[PPT] - Many - body localization edge in the random - field Heisenberg chain PowerPoint Presentation

SLIDE 1

Many-body localization edge in the random-field Heisenberg chain

Fabien Alet

In collaboration with : David Luitz, Nicolas Laflorencie

Laboratoire de Physique Théorique Toulouse

W

rkshop Quantum chaos,

Luchon, March 2015

Ref. :
Phys. Rev. B 91, 081103 (2015)

SLIDE 2

Outline

Broad question: What happens to Anderson localization in presence
f interactions?

«Many-body localization» (MBL) A new distinct dynamical phase of matter, which does not thermalize

Present in simple toy models
Distinct features from a «thermal» state, and Anderson insulator
Many open questions...

Part 1 : Mini-Review on Many-Body localization Part 2 : «Large»-scale numerics on a MBL Hamiltonian

Energy-resolved phase diagram
Computational issues

SLIDE 3

Part 1 : Mini-Review on Many- Body Localization

Reviews on MBL : Nandkishore & Huse, arxiv:1404.0686 Altman & V

sk, arxiv:1408.2834

my current understanding of

SLIDE 4

Thermalisation & ETH

Initial wave-function expressed in the eigenbasis of

|Ψ0i = X

i

an|ni

Time-evolved observable (generic Hamiltonian)

hO(t)i = X

n,n0

a∗

n0ane−i(En0−En)tOnn0 t→∞

!

X

n

|an|2Onn

Eigenstate thermalization hypothesis (ETH)

in the thermodynamic limit and for local observable

hn|O|ni ' hn0|O|n0i = O(E)

hn|O|n0i vanish

in the same energy shell Deutsch, Srednicki, Rigol & many authors

|ni, |n0i

«Diagonal ensemble»

ETH + (some other minimal assumptions) implies thermalisation

hO(t ! 1)i = O(E) = O(T) E = hΨ0|H|Ψ0i E = hHiT

H = X

n

En|nihn|

SLIDE 5

Consequences

Each eigenstate is thermal, «knows» equilibrium

ρ(0) = |nihn| = ρ(t) = ρeq(Tn) En = hHiTn

ETH is a «justification» of the microcanonical ensemble at the invididual eigenstate level
Memory of initial conditions is lost

many authors...

Integrable systems : May have their own ETH, relaxation to a Generalized Gibbs Ensemble
Many-Body Localized systems

t → ∞

Localized systems : single-particle localization
ETH seems to work (analytics+numerics) for most many-body quantum systems, except

SLIDE 6

Many-body localization

Beyond perturbation : numerics (including this talk) indicate that the localized phase survives
Crucial to work in the «eigenstate ensemble», not (micro-)canonical ensemble
Old problem revived by an enormous amount of contributions!

Nandkishore, Huse, arxiv:1404.0686, Altman, V

sk 1408.2834
Infinite disorder : eigenstates are fully localized product-states, no entanglement
Typical example : XXZ chain with random fields

H = X

i

σz

i σz i+1 + σz i σz i+1 + ∆σz i σz i+1 −

X

i

hiσz

i

hi ∈ [−h, h]

Branch small interaction : perturbative calculations indicate that

thermalization does not occur: states keep localized, no spin or energy transport Gornyi et al. Basko et al. MBL Thermal, «ergodic»

h

Anderson, Fleishmann, Shepelyansky...

Phase transition (?) is dynamical can’t be seen in thermodynamics

hc

SLIDE 7

Phenomenology of MBL systems

Consider only fully MBL systems (all eigenstates are localized)
Exact results , phenomenology , perturbative results, strong disorder RG

Ji1,...,ik

Idea: Quasi-local unitary transform can «diagonalize» the Hamiltonian

Imbrie

Huse. Oganesyan,

Abanin et al. Ros, Müller Scardicchio Altman, V

sk

U †HU = − X

i

hiτ z

i −

X

i<j

Ji,jτ z

i τ z j +

X

i<j<k

Ji,j,kτ z

i τ z j τ z k + ...,

decay exponentially

Other sets of local integrals of motion can be constructed, which may have a better physical

interpretation

Ros, Müller Scardicchio Abanin et al.

