Strong-disorder ferromagnetic quantum phase transitions

Thomas VojtaDepartment of Physics, Missouri University of Science and Technology

Dresden, May 6, 2019

Collective modes at a disordered quantum phase transition

Thomas Vojta

Department of Physics, Missouri University of Science and Technology, USA

Dresden, May 6, 2019

Outline

• Collective modes: Goldstone and

amplitude (Higgs)

• Superfluid-Mott glass quantum phase

transition

• Fate of the collective modes at the

superfluid-Mott glass transition

• Conclusions

Martin PuschmannJose Hoyos

Jack CrewseCameron Lerch

Broken symmetries and collective modes

• collective excitation in systems with brokencontinuous symmetry, e.g.,− planar magnet breaks O(2) rotation symmetry− superfluid wave function breaks U(1) symmetry

• Amplitude(Higgs) mode: corresponds tofluctuations of order parameter amplitude

• Goldstone mode: corresponds to fluctuations oforder parameter phase

• Amplitude mode is condensed matter analogueof famous Higgs boson

effective potential for order parameter

in symmetry-broken phase

What is the fate of the Goldstone and amplitude modes near adisordered quantum phase transition?

• Collective modes: Goldstone and amplitude (Higgs)

• Superfluid-Mott glass quantum phase transition

• Fate of the collective modes at the superfluid-Mott glass transition

• Conclusions

Disordered interacting bosons

Ultracold atoms in opticalpotentials:

• disorder: speckle laser field

• interactions: tuned byFeshbach resonance and/ordensity

F. Jendrzejewski et al., Nature Physics 8, 398 (2012)

-2 -1 0 1 2 3 4

1

2

3

-4 -3 -2 -1 0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

GH

=0(V

)/GH

=11T

(V)

V(mV)

G(V

)/G(4

mV

)

V(mV)

Sherman et al., Phys. Rev. Lett. 108, 177006 (2012)

Disordered superconducting films:

• energy gap in insulating as well assuperconducting phase

• preformed Cooper pairs ⇒ superconductingtransition is bosonic

Disordered interacting bosons

Bosonic quasiparticles in doped quantum magnets:

Yu et al., Nature 489, 379

(2012)

• bromine-doped dichloro-tetrakis-thiourea-nickel (DTN)

• coupled antiferromagnetic chains of S = 1 Ni2+ ions

• S = 1 spin states can be mapped onto bosonic states with n = ms + 1

Bose-Hubbard model

Bose-Hubbard Hamiltonian in two dimensions:

H =U

2

∑i

(n̂i − n̄i)2 −∑〈i,j〉

Jij(a†iaj + h.c.)

• superfluid ground state if Josephson couplings Jij dominate

• insulating ground state if charging energy U dominates

• chemical potential µi = Un̄i

Particle-hole symmetry:

• large integer filling n̄i = k with integer k � 1⇒ Hamiltonian invariant under (n̂i − n̄i)→ −(n̂i − n̄i)

Phase diagrams

/U0

(µ)~

0

0

0

~

<n>=1

<n>=−1

MI

n=0

SF

SF

J

/Uµ

1

0<n>=0

(a)

−1MI

MI

n=−1

n=1

1−

_23−

_2

2

23_

U0,cJ______

_

1

/U0

J0

U0

______~

+− +−δµ ( )

0~

0

<n>=−1

<n>=1

<n>=0

SF

MI

MI

n=1MI

n=0

n=−1

BG

BG

SF

(b)

1

0−δ

+δ

J

µ /U

−1

12_

12_

_32

_−23

−12_

_12

+(1+δ )0

(µ)~/U

~0

0 0

<n>=0

<n>=1

<n>=−1

MI

MI

________J0,cBG

BG

SF

MG

SF

J

/Uµ(c)

0

−1

1n=1

n=0

n=−1MI

U

12_

−12_

−32_

_32

clean random potentials random couplingsWeichman et al., Phys. Rev. B 7, 214516 (2008)

Stability of clean quantum critical point against dilution

Site dilution:

• randomly remove a fraction p of lattice sites

• superfluid phase possible for 0 ≤ p ≤ pc (percolation threshold)

Harris criterion:

• for dilution p = 0, quantum critical point is in 3D XY universality class

• correlation length critical exponent ν ≈ 0.6717

• clean ν violates Harris criterion dν > 2 with d = 2

⇒ clean critical behavior unstable against disorder (dilution)

Critical behavior of superfluid-Mott glass transition must be in newuniversality class

