stable structurePlateau at M/Msat = 0 is considered as spin liquid(no long range order).At M/Msat = 3/9,5/9,7/9 hexagram pattern appeared.

Magnon picture from all up state is useful

Nishimoto et al.,Nature communications 4 (2013).


Magnon picture from all up state

Fluctuations of spins can be described as bosons called magnons Ordered state(FMs:all up, AFs: Neel state) is magnon vacuum

S : spin value: boson creation, annihilation operator

Magnon pictureSpin picture

Isotropic kagome


Spin wave Hamiltonian1 magnon Hamiltonian(no approximation)

By discrete Fourier transformation, dispersion relation can be obtained for 1 magnon state.The saturation field is given by the point where the minimum of energy reaches zero.Magnetization M= commutes with Hamiltonian.

Eigenstates are characterized by M

Heisenberg Hamiltonian with magnetic field

1 magnon dispersion for isotropic KAFh=hsat=3J, J=1


A localized magnon in a kagome antiferromagnet

This exact eigenstate is called localized magnon.The flat band is a Fourier transform of such a magnon.Effective magnon interaction is repulsive.At h=hs, localized magnons are excited without costing

energy.All up state changes suddenly to fully packed localized

magnons .This explains the magnetization jump from M/Msat =

1 to 7/9


Hexagram pattern in 7/9, 5/9, 1/3 plateaus

1,2,3 magnons in the red circle= 7/9,5/9,1/3 plateau

Magnetization plateau at 1/3, 5/9,7/9 can be understood by periodic hexagram structure.

At the plateaus, magnons are in the interior of each hexagram.

These plateaus appear in a distorted kagome?


Magnon from all up state: distorted kagome case

JJ’

𝑎1

𝑎2

: boson creation operator of atoms in sub lattice A, B, C

Saturation Field: 1 magnon ground energy touches zero.

Magnon Hamiltonian with magnetic field

-J’,J=13.4K→ hsat=40T

Unlike other kagome(vesigniete:hsat~165T), experimentally achieved H


Exact diagonalization analysis

J is fixed to 1 and J’ is changed from -2 to 2.In the subspace of M =const. the ground state energy was calculated. Main program is based on TITpack ver.2(exact diagonalization)

For small dimension Householder methodFor larger dimensions Lanczos method

Periodic boundary condition was imposed.The cases of N=18,27,and N=36 were treated.

Red: J’Blue: J = -1


Magnetization process

Energy is calculated for each M>0Magnetization as a function of magnetic

field is given like this:

Jump will appear if E(M) is linear(yellow region) or concave(blue region).

Plateaus(green region) appear as a cusp in E-M plot.


Numerical results(1)

J’/J=1 J’/J=-1

1/3 plateau

1 to7/9 jump

5/9 plateau

7/9 plateau

1 to 7/9 jumpSome features of the isotropic case

can also be seen in the mixed FM/AFM case.

Similar to isotropic KAF, several plateau was observed in a distorted kagome model.

5/9 plateau is much more robust for J’<0.

5/9 plateau


DiscussionSurprisingly, robust M/Msat=5/9 plateau appears in almost all J’<0. Consequence of attractive interaction

between magnons?

Around J’/J=-1 there is a jump from 1 to 7/9.This jump can be described as crystallization

of quasi localized magnons?Neglecting term in Heisenberg model, this

localized state is also eigen state.

J=-J’>0


Numerical results(2)The shape and position of each plateau are very similar for N=18,27,36


SummaryA robust magnetization plateau of 5/9 was observed in almost all J’<0. Jump from 1 to 7/9 appeared around J=-J’ ,which could be interpreted as crystallization of quasi localized magnons. In a distorted kagome mineral , these plateaus and jumps may be observed experimentally.

Education

Magnetization process of the kagome magnets