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

Many - body localization edge in the random - field Heisenberg chain Fabien Alet Laboratoire de Physique Théorique Toulouse In collaboration with : David Luitz, Nicolas Laflorencie Ref. : Phys. Rev. B 91 , 081103 ( 2015 ) W orkshop Quantum chaos, Luchon, March 2015

Outline • Broad question: What happens to Anderson localization in presence of interactions? «Many - body localization» ( MBL ) A new distinct dynamical phase of matter, which does not thermalize Part 1 : Mini - Review on Many - Body localization • Distinct features from a «thermal» state, and Anderson insulator • Present in simple toy models • Many open questions... Part 2 : «Large» - scale numerics on a MBL Hamiltonian • Computational issues • Energy - resolved phase diagram

my current understanding of Part 1 : Mini - Review on Many - Body Localization Reviews on MBL : Nandkishore & Huse, arxiv:1404.0686 Altman & V osk, arxiv:1408.2834

Thermalisation & ETH X X | Ψ 0 i = a n | n i E n | n ih n | H = • Initial wave - function expressed in the eigenbasis of i n • Time - evolved observable ( generic Hamiltonian ) t →∞ X X n 0 a n e − i ( E n 0 − E n ) t O nn 0 | a n | 2 O nn a ∗ h O ( t ) i = � � � ! n n,n 0 «Diagonal ensemble» • Eigenstate thermalization hypothesis ( ETH ) Deutsch, Srednicki, Rigol & many authors h n |O| n i ' h n 0 |O| n 0 i = O ( E ) | n i , | n 0 i in the same energy shell h n |O| n 0 i vanish in the thermodynamic limit and for local observable • ETH + ( some other minimal assumptions ) implies thermalisation h O ( t ! 1 ) i = O ( E ) = O ( T ) E = h Ψ 0 | H | Ψ 0 i E = h H i T

Consequences • Each eigenstate is thermal, «knows» equilibrium ρ (0) = | n ih n | = ρ ( t ) = ρ eq ( T n ) E n = h H i T n • Memory of initial conditions is lost • ETH is a «justification» of the microcanonical ensemble at the invididual eigenstate level • ETH seems to work ( analytics+numerics ) for most many - body quantum systems, except • Integrable systems : May have their own ETH, relaxation to a Generalized Gibbs Ensemble many authors... • Localized systems : single - particle localization t → ∞ • Many - Body Localized systems

Many - body localization • Old problem revived by an enormous amount of contributions! Anderson, Fleishmann, Nandkishore, Huse, arxiv:1404.0686, Shepelyansky... Altman, V osk 1408.2834 • Typical example : XXZ chain with random fields X X σ z i σ z i +1 + σ z i σ z i +1 + ∆ σ z i σ z h i σ z H = h i ∈ [ − h, h ] i +1 − i i i • Infinite disorder : eigenstates are fully localized product - states, no entanglement • Branch small interaction : perturbative calculations indicate that Gornyi et al. Basko et al. thermalization does not occur: states keep localized, no spin or energy transport • Beyond perturbation : numerics ( including this talk ) indicate that the localized phase survives h c 0 Thermal, «ergodic» MBL h Phase transition ( ? ) is dynamical can’t be seen in thermodynamics • Crucial to work in the «eigenstate ensemble», not ( micro -) canonical ensemble

Phenomenology of MBL systems • Consider only fully MBL systems ( all eigenstates are localized ) • Exact results , phenomenology , perturbative results, strong disorder RG Imbrie Huse. Oganesyan, Ros, Müller Altman, V osk Abanin et al. Scardicchio • Idea: Quasi - local unitary transform can «diagonalize» the Hamiltonian X X X U † HU = − h i τ z J i,j τ z i τ z J i,j,k τ z i τ z j τ z k + ..., j + i − i i<j i<j<k J i 1 ,...,i k decay exponentially i ' ˆ τ z Z i ( h ) σ z i + tail • Set of localized bits: i U † U τ z • = complete set of local integral of motions • Useful to describe properties of MBL, to detect MBL through spin - echo experiments Serbin et al. • Other sets of local integrals of motion can be constructed, which may have a better physical interpretation Ros, Müller Abanin et al. Scardicchio

Entanglement & MBL • ETH : Entanglement entropy of eigenstates is extensive : V olume law B A ρ A = Tr B ρ = ρ eq A ( T n ) N A S A = − Tr A ρ A log ρ A ∝ N A T n 6 = 0 if System is its own bath: B acts a thermal bath for A MBL states have low entanglement S A /N A → 0 • • MBL states e ffi ciently represented as matrix - product states Abanin, Vidal et al. Eisert et al. Pekker, Clark • Entanglement spreads logarithmically Can be understood with the Znidaric et al. localized - bits picture Bardarson et al.

Summary of MBL Thermal phase Single-particle localized Many-body localized from Nandkishore, Huse 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 arxiv:1404.0686, at long times observables at long times observables 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 • 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, • MBL in systems with no disorder? Garrahan et al. , Y ao et al. • Many - body mobility edge: Gri ffi ths e ff ects? sub - di ff usive ergodic phase? Agarwal et al. • Nature of many - body localization transition?

� � � � � � � � � � � � � � Experimental realization Schreiber et al. , arXiv:1501.05661 X ⇣ ⌘ • Cold - atomic gas realization of interacting Aubry - André model: ⇣ ⌘ X X c † ˆ X c † + ∆ cos(2 πβ i + φ )ˆ i, σ ˆ c i, σ + U n i, ↑ ˆ ˆ n i, ↓ . H = − J ˆ i, σ ˆ c i +1 , σ + h.c. i, σ i i, σ • ( Non -) Equilibration of a quenched initial state measured by imbalance X I = N e − N o initial state N e + N o , U/J =4.7(1) , U/J =10.3(1) 0.8 � /J =8 e o e o e o e o 0.6 Imbalance 0.4 � /J =3 0.2 3 B � /J =0 0 AA localized 0 10 20 30 Time ( � ) non-ergodic localized 2 attractive ∆ /J 0.6 0.3 non- � / J AA extended interacting repulsive 0.5 ergodic delocalized 0.2 Imbalance -4 0 4 0.4 0 0.3 � � 0 5 10 15 20 U/J 0.2 -20 -10 0 10 20 U/J Imbalance � �

Part 2 : «Large» - scale numerics on a MBL Hamiltonian Phys. Rev. B 91 , 081103 ( 2015 ) • 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?

MBL & Numerics • Prototypical MBL Hamiltonian : Heisenberg S=1/2 spin chain in a random field L X X h i S z H = S i · S i +1 − h i ∈ [ − h, h ] i i =1 i S z • Total magnetization is conserved, no other symmetries • A tough computational problem • 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 DMRG, power method, Lanczos, T=0 series expansion, etc • W e want eigenstates of a closed system: cannot impose a bath ! Quantum Monte Carlo, high - T series expansion ... • W e are left with • Time evolution after a quench: time - dependent DMRG, TEBD ... ( but no eigenstates ) • Real Space Renormalization Group: for excited states, only in the strong disorder limit • Brute - force methods: Full diagonalization, or ( slightly better ) spectral transforms

Computational details • Obtain eigenstates in the middle of the spectrum with shift - invert σ H R − 1 R = ( H − σ I ) R − 1 • Extremal eigenvalue problem for ( Lanczos - like methods ... ) • W R − 1 R − 1 a = b e cannot compute ! However just need to know how to apply • Three steps method: ‣ LU decomposition Bottleneck ! R = LU ‣ Solve for in a = Lc c «Simple» linear equations ‣ Solve for in b c = Ub • LU decomposition done by the massively parallel solver Mumps

Computational details • Obtain eigenstates in the middle of the spectrum with shift - invert σ H R − 1 R = ( H − σ I ) • Further details: ‣ Method typical for Anderson localization, however LU much more di ffi cult here ! ‣ L=22 ( matrix size ) instead of L=16 ( matrix size ) ∼ 12 . 000 ∼ 700 . 000 ‣ Obtain eigenstates for disorder realizations for each field, energy, L ∼ 50 ∼ 1000 ‣ Strong correlations between eigenstates of the same disorder sample! ✏ = 1 ‣ All results in terms of ✏ = ( E − E max ) / ( E min − E max ) ✏ = 0