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: 
 Ergodic 
 Integrable ergodicity from chaos stable to weak perturbations [Kolmogorov-Arnold-Moser theorem] • Quantum systems: Thermalizing 
 Many-body localized emergent integrability 
 ETH mechanism stable to weak perturbations < O ＞ E


 
 
 
 Thermalizing systems: ETH • Thermalization: 
 e -iHt subsystem in thermal state 
 time • Mechanism: 
 [Deutsch’91] [Srednicki’94] 
 Eigenstate Thermalization Hypothesis : 
 [Rigol,Dunjko,Olshanii’08] property of eigenstates 
 thermal density matrix 
 | λ >= ρ λ = e -H/T 
 • Works in many cases 
 Many open questions: timescale, other mechanisms?..

Many-body localized phase t ⚡ V • MBL = localized phase with interactions E i • Perturbative arguments for existence of MBL phase: 
 [Basko,Aleiner,Altshuler’05][Gornyi,Polyakov,Mirlin’05] 
 • Numerical evidence for MBL: 
 [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] Non-thermalizing MBL phase exists! Properties of MBL phase? Why thermalization breaks down?


 
 
 
 
 Universal Hamiltonian of MBL phase • If model is in MBL phase, rotate basis 
 J ⟂ ⚡ J z h i X S i · ~ ~ S i +1 + h i S z H = i i • New spins: τ i = U † S i U are quasi-local; form complete set 
 H ij ∝ exp( − | i − j | / ξ ) τ jz τ iz S i S j • Consequences: no transport, ETH breakdown, 
 universal dynamics [MS, Papic, Abanin, PRL’13] 
 [Huse, Oganesyan, PRB’14] [Imbrie, arXiv:1403.7837]

͠ 
 
 
 
 
 Properties of MBL phase • Transport: 
 Diffusion 
 ?? Entanglement light cone • Matrix elements: 
 • Eigenstate properties: 
 Thermalizing phase MBL phase disorder W


 
 Dynamics in MBL phase H ij ∝ Je − | i − j | / ξ time • Dephasing dynamics + ) ( + ) + ) ( ( + ) ( ( + ) • Phases randomize 
 x ( t ) = ξ log( Jt ) on distance x ( t ) : 
 tH ij = tJ exp( − x/ ξ ) ∼ 1 ( + ) ( + ) ( + ) ( + ) ( + ) distance • Explains logarithmic growth of entanglement [MS, Papic, Abanin, PRL’13] • Dynamics of local observables? 


Local observables in a quench ⚡ e -iHt measure < S x ＞ or !< S z ＞ • < τ z ( t ) ＞ = const x ( t ) ∝ ln( t ) • < τ x ( t ) ＞ = = [sum of N(t) = 2 x(t) oscillating terms] ρ ↑↓ ( t ) 1 1 • Decay of oscillations of < τ x ( t ) ＞ : | h τ x k ( t ) i | / = p ( tJ ) a N ( t ) � ⇠ 1 � h ˆ � � O ( t ) i � h O ( 1 ) i S x t a W =5 memory of 
 initial state S z [MS, Papic, Abanin, PRB’14] t


 
 
 
 ͠ Properties of MBL phase • Transport: 
 No transport 
 Diffusion 
 Log-growth of entanglement Entanglement lightcone • Matrix elements 
 ?? ETH ansatz, typicality Thermalizing phase MBL phase disorder W


 
 Structure of many-body wave function • Single-particle localization: 
 𝜔 (x) • Many-body wave function: 
 | 𝜔 >= P σ = " , # ψ σ 1 σ 2 ... | σ 1 σ 2 . . . i Problems : basis-dependent, 
 not related to observables • Alternative: “wave function” created by V 
 V V nm H | n i = E n | n i ψ n ( m ) = h m | V | n i

͠ Matrix elements of local operators local 
 perturbation V R ETH ansatz 
 Local integrals of motion 
 S z = τ { α } ˆ B { α } [ τ z ] h i | S z | j i = e � S ( E,R ) / 2 f ( E i , E j ) R ij X ˆ { α } h i | S z | j i [Srednicki’99] narrow distribution: broad distribution: h i | S z | j i ⇠ 1 / h i | S z | j i ⇠ exp( � κ 0 R ) p 2 R Thermalizing phase disorder W MBL phase


 
 
 
 
 
 
 
 
 
 
 
 Fractal analysis of matrix elements h | V nm | 2 q i / 1 • Fractal dimensions from scaling of 
 X P q = D τ q m τ q τ q = q − 1 Ergodic phase τ q>q c = 0 q MBL phase • “Frozen” fractal spectrum in MBL: 
 h ln V nm i / � κ L


 
 
 
 
 
 
 
 