Monte Carlo simulations

10 100

1

2

3

4

56

L

Lm

ax

τ/L

1/81/52/71/3

9/25

p =

(2+1)D exponents

exponent clean disorderedz 1 1.52ν 0.6717 1.16β/ν 0.518 0.48γ/ν 1.96 2.52

• large-scale Mote Carlo simulations in2d and 3d

• conventional critical behavior withuniversal exponents for dilutions0 < p < pc

• Griffiths singularities exponentiallyweak (see classification in J. Phys. A 39,

R143 (2006), PRL 112, 075702 (2014))

(3+1)D exponents

exponent clean disorderedz 1 1.67ν 0.5 0.90β/ν 1 1.09γ/ν 2 2.50

• Collective modes: Goldstone and amplitude (Higgs)

• Superfluid-Mott glass quantum phase transition

• Fate of the collective modes at the superfluid-Mott glass transition

• Conclusions

Amplitude mode: scalar susceptibility

• parameterize order parameter fluctuations intoamplitude and direction

~φ = φ0(1 + ρ)n̂

• Amplitude mode is associated with scalarsusceptibility

χρρ(~x, t) = iΘ(t) 〈[ρ(~x, t), ρ(0, 0)]〉

• Monte-Carlo simulations compute imaginary time correlation function

χρρ(~x, τ) = 〈ρ(~x, τ)ρ(0, 0)〉

• Wick rotation required: analytical continuation from imaginary to realtimes/frequencies ⇒ maximum entropy method

Amplitude mode in clean undiluted system

Scaling form of the scalar susceptibility: [Podolsky + Sachdev, PRB 86, 054508 (2012)]

χρρ(ω) = |r|3ν−2X(ω|r|−ν)

0 0.5 1 1.5 2 2.5 3ω

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

A(ω

)

0.01 0.03 0.1Tc-T

0.2

0.5

1

ωΗ

T=2.0022.0522.0822.1022.1222.1422.1622.1822.192

ν=0.664

• sharp Higgs peak in spectral function• Higgs energy (mass) ωH scales as expected with distance from criticality r

Amplitude mode in disordered system

0 0.5 1 1.5 2 2.5 3ω

0

0.01

0.02

0.03

0.04

0.05

A(ω

)

T=1.2001.3001.3501.4001.4501.5001.5251.5501.575

dilution p=1/3Tc =1.577

• spectral function shows broad peak near ω = 1

• peak is noncritical: does not move as quantum critical point is approached

• amplitude fluctuations not soft at criticality

• violates expected scaling form χρρ(ω) = |r|(d+z)ν−2X(ω|r|−zν)

What is the reason for the absence of a sharp amplitude mode at thesuperfluid-Mott glass transition?

Quantum mean-field theory

H =U

2

∑i

εi(n̂i − n̄i)2 − J∑〈i,j〉

εiεj(a†iaj + h.c.)

• truncate Hilbert space: keep only states |n̄− 1〉, |n̄〉, and |n̄+ 1〉 on each site

Variational wave function:

|ΨMF 〉 =∏i

|gi〉 =∏i

[cos

(θi2

)|n̄〉i + sin

(θi2

)1√2

(eiφi|n̄+ 1〉i + e−iφi|n̄− 1〉i

)]

• locally interpolates between Mott insulator, θ = 0, and superfluid limit, θ = π/2

Mean-field energy:

E0 = 〈ΨMF |H|ΨMF 〉 =U

2

∑i

εi sin2

(θi2

)− J

∑〈ij〉

εiεj sin(θi) sin(θj) cos(φi − φj)

• solved by minimizing E0 w.r.t. θi ⇒ coupled nonlinear equations

Diluted lattice: order parameter

• local order parameter: mi = 〈ai〉 = sin(θi)eiφi (dilution p = 1/3)

U = 8 U = 10

U = 12 U = 14

0.0 0.2 0.4 0.6 0.8 1.0mi

7 8 9 10 11 12 13 140.0

0.2

0.4

0.6

U

m

typicalmean

Mean-field theory: excitations

• define local excitations (orthogonal to |gi〉, OP phase fixed at 0)

|gi〉 = cos

(θi2

)|n̄〉i + sin

(θi2

)1√2

(|n̄+ 1〉i + |n̄− 1〉i)

|θi〉 = sin

(θi2

)|n̄〉i − cos

(θi2

)1√2

(|n̄+ 1〉i + |n̄− 1〉i)

|φi〉 =1√2

(|n̄+ 1〉i − |n̄− 1〉i)

