[PPT] - Collective modes at a disordered quantum phase transition Thomas PowerPoint Presentation

SLIDE 1

Collective modes at a disordered quantum phase transition

Thomas Vojta Department of Physics, Missouri University of Science and Technology

Los Alamos, January 27, 2020

SLIDE 2

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 Puschmann Jose Hoyos Jack Crewse Cameron Lerch

SLIDE 3

Spontaneous symmetry breaking

Does a symmetric Hamiltonian imply a symmetric equilibrium state?

world of this pencil is completely isotropic, all directions are equal
symmetry is lost when pencil falls over, now only one direction holds
state of lowest energy has lower symmetry than system

Rotational symmetry has been broken spontaneously!

SLIDE 4

Broken symmetries and collective modes

systems with broken continuous symmetry:

− planar magnet breaks O(2) rotation symmetry − superfluid wave function breaks U(1) symmetry

Higgs (amplitude) mode: corresponds to

fluctuations of order parameter amplitude

Goldstone (phase) mode: corresponds to

fluctuations of order parameter phase

Amplitude mode is condensed matter analogue
f famous Higgs boson

Goldstone theorem: When a continuous symmetry is spontaneously broken, massless Goldstone modes appear.

”Mexican hat” potential for order parameter in symmetry-broken phase F = t m2 + u m4

SLIDE 5

Higgs (amplitude) mode in condensed matter?

Is the Higgs mode a sharp, particle-like excitation or is it overdamped because it

decays into other modes? Raman scattering data for NbSe2

[from Measson et.al., Phys. Rev. B 89, 060503 (2014)]

SLIDE 6

What is the fate of the Goldstone and Higgs modes near a disordered quantum phase transition?

SLIDE 7

Collective modes: Goldstone and Higgs
Superfluid-Mott glass quantum phase transition
Fate of the collective modes at the superfluid-Mott glass transition
Conclusions

SLIDE 8

Disordered interacting bosons

Ultracold atoms in optical potentials:

disorder: speckle laser field
interactions: tuned by

Feshbach resonance and/or density

F. Jendrzejewski et al., Nature Physics 8, 398 (2012)
2
1

1 2 3 4 1 2 3

4
3
2
1

1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 G H=0 (V)/G H=11T (V) V(mV) G(V)/G(4mV) V(mV)

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

Disordered superconducting films:

energy gap in insulating as well as

superconducting phase

preformed Cooper pairs ⇒ superconducting

transition is bosonic

SLIDE 9

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

SLIDE 10

Bose-Hubbard model

Bose-Hubbard Hamiltonian in two dimensions: H = U 2

i

(ˆ ni − ¯ ni)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 = U ¯

ni Particle-hole symmetry:

large integer filling ¯

ni = k with integer k ≫ 1 ⇒ Hamiltonian invariant under (ˆ ni − ¯ ni) → −(ˆ ni − ¯ ni)

SLIDE 11

Phase diagrams /U0

(µ) ~

~ <n>=1 <n>=−1 MI n=0 SF SF J /U µ 1 <n>=0 (a) −1 MI MI n=−1 n=1

1 − _ 2 3 − _ 2 2 2 3 _ U

0,c

J ______ _ 1

/U0

J0 U0 ______ ~

+ −

+ −δ

µ ( )

~ <n>=−1 <n>=1 <n>=0 SF MI MI n=1 MI n=0 n=−1 BG BG SF (b) 1 −δ +δ J µ/U −1

1 2 _ 1 2 _ _ 3 2 _ −2 3 −1 2 _ _ 1 2

+

(1+δ ) (µ) ~

/U ~ <n>=0 <n>=1 <n>=−1 MI MI

________ J0,c

BG BG SF MG SF J /U µ (c) −1 1 n=1

n=0 n=−1

MI

U 1 2 _ −1 2 _ −3 2 _ _ 3 2

clean random potentials random couplings

Weichman et al., Phys. Rev. B 7, 214516 (2008)