 Energy structure of matrix elements • Spectral function 
 f 2 ( ω ) = e S ( E ) h | V nm | 2 δ ( ω � ( E m � E n )) i Z 1 • Related to dynamics: 
 d ω e � i ω t f 2 ( ω ) h α | V ( t ) V (0) | α i c ⇡ �1 • Thermalizing phase: 
 ln f 2 ( ω ) 1 h α | V ( t ) V (0) | α i c / t 1 � φ 1 1 ω φ E Th ∝ L 1 / (1 − φ ) ln ω more details: E T h [arXiv:1610.02389]

͠ Numerical results for spectral function: (c) (a) ln f 2 ( ω ) ln f 2 ( ω ) ω / ∆ ω / ∆ • MBL phase: Thouless energy < level spacing • Breakdown of typicality: log h V nm i 6 = h log V nm i Thermalizing phase disorder W MBL phase

͠ 
 
 
 
 Properties of MBL phase • Transport: 
 No transport 
 Diffusion 
 Log-growth of entanglement Entanglement lightcone • Matrix elements: 
 broad distribution ETH ansatz, typicality strong fractality • Eigenstate properties: 
 volume-law entanglement ?? “flat” entanglement spectrum Thermalizing phase MBL phase disorder W


 
 Beyond entanglement • Gapped ground states: area-law 
 S ent ( L ) ~ const in 1d 
 E • Excited eigenstates: volume-law 
 S ent ( L ) ~ L in 1d 
 Ground state • MBL: area-law entanglement 
 Q: Difference with gapped ground states? • Entanglement spectrum { 𝜇 i } S ent = − P i λ i log λ i • “Flat” in ergodic states: 
 ln λ k [Marchenko&Pastur'67] 
 [Yang,Chamon,Hamma&Muciolo’15]


 
 
 
 
 Entanglement spectrum: probes boundary • Quantum Hall wave function: 
 k y 
 k y to organize ES [Li & Haldane] k x • MBL phase: conserved quantities label ES 
 #" | "#i | #"i +… | """"i = "" | ""i | ""i "" | "#i | ""i + e − 2 κ + e − κ c 0 r =2 r =1 e − 4 κ ## | ##i | ##i + + ….. r =4 • Coefficients decay as ↑ ... ↑ | ∝ e − κ r | C ↑ ... ↑↓↓↑↑↑↓ | {z } r


 
 
 
 Power-law entanglement spectrum • Hierarchical structure of 
 r =0 | ψ ( r ) ih ψ ( r ) | ρ L = P L h ψ ( r ) | ψ ( r ) i / e � 2 κ r but non-orthogonal λ (0) • Orthogonalize perturbatively 
 λ (1) λ (1) λ ( r ) ∝ e − 4 κ r λ (2) λ (2) multiplicity is 2 r λ (2) λ (2) • Power-law entanglement spectrum 
 λ k ∝ 1 γ ≈ 4 κ k γ ln 2


 
 Numerics for XXZ spin chain • Numerical studies for XXZ spin chain, J ⟂ = J z =1 
 X i + J ⊥ S + ( h i S z i S − H = i +1 + h.c. ) i X J z S z i S z + i +1 i • Power law entanglement spectrum: disorder W = 5 λ k ∝ 1 k γ more details in: [arXiv:1605.05737]

Estimates for the bond dimension • Large 𝛿 → MPS error can be small ∝ 1 / χ γ − 1 • Implementation of DMRG for highly excited states: disorder W = 5 χ more details: 
 also: [Yu et al arXiv:1509.01244] [Lim&Sheng arXiv:1510.08145] 
 [arXiv:1605.05737] [Pollmann et al arXiv:1509.00483] [Kennes&Karrasch arXiv:1511.02205]

͠ 
 
 
 
 Properties of MBL phase • Transport: 
 No transport 
 Diffusion 
 Log-growth of entanglement Entanglement lightcone • Matrix elements: 
 broad distribution ETH ansatz, typicality strong fractality • Eigenstate properties: 
 volume-law entanglement area-law entanglement “flat” entanglement spectrum power-law entanglement spectrum Thermalizing phase MBL phase disorder W ??


 
 
 
 
 Summary and outlook • MBL: new universality class of non-thermalizing systems • Properties: dynamics, matrix elements, entanglement 
 � ⇠ 1 � h ˆ � � O ( t ) i � h O ( 1 ) i t a h i | S z | j i ⇠ exp( � κ 0 R ) 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 • Questions: MBL in d>1, symmetries, MPS/MPO description breakdown of MBL, mobility edge

Acknowledgments Alexios Michailidis 
 Zlatko Papic 
 Dima Abanin 
 Joel Moore 
 Nottingham Leeds Univ. of Geneva UC Berkeley


 
 
 
 
 Summary and outlook • MBL: new universality class of non-thermalizing systems • Properties: dynamics, matrix elements, entanglement 
 � ⇠ 1 � h ˆ � � O ( t ) i � h O ( 1 ) i t a PRL 110, 260601 (2013) h i | S z | j i ⇠ exp( � κ 0 R ) 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 • Questions: MBL in d>1, symmetries, MPS/MPO description breakdown of MBL, mobility edge