• expand H to quadratic order in excitations: H = E0 +Hθ +Hφ

Hθ =∑i

U2

+ 2J∑j′

sin(θi) sin(θj)

εib†θibθi−J

∑〈ij〉

cos(θi) cos(θj)εiεj(b†θi + bθi)(b

†θj + bθj)

Hφ has similar structure but different coefficients

Clean system: results

• mean-field quantum phase transitionat U = 16J

• all excitations are spatially extended(plane waves)

Mott insulator

• all excitations are gapped

Superfluid

• Goldstone mode is gapless

• amplitude (Higgs) modes is gapped,gap vanishes at QCP

1,--�

0.5

• Computation-- Superfluid - w = v�l-�(V�/1-6�)�2

Insulator - W = 0

o ............ --............. � ............................................................. �---.......................... � ..... --� ..... 0 5 10 15 20

U

m

- -0.- -0.

200 ...

€>,

100

0 5

•

0

- - -

'

'6) '

10

' ' �,

Goldstone (computation) Higgs ( computation)

Goldstone

\

'

\

\

15

Higgs

20 25

ω02

U

Diluted lattice: Goldstone mode

• Goldstone mode becomes massless insuperfluid phase, as required by Goldstone’stheorem

• wave function of lowestexcitation for U = 8 to15

• localized in insulator,delocalizes insuperfluid phase

Goldstone mode: localization properties

• inverse participation ratio:

P−1 = N∑i |ψi|4

P → 1 for delocalized statesP → 0 for localized states

10° 10°

L top to bottom

Goldstone 32

32 64

10-1 • 64 128

128Higgs 10-1

-�- 32

� 10-2--o-- 64 /J

� --- 128

""'Cl

0-t 0 0-t p'cf ,If

,Jl'rz .cl'

'2 c,.oO

p" ,ti' 10-2 10-3 a ....

"' .� ....

a,Da

,ti'

• �,

5 10 15 0 2 4 6

u w

• wave function atU = 8 as function ofexcitation energy

• delocalized at ω = 0,localized for higherenergies

Amplitude (Higgs) mode

• amplitude mode strongly localized for allU and all excitation energies

10°

Goldstone 32

10-1 • 64128

Higgs -�- 32

� 10-2--o-- 64 /J

--- 128 ""'Cl

0-t 0 p'cf ,If

,Jl'rz .cl'

'2 c,.oO

p" ,ti' 10-3a .. ..

"' .� ....

a,Da

,ti'

• �,

5 10 15 u

• wave function of lowestexcitation for U = 8 to15

Longitudinal and transverse susceptibilities (q = 0)

U = 15

U = 14

U = 13

U = 12

U = 11

U = 10

0 2 4 60

2

4

6

ω

χ′′

U = 17

U = 16

U = 15

U = 14

U = 13

U = 12

0 2 4 60

2

4

6

ω

χ′′

diluted, p = 1/3 clean

Conclusions

• disordered interacting bosons undergo quantum phase transition betweensuperfluid state and insulating Mott glass state

• conventional critical behavior with universal critical exponents

• Griffiths effects exponentially weak [see classification in T.V., J. Phys. A 39, R143 (2006)]

• collective modes in superfluid phase show striking localization behavior

• Goldstone mode is delocalized at ω = 0 but localizes with increasing energy

• amplitude (Higgs) mode is strongly localized for all energies

• broad incoherent scalar response at q = 0, violates naive scaling

Exotic collective mode dynamics even if critical behavior is conventional

T.V., Jack Crewse, Martin Puschmann, Daniel Arovas, and Yury Kiselev, PRB 94, 134501 (2016)

Jack Crewse, Cameron Lerch and T.V., PRB 98, 054514 (2018)

Cameron Lerch and T.V., Eur. Phys. J. ST 227, 22753 (2019)

Analytic continuation - maximum entropy method

• Matsubara susceptibility χρρ(iωm) vs. spectral function A(ω) = χ′′ρρ(ω)/π

χρρ(iωm) =

∫ ∞0

dωA(ω)2ω

ω2m + ω2

.

Maximum entropy method:

• inversion is ill-posed problem, highly sensitiveto noise

• fit A(ω) to χρρ(iωm) MC data by minimizing

Q = 12σ

2 − αS

• parameter α balances between fit error σ2

and entropy S of A(ω), i.e., between fittinginformation and noise

• best α value chosen by L-curve method [see

Bergeron et al., PRE 94, 023303 (2016)]

4 6 8 10 12 14 16ln α

101

102

103

104

105

σ2

L-curve