SLIDE 12

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 new universality class

SLIDE 13

Monte Carlo simulations

quantum fluctuations (~U/J)

(2+1)D exponents exponent clean disordered z 1 1.52 ν 0.6717 1.16 β/ν 0.518 0.48 γ/ν 1.96 2.52

large-scale Monte Carlo simulations in

2d and 3d

conventional power-law critical

behavior

universal critical exponents for

dilutions 0 < p < pc

Griffiths singularities exponentially

weak (see classification in J. Phys. A 39,

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

(3+1)D exponents exponent clean disordered z 1 1.67 ν 0.5 0.90 β/ν 1 1.09 γ/ν 2 2.50

SLIDE 14

Collective modes: Goldstone and amplitude (Higgs)
Superfluid-Mott glass quantum phase transition
Fate of the collective modes at the superfluid-Mott glass transition
Conclusions

SLIDE 15

Amplitude mode: scalar susceptibility

parameterize
rder

parameter fluctuations into amplitude and direction

φ = φ0(1 + ρ)ˆ

n

Amplitude

mode is associated with scalar susceptibility 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 real

times/frequencies ⇒ maximum entropy method

SLIDE 16

Analytic continuation - maximum entropy method

Matsubara susceptibility χρρ(iωm) vs. spectral function A(ω) = χ′′

ρρ(ω)/π

χρρ(iωm) = ∞ dωA(ω) 2ω ω2

m + ω2

. Maximum entropy method:

inversion is ill-posed problem, highly sensitive

to noise

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

Q = 1

2σ2 − αS

parameter α balances between fit error σ2

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

best α value chosen by L-curve method [see

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

4 6 8 10 12 14 16 ln α 101 102 103 104 105 σ2 L-curve

SLIDE 17

Amplitude mode in clean undiluted system

Scaling form of the scalar susceptibility: χρρ(ω) = |r|3ν−2X(ω|r|−ν)

[Podolsky + Sachdev, PRB 86, 054508 (2012)]

0.5 1 1.5 2 2.5 3 ω 0.02 0.04 0.06 0.08 0.1 0.12 0.14 A(ω)

0.01 0.03 0.1 0.2 0.5 1

ωΗ ν=0.664

0.0045 0.0090 0.0181 0.0363 0.0474 0.0545 0.0682 0.0909 0.0272 r |r | =

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

SLIDE 18

Dispersion of the clean amplitude mode

1 2 3 4

ω

0.025 0.05 0.075 0.1 0.125 A(ω)

q= 0.05 0.1 0.25 0.49 0.98 1.96

spectral density at different q for r = −0.0045 Higgs mode dispersion

SLIDE 19

Amplitude mode in disordered system

0.5 1 1.5 2 2.5 3 ω 0.01 0.02 0.03 0.04 0.05 A(ω) dilution p=1/3 0.0013 0.0171 0.0330 0.0488 0.0805 0.1122 0.1439 0.1756 0.2391 |r |=

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)

SLIDE 20

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

SLIDE 21

Quantum mean-field theory

H = U 2

i

ǫi(ˆ ni − ¯ ni)2 − J

i,j

ǫiǫj(a†

iaj + h.c.)

truncate Hilbert space: keep only states |¯

n − 1, |¯ n, and |¯ n + 1 on each site Variational wave function: |ΨMF =

i

|gi =

i
cos

θi 2

|¯

ni + sin θi 2 1 √ 2

eiφi|¯

n + 1i + e−iφi|¯ n − 1i

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

Mean-field energy: E0 = ΨMF|H|ΨMF = U 2

i

ǫi sin2 θi 2

− J
ij

ǫiǫj sin(θi) sin(θj) cos(φi − φj)

