Properties of many-body localized phase Maksym Serbyn UC Berkeley - - PowerPoint PPT Presentation

Properties of many-body localized phase

Maksym Serbyn UC Berkeley
 →IST Austria

QMATH13 Atlanta, 2016

Motivation: MBL as a new universality class

• Classical systems:


• Quantum systems:

Integrable

stable to weak perturbations

[Kolmogorov-Arnold-Moser theorem]

Ergodic


ergodicity from chaos

Thermalizing


ETH mechanism

E <O＞

Many-body localized

emergent integrability
 stable to weak perturbations

Thermalizing systems: ETH

• Thermalization:


• Mechanism:


Eigenstate Thermalization Hypothesis: 
 property of eigenstates
 
 


• Works in many cases


Many open questions: timescale, other mechanisms?..

e-iHt

time

[Deutsch’91] [Srednicki’94]
 [Rigol,Dunjko,Olshanii’08]

|λ>=


 subsystem in thermal state
 thermal density matrix


ρλ =e-H/T


• MBL = localized phase with interactions
• Perturbative arguments for existence of MBL phase:

• Numerical evidence for MBL: 


Many-body localized phase

[Basko,Aleiner,Altshuler’05][Gornyi,Polyakov,Mirlin’05]
 [Oganesyan,Huse’08] [Pal,Huse’10] [Znidaric,Prosen’08]
 [Monthus, Garel’10][Bardarson,Pollman,Moore’12] 
 [MS, Papic, Abanin’13,’14] [Kjall et al’14]

Properties of MBL phase? Why thermalization breaks down? Non-thermalizing MBL phase exists!

t

Ei

V

⚡

Universal Hamiltonian of MBL phase

• If model is in MBL phase, rotate basis


• New spins: τi = U† Si U are quasi-local; form complete set


• Consequences: no transport, ETH breakdown, 


universal dynamics

Hij ∝ exp(−|i − j|/ξ)

τiz τjz Si Sj

H =

X

i

~ Si · ~ Si+1 + hiSz

i

hi J⟂ Jz

⚡

[MS, Papic, Abanin, PRL’13] 
 [Huse, Oganesyan, PRB’14] [Imbrie, arXiv:1403.7837]

Properties of MBL phase

Thermalizing phase MBL phase

disorder W

͠

Diffusion
 Entanglement light cone

??

• Transport: 


• Matrix elements:


• Eigenstate properties:


Dynamics in MBL phase

• Dephasing dynamics
• Phases randomize 

• n distance x(t):


• Explains logarithmic growth of entanglement
• Dynamics of local observables?


[MS, Papic, Abanin, PRL’13]

tHij = tJ exp(−x/ξ) ∼ 1

Hij ∝ Je−|i−j|/ξ

distance time

+ ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) (

x(t) = ξ log(Jt)

Local observables in a quench

e-iHt

measure <Sx＞or !<Sz＞

⚡

ρ↑↓(t)

x(t) ∝ ln(t)

|hτ x

k (t)i| /

1

p

N(t) = 1 (tJ)a

• h ˆ

O(t)i hO(1)i

• ⇠ 1

ta

• <τz(t)＞= const
• <τx(t)＞= = [sum of N(t) = 2x(t) oscillating terms]
• Decay of oscillations of <τx(t)＞:

t

W=5 Sz Sx

[MS, Papic, Abanin, PRB’14]

memory of 
 initial state

Properties of MBL phase

• Transport: 


• Matrix elements


Thermalizing phase MBL phase

disorder W

͠

Diffusion
 Entanglement lightcone No transport
 Log-growth of entanglement ETH ansatz, typicality

??

Structure of many-body wave function

• Single-particle localization:

• Many-body wave function:


• Alternative: “wave function” created by V


𝜔(x)

P

σ=",# ψσ1σ2...|σ1σ2 . . .i

|𝜔>= Problems: basis-dependent, 
 not related to observables

H|ni = En|ni

ψn(m) = hm|V |ni

V

Vnm

Matrix elements of local operators

local 
 perturbation V

ETH ansatz 


[Srednicki’99]

hi|Sz|ji = eS(E,R)/2f(Ei, Ej)Rij

Local integrals of motion


Sz =

X

{α}

ˆ τ{α} ˆ B{α}[τz]

hi|Sz|ji

R

narrow distribution:

hi|Sz|ji ⇠ 1/

p

2R

broad distribution:

hi|Sz|ji ⇠ exp(κ0R)

Thermalizing phase

disorder W

͠

MBL phase

• Fractal dimensions from scaling of


• “Frozen” fractal spectrum in MBL: 


Fractal analysis of matrix elements

MBL phase Ergodic phase

Pq =

X

m

h|Vnm|2qi / 1 Dτq

τq q

τq = q − 1

τq>qc = 0

hln Vnmi / κL

• Spectral function

• Related to dynamics:

• Thermalizing phase: 