Useful to describe properties of MBL, to detect MBL through spin-echo experiments

Serbin et al.

Set of localized bits:
= complete set of local integral of motions

Uτ z

i U †

τ z

i ' ˆ

Zi(h)σz

i + tail

SLIDE 8

Entanglement & MBL

ETH : Entanglement entropy of eigenstates is extensive : V
lume law

System is its own bath: B acts a thermal bath for A

A B

NA

ρA = TrBρ = ρeq

A (Tn)

SA = −TrAρA log ρA ∝ NA Tn 6= 0

if Can be understood with the localized-bits picture

Entanglement spreads logarithmically

Bardarson et al.

MBL states have low entanglement SA/NA → 0

Eisert et al. Abanin, Vidal et al.

MBL states efficiently represented as matrix-product states

Znidaric et al. Pekker, Clark

SLIDE 9

Summary of MBL

MBL in systems with no disorder?
Many-body mobility edge: Griffiths effects? sub-diffusive ergodic phase?
Nature of many-body localization transition?

Thermal phase Single-particle localized Many-body localized Memory of initial conditions Some memory of local initial Some memory of local initial ‘hidden’ in global operators conditions preserved in local conditions preserved in local at long times

bservables at long times
bservables at long times.

ETH true ETH false ETH false May have non-zero DC conductivity Zero DC conductivity Zero DC conductivity Continuous local spectrum Discrete local spectrum Discrete local spectrum Eigenstates with Eigenstates with Eigenstates with volume-law entanglement area-law entanglement area-law entanglement Power-law spreading of entanglement No spreading of entanglement Logarithmic spreading of entanglement from non-entangled initial condition from non-entangled initial condition Dephasing and dissipation No dephasing, no dissipation Dephasing but no dissipation

from Nandkishore, Huse arxiv:1404.0686,

MBL also found in quasi-periodic systems
MBL states can host «forbidden» (discrete-symmetry breaking, topological) order, in 1d at finite E
Coupling with a bath: MBL physics can still be detected

Some open questions

Müller et al., Grover & Fisher; De Roeck & Huveneers, Garrahan et al., Y ao et al. Agarwal et al.

SLIDE 10

Experimental realization

Schreiber et al., arXiv:1501.05661

e e e e

initial state

2 3 5 10 15 20

/J U/J

non-ergodic localized

ergodic delocalized

AA localized AA extended

B

U/J=4.7(1) U/J=10.3(1) , /J=8 /J=3 /J=0

Imbalance

20 30 Time () 0.2 0.8 0.4 0.6 10

Imbalance
20
10

10 20

U/J

0.2 0.3 0.4 0.5 0.6

Imbalance

4

4 0.2 0.3

attractive non- interacting repulsive

Cold-atomic gas realization of interacting Aubry-André model:

X ⇣ ⌘ + ∆ X

i,σ

cos(2πβi + φ)ˆ c†

i,σˆ

ci,σ + U X

i

ˆ ni,↑ˆ ni,↓.

ˆ

H = − J X

i,σ

⇣ ˆ c†

i,σˆ

ci+1,σ + h.c. ⌘ X

(Non-)Equilibration of a quenched initial state measured by imbalance

I = Ne − No Ne + No ,

∆/J

SLIDE 11

Part 2 : «Large»-scale numerics on a MBL Hamiltonian

How to detect MBL states in numerics?
Presence of a many-body mobility edge?
Nature of the MBL transition? First fingerprints of universality class...
Is MBL a true localization in Hilbert space?
Phys. Rev. B 91, 081103 (2015)

SLIDE 12

MBL & Numerics

Prototypical MBL Hamiltonian : Heisenberg S=1/2 spin chain in a random field

hi ∈ [−h, h] H =

L

X

i=1

Si · Si+1 − X

i

hiSz

i

Total magnetization is conserved, no other symmetries

Sz

Almost no symmetries, average over disorder
MBL physics is located at high-energy: eigenstates in the middle of the spectrum
Ground-state methods are not well adapted
A tough computational problem

DMRG, power method, Lanczos, T=0 series expansion, etc Quantum Monte Carlo, high-T series expansion ...

W

e want eigenstates of a closed system: cannot impose a bath !