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

SLIDE 22

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 1.0 mi

7 8 9 10 11 12 13 14 0.0 0.2 0.4 0.6

U m

typical mean

SLIDE 23

Mean-field theory: excitations

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

|gi = cos θi 2

|¯

ni + sin θi 2 1 √ 2 (|¯ n + 1i + |¯ n − 1i) |θi = sin θi 2

|¯

ni − cos θi 2 1 √ 2 (|¯ n + 1i + |¯ n − 1i) |φi = 1 √ 2 (|¯ n + 1i − |¯ n − 1i)

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

Hθ =

i

 U 2 + 2J

j′

sin(θi) sin(θj)   ǫib†

θibθi

−J

ij

cos(θi) cos(θj)ǫiǫj(b†

θi + bθi)(b† θj + bθj)

Hφ has similar structure but different coefficients

SLIDE 24

Clean system: results

mean-field quantum phase transition

at 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
- Superfmuid - w = vl-(

V/1-6)2 Insulator - W = 0

..../....../
......./...../
...../....../..../....../...../...../...../....../..../...../
-......./....../...../...../
..../
.

..../

5 10 15 20

U m

-0/ - -0/

200

../

€?,

100 5

'

'6* '

10

' ' ,

Goldstone (computation) Higgs ( computation) Goldstone

\ ' \ \

15

Higgs

20 25

ω02 U

SLIDE 25

Diluted lattice: Goldstone mode

Goldstone mode becomes massless in

superfluid phase, as required by Goldstone’s theorem

wave function of lowest

excitation for U = 8 to 15

localized in insulator,

delocalizes in superfluid phase

SLIDE 26

Goldstone mode: localization properties

inverse participation ratio:

P −1 = N

i |ψi|4

P → 1 for delocalized states P → 0 for localized states

10° 10°

L top to bottom

Goldstone

32 32 64

10-1

64

128 128

Higgs

10-1

- 32

10-2

-o.- 64

/K

-- 128 ""(Cm

0-u 0-u

p(cg

,Ig

,Jl(r{

.dl'

'3

c,/oO

p"

,ti(

10-2 10-3

a

././

"( .

././

a-Da ,ti(

,

5 10 15

2

4

6

u

w

wave function at

U = 8 as function of excitation energy

delocalized at ω = 0,

localized for higher energies

SLIDE 27

Amplitude (Higgs) mode

amplitude mode strongly localized for all

U and all excitation energies

10°

Goldstone 32

10-1

64

128 Higgs

- 32

10-2

o.
6

4

/K

-- 128 ""(Cm

0-u

p(cg

,Ig

,Jl(r{

.dl'

'3

c,/oO

p"

,ti(

10-3

a

././

"( .

././

a-Da ,ti(

,

5 10 15

u

wave function of lowest

excitation for U = 8 to 15

SLIDE 28

Longitudinal and transverse susceptibilities (q = 0)

U = 15 U = 14 U = 13 U = 12 U = 11 U = 10

2 4 6 2 4 6

ω χ′′

U = 17 U = 16 U = 15 U = 14 U = 13 U = 12

2 4 6 2 4 6

ω χ′′

diluted, p = 1/3 clean

SLIDE 29

Conclusions

disordered interacting bosons undergo quantum phase transition between superfluid

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) Martin Puschmann, Jack Crewse, Jos´ e A. Hoyos and T.V., arXiv:1911.04452

SLIDE 30

From quantum rotors to classical XY model

site-diluted quantum rotor model on square lattice

(ǫi = 0 or 1 with probabilities p and 1 − p) H = U 2

i

ǫi(ˆ ni − ¯ ni)2 − J

ij

ǫiǫj cos(ˆ θi − ˆ θj)

can be mapped onto classical (2+1)-dimensional XY

model for ¯ ni = k (particle-hole symmetric case) Hcl = −Jτ

i,t

ǫiSi,t · Si,t+1−Js

i,j,t

ǫiǫjSi,t · Sj,t Jτ = Js = 1 (values not important for critical behavior because of universality )

