[PPT] - Strong-disorder ferromagnetic quantum phase transitions Thomas Vojta PowerPoint Presentation

SLIDE 1

Strong-disorder ferromagnetic quantum phase transitions

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

Dresden, May 6, 2019

SLIDE 2

Collective modes at a disordered quantum phase transition

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

Dresden, May 6, 2019

SLIDE 3

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 4

Broken symmetries and collective modes

collective excitation in systems with broken

continuous symmetry, e.g., − planar magnet breaks O(2) rotation symmetry − superfluid wave function breaks U(1) symmetry

Amplitude(Higgs) mode: corresponds to

fluctuations of order parameter amplitude

Goldstone mode: corresponds to fluctuations of
rder parameter phase
Amplitude mode is condensed matter analogue
f famous Higgs boson

effective potential for order parameter in symmetry-broken phase

SLIDE 5

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

SLIDE 6

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 7

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 8

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 9

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 10

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 11

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 12

Monte Carlo simulations

10 100 1 2 3 4 5 6 L Lmax

τ

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

(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 Mote Carlo simulations in

2d and 3d

conventional critical behavior with

universal 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 13

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 14

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 15

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)

0.5 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 Tc-T 0.2 0.5 1

ωΗ

T= 2.002 2.052 2.082 2.102 2.122 2.142 2.162 2.182 2.192

ν=0.664

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

SLIDE 16

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) T= 1.200 1.300 1.350 1.400 1.450 1.500 1.525 1.550 1.575 dilution p=1/3 Tc =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)

SLIDE 17

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

SLIDE 18

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 19

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 20

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 21

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 22

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 23

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 24

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 25

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 26

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)

SLIDE 27

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