W

e are left with

Time evolution after a quench: time-dependent DMRG, TEBD ... (but no eigenstates)
Brute-force methods: Full diagonalization, or (slightly better) spectral transforms
Real Space Renormalization Group: for excited states, only in the strong disorder limit

SLIDE 13

Computational details

Obtain eigenstates in the middle of the spectrum with shift-invert

σ H

R = (H − σI)

R−1

Extremal eigenvalue problem for (Lanczos-like methods ...)

R−1

«Simple» linear equations Bottleneck !

LU decomposition done by the massively parallel solver Mumps
W

e cannot compute ! However just need to know how to apply

R−1

R−1a = b

LU decomposition

R = LU c

a = Lc

Three steps method:
Solve for in
Solve for in

b

c = Ub

SLIDE 14

Computational details

Obtain eigenstates in the middle of the spectrum with shift-invert

σ H

R = (H − σI)

R−1

Further details:
Method typical for Anderson localization, however LU much more difficult here !
L=22 (matrix size ) instead of L=16 (matrix size )

∼ 700.000

∼ 12.000

Obtain eigenstates for disorder realizations for each field, energy, L

∼ 1000

All results in terms of

∼ 50

✏ = (E − Emax)/(Emin − Emax)

Strong correlations between eigenstates of the same disorder sample!

✏ = 0

✏ = 1

SLIDE 15

Gap ratios

Level statistics: natural tool to check for localization
Thermal (ETH) phase: expect Random Matrix Theory (in particular GOE) to correctly

capture highly-excited eigenvalues

MBL phase: expect Poisson statistics (no correlation, no level repulsion)
However, unfolding necessary due to d.o.s. effects

−30 −20 −10 10 20 30 1 · 10−2 2 · 10−2 3 · 10−2 4 · 10−2 Energy

a

Gap ratio

Oganesyan, Huse

gn = |En − En−1|

r = min (gn, gn+1)/ max (gn, gn+1) hriGOE ' 0.5307

hriPoisson ' 0.3863

SLIDE 16

Gap ratios

Energy-resolved data

0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 r rGOE rPoisson =0.5

−80 −40 40 0.40 0.45 0.50 0.55

hc =3.72(6) ν =0.91(7)

1 2 3 4 5

12 14 15 16 17 18 19 20 22

L =

h

More...

−80 −40 40 0.40 0.45 0.50 0.55

hc =3.72(6) ν =0.91(7)

Finite-size scaling ansatz

r = fr(|h − hc|.L1/ν)

SLIDE 17

Eigenstate correlations

Beyond level statistics: correlations between eigenstates
Thermal (ETH) phase: expect eigenstates to be «similar»
MBL phase: expect eigenstates to be «different»
Kullback-Leibler divergence quantify similarity between eigenstates (in a basis)

KL = X

i

pi ln(pi/qi)

pi = |hn|ii|2 qi = |hn0|ii|2

{|ii} = {Sz} basis

Identical states : , GOE : , diverges for very different states

KL = 0 KL = 2

More... 0.38

12 14 15 16 17 18 19 20 22

1 2 3 4 5 5 10 15 20 25 30 KL divergence KLGOE

1.8 2.0 2.2

KL

10 20 30

h=1

30 60

KL

0.00 0.02 0.04

h(KL)

h=4.8

h

L =

✏ = 0.5

SLIDE 18

Phase diagram (1)

Energy-resolved phase diagram

1 2 3 4

h

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ε

r

GOE Poisson

SLIDE 19

Entanglement entropy

Area vs. volume law scaling of entanglement entropy distinguishes the two phases

12 16 20 24 L 0.01 0.1 SE/L 12 16 20 24 L 0.01 0.1

✏ = 0.5 ✏ = 0.8

h h

1 2 3 4 5

h

area law volume law

Collapse to finite-size scaling form:

1 10 L|h − hc|ν 0.01 0.1 SE/L 1 10 L|h − hc|ν 0.01 0.1

SE/L = fS(|h − hc|.L1/ν)

a r e a l a w b r a n c h area law branch volume law volume law

SLIDE 20

1 2 3 4

h

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ε

r S

E

S

E (visual)

Phase diagram (2)