columnar disorder in classical XY model, correlated in imaginary time

SLIDE 31

Monte Carlo simulations

combine Wolff cluster algorithm and conventional

Metropolis updates

Wolff algorithm greatly reduces critical slowing down
Metropolis updates equilibrate small disconnected

clusters (important for high dilutions)

system sizes up to L = 150 and Lτ = 1792
dilutions p = 0, 1/8, 1/5, 2/7, 1/3, 9/25 and

percolation threshold pc = 0.407253

averages over 10 000 to 50 000 disorder

configurations

Pegasus IV Cluster:

web.mst.edu/∼vojtat/pegasus/home.htm

SLIDE 32

Finite-size scaling

Binder cumulant: gav =

1 − |m|4

3|m|22

dis

Isotropic systems:

scaling form: gav(r, L) = X(rL1/ν)

[r = (T − Tc)/Tc]

gav vs. T curves for different L cross at Tc with

value gav(0, L) = X(0)

2.1 2.2 2.3 T 0.3 0.4 0.5 0.6 0.7 g L 10 14 20 28 40

Anisotropic systems:

L and Lτ are not equivalent, Lτ scales like Lτ ∼ Lz (or even as ln Lτ ∼ Lψ)
conventional scaling: gav(r, L, Lτ) = X(rL1/ν, Lτ/Lz)

activated scaling: gav(r, L, Lτ) = X(rL1/ν, ln(Lτ)/Lψ)

How to choose correct sample shapes if dynamical exponent z (or tunneling

exponent ψ) is not known?

SLIDE 33

Anisotropic finite-size scaling

0.5 1 1.5 2 0.96 0.98 1.00 Lτ/Lmax

τ

gav/gmax

av

10 14 20 28 36 44 56 70 84 100 L = 101 102 103 0.56 0.57 0.58 0.59 Lτ gav

gav vs Lτ has maximum at “optimal” shape
at criticality, Lmax

τ

∼ Lz (for activated scaling: ln(Lmax

τ

) ∼ Lψ)

once optimal shapes are found, FSS works as usual
ptimal gav vs. T curves cross at Tc:

gav(0, L, Lmax

τ

) = X(0, const)

SLIDE 34

Phase diagram

classical temperature T represents ratio U/J of quantum rotor Hamiltonian

(physical temperature of quantum system is zero)

SLIDE 35

Dynamical critical exponent z

10 100 1 2 3 4 5 6 L Lmax

τ

/L 1/8 1/5 2/7 1/3 9/25 p =

significant deviations from pure power laws ⇒ corrections to scaling
fit to ansatz

Lmax

τ

= aLz(1 + bL−ω) with universal z and ω

dynamical critical exponent

z = 1.52(3)

SLIDE 36

Order parameter φ and susceptibility χ

10 100 0.1 0.2 0.3 L m 1/8 1/5 2/7 1/3 9/25 p = 10 100 10 100 L χ 1/8 1/5 2/7 1/3 9/25 p =

Scaling forms: φ = L−β/νXφ(rL1/ν, Lτ/Lz) , χ = Lγ/νXχ(rL1/ν, Lτ/Lz)

fit data at criticality to

φ = aL−β/ν(1 + bL−ω) χ = aLγ/ν(1 + bL−ω)

critical exponents:

β/ν = 0.48(2) and γ/ν = 2.52(4)

SLIDE 37

Correlation length critical exponent ν

1.56 1.57 1.58 1.59 0.58 0.59 0.60 T gav 1.56 1.57 1.58 1.59 0.50 0.55 0.60 T ξτ/Lτ

(2+1)D exponents exponent clean disordered z 1 1.52 ν 0.6717 1.16 β/ν 0.518 0.48 γ/ν 1.96 2.52

slopes at criticality:

(d/dT)gav ∼ (d/dT)ξτ/Lτ ∼ L1/ν

fit data to L1/ν(1 + bL−ω)
critical exponent: ν = 1.16(5)

(dilution p = 1/3)