Energy structure of matrix elements

f 2(ω) = eS(E)h|Vnm|2δ(ω (Em En))i

hα|V (t)V (0)|αic ⇡

Z 1

1

dω eiωtf 2(ω)

ln f 2(ω) ln ω

ET h 1 ωφ

ETh ∝ 1 L1/(1−φ)

hα|V (t)V (0)|αic / 1 t1φ

[arXiv:1610.02389] more details:

• MBL phase: Thouless energy < level spacing
• Breakdown of typicality:

Numerical results for spectral function:

Thermalizing phase

disorder W

͠

MBL phase

loghVnmi 6= hlog Vnmi

(a)

ω/∆

(c)

ω/∆

ln f 2(ω)

ln f 2(ω)

Properties of MBL phase

• Transport: 


• Matrix elements:


• Eigenstate properties:


Thermalizing phase MBL phase

disorder W

͠

Diffusion
 Entanglement lightcone No transport
 Log-growth of entanglement ETH ansatz, typicality broad distribution strong fractality volume-law entanglement “flat” entanglement spectrum

??

• Gapped ground states: area-law

• Excited eigenstates: volume-law


• MBL: area-law entanglement


Q: Difference with gapped ground states?

• Entanglement spectrum {𝜇i}
• “Flat” in ergodic states:


Beyond entanglement

Sent(L) ~ L in 1d
 E

Ground state

Sent(L) ~ const in 1d


Sent = − P

i λi log λi [Marchenko&Pastur'67]
 [Yang,Chamon,Hamma&Muciolo’15] ln λk

• Quantum Hall wave function: 


• MBL phase: conserved quantities label ES


• Coefficients decay as

Entanglement spectrum: probes boundary

ky kx

""|"#i|""i

+

r=1

e−κ

|""""i =

""|""i|""i

c0

+ …..

##|##i|##i

+

r=4

e−4κ e−2κ

#"|"#i|#"i +…

+

r=2

|C↑...↑↓↓↑↑↑↓

| {z }

r

↑...↑| ∝ e−κr

[Li & Haldane]

ky to organize ES

Power-law entanglement spectrum

• Hierarchical structure of 


• Orthogonalize perturbatively


• Power-law entanglement spectrum


ρL = PL

r=0 |ψ(r)ihψ(r)|

λ(r) ∝ e−4κr

hψ(r)|ψ(r)i / e2κr

but non-orthogonal

2r

multiplicity is

λ(0) λ(1) λ(1) λ(2) λ(2) λ(2) λ(2)

λk ∝ 1

kγ

γ ≈ 4κ

ln 2

Numerics for XXZ spin chain

• Numerical studies for XXZ spin chain, J⟂=Jz =1


• Power law entanglement spectrum:

H =

X

i

(hiSz

i + J⊥S+ i S− i+1 + h.c.)

+

X

i

JzSz

i Sz i+1

λk ∝ 1

kγ disorder W = 5 [arXiv:1605.05737] 
 more details in:

Estimates for the bond dimension

χ

disorder W = 5

also: [Yu et al arXiv:1509.01244] [Lim&Sheng arXiv:1510.08145]
 [Pollmann et al arXiv:1509.00483] [Kennes&Karrasch arXiv:1511.02205] more details: 
 [arXiv:1605.05737]

∝ 1/χγ−1

• Large 𝛿 → MPS error can be small
• Implementation of DMRG for highly excited states:

Properties of MBL phase

• Transport: 


• Matrix elements:


• Eigenstate properties:


Thermalizing phase MBL phase

disorder W

͠

Diffusion
 Entanglement lightcone No transport
 Log-growth of entanglement ETH ansatz, typicality broad distribution strong fractality volume-law entanglement “flat” entanglement spectrum area-law entanglement power-law entanglement spectrum

??

Summary and outlook

• MBL: new universality class of non-thermalizing systems
• Properties: dynamics, matrix elements, entanglement


• Questions: MBL in d>1, symmetries, MPS/MPO description

breakdown of MBL, mobility edge

PRL 110, 260601 (2013) PRL 111, 127201 (2013) PRL 113, 147204 (2014) PRB 90, 174302 (2014) PRX 5, 041047 (2015) PRB 93, 041424 (2016)
 arXiv:1605.05737 arXiv:1610.02389

• h ˆ

O(t)i hO(1)i

• ⇠ 1

ta

hi|Sz|ji ⇠ exp(κ0R)

Acknowledgments

Dima Abanin


• Univ. of Geneva

Zlatko Papic
 Leeds Joel Moore
 UC Berkeley Alexios Michailidis 
 Nottingham

Summary and outlook

• MBL: new universality class of non-thermalizing systems
• Properties: dynamics, matrix elements, entanglement


• Questions: MBL in d>1, symmetries, MPS/MPO description

breakdown of MBL, mobility edge

PRL 110, 260601 (2013) PRL 111, 127201 (2013) PRL 113, 147204 (2014) PRB 90, 174302 (2014) PRX 5, 041047 (2015) PRB 93, 041424 (2016)
 arXiv:1605.05737 arXiv:1610.02389

• h ˆ

O(t)i hO(1)i

• ⇠ 1

ta

hi|Sz|ji ⇠ exp(κ0R)