Energy-resolved phase diagram

GOE Poisson V

lume law

Area law

SLIDE 21

Memory of initial magnetization

Consider relaxation of an initial spin inhomogeneity ˆ

M = X

r

Sz

rei2πr/L

Prepare the initial state ρ0 = (1 + x ˆ

M †)/Z

Initial spin polarization h ˆ

Mi0 = Trρ0 ˆ M = x Z X

n

hn| ˆ M † ˆ M|ni

Final spin polarization

following Pal & Huse

h ˆ Mi∞ = X

n

ρnn

0 hn| ˆ

M|ni = x Z X

n

hn| ˆ M †|nihn| ˆ M|ni diagonal ensemble

1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f =0.5

L =12 L =14 L =15 L =16 L =17 L =18 L =19 L =20 L =22

−200 0 100 (h − hc)L1/ν 0.2 0.4 0.6 0.8 1.0

hc =3.42(6) ν =0.62(5)

«Memory» «No memory»

f = 0 f = 1

f = ff(|h − hc|.L1/ν)

Contribution of eigenstate to depolarization

|ni

fn = 1 hn| ˆ M †|nihn| ˆ M|ni hn| ˆ M † ˆ M|ni

SLIDE 22

Phase diagram (4)

Energy-resolved phase diagram

GOE Poisson

1 2 3 4

h

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ε

r S

E

S

E (visual)

F f

GOE Poisson V

lume law

Area law Extensive fluctuations No fluctuations No memory Memory

SLIDE 23

Localization in Hilbert space ?

= ln (IPR)

How much a many-body wave-function is localized in a given basis?

Participation entropies

pi = |hn|ii|2

{|ii} = {Sz} basis

|ni = X

i

ni|ii

Sp

1 = −

X

i

pi ln(pi)

Sp

q =

1 1 − q ln X

i

pq

i

Scaling of participation entropy

7 8 9 10 11 12 13 14 SP 2 4 6 8 10 12 14 = 0.4 h = 1.8 h = 4.8 SP

1

SP

2

SP

1

SP

2 a1 =1.00±0.02, l1 =-0.56±0.69 a2 =1.00±0.01, l2 =-1.42±1.16 a1 =0.07±0.09, l1 =1.61±0.79 a2 =0.00±0.07, l2 =1.66±0.60

= ln (Hilbert space size)

Thermal:

aq = 1 Sp

q = aqSp 0 − ...

MBL: aq ⌧ 1 Sp

q = lq ln(Sp

) + ...
r

No true localization in Hilbert space

SLIDE 24

Final phase diagram

Evidence for a many-body localization edge

Sp

1 = a1Sp 0 + ...

h

SLIDE 25

Critical exponent ?

Systematic study of fit qualities to finite-size scaling ansätze
Different quantities
Different fit windows
Starting from minimal size
Including or not corrections to scaling ...

Lmin

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ⌫

Lmin=12 w =0.4 Lmin=12 w =0.6 Lmin=12 w =0.8 Lmin=12 w =1.0 Lmin=12 w =1.2 Lmin=14 w =0.4 Lmin=14 w =0.6 Lmin=14 w =0.8 Lmin=14 w =1.0 Lmin=14 w =1.2 Lmin=16 w =0.4 Lmin=16 w =0.6 Lmin=16 w =0.8 Lmin=16 w =1.0 Lmin=16 w =1.2 f r SE /L F/L

ν = 0.8(3)

Violation of Harris criterion ??

ν ≥ 2/d

Wrong form of ansätze ??
Rare events ??
too small ??

L = 22

Effective models find ν ' 3

V

sk et al., Potter et al.

SLIDE 26

Conclusions & outlooks

Message 2 : MBL is a computational challenge
Message 1: MBL is an active interesting field! Revisits usual stat-mech,

connections to different fields (quantum chaos, information...)

Message 3 : Many open questions

MBL in translation-invariant systems? Many-body edge: Griffiths effects? sub-diffusive ergodic phase? Nature of the transition? New type of fixed point? Experiments?

Based on improved numerics, evidence for:
Presence of a many-body localization edge
No true localization in Hilbert space
Finite-size signatures of many-body localization
Apparent violation of Harris criterion (?)

within our system sizes