Properties of many-body localized phase

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

Thermalizing systems: ETH Thermalization: e -iht time subsystem in thermal state Mechanism: Eigenstate Thermalization Hypothesis: property of eigenstates λ>= [Deutsch 91] [Srednicki 94] [Rigol,Dunjko,Olshanii 08] thermal density matrix ρ λ =e -H/T Works in many cases Many open questions: timescale, other mechanisms?..

Many-body localized phase MBL = localized phase with interactions Ei t V Perturbative arguments for existence of MBL phase: Numerical evidence for MBL: [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] 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 X H = ~S i ~S i+1 + h i Si z i hi J Jz New spins: τ i = U Si U are quasi-local; form complete set τi z τj z H ij / exp( i j / ) S j S i 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 / Dephasing dynamics time ( + )( + )( + )( + )( + ) Phases randomize on distance x(t): th ij = tj exp( x/ ) 1 x(t) = log(jt) Explains logarithmic growth of entanglement [MS, Dynamics of local observables? ( + )( + )( + )( + )( + ) Papic, Abanin, PRL 13] distance

Local observables in a quench e -iht measure <S x >or <S z > <τz (t)>= const <τx (t)>= "# (t) Decay of oscillations of <τx (t)>: x(t) / ln(t) = [sum of N(t) = 2 x(t) oscillating terms] h x k (t)i / 1 p N(t) = 1 (tj) a hô(t)i memory of initial state ho(1)i 1 t a [MS, Papic, Abanin, PRB 14] S x S z t W=5

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

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

Matrix elements of local operators local perturbation V ETH ansatz R Local integrals of motion hi S z ji = e S(E,R)/2 f(e i,e j )R ij [Srednicki 99] narrow distribution: hi S z ji 1/ p 2 R S z = X { } ˆ { } ˆB { } [ z ] hi S z ji broad distribution: hi S z ji exp( apple 0 R) Thermalizing phase MBL phase disorder W

Fractal analysis of matrix elements Fractal dimensions from scaling of q q = q 1 Ergodic phase MBL phase q>qc =0 P q = X m h V nm 2q i/ 1 D q q Frozen fractal spectrum in MBL: hln V nm i/ applel

Energy structure of matrix elements Spectral function f 2 (!) =e S(E) h V nm 2 (! (E m E n ))i Related to dynamics: h V (t)v (0) i c Z 1 1 d! e i!t f 2 (!) Thermalizing phase: ln f 2 (!) E Th 1! E Th / ln! h V (t)v (0) i c / 1 1 L 1/(1 ) more details: [arxiv:1610.02389] t 1

Numerical results for spectral function: ) f 2 (!) f 2 (!) )!/!/ MBL phase: Thouless energy < level spacing Breakdown of typicality: loghv nm i6= hlog V nm i Thermalizing phase MBL phase disorder W

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

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

Entanglement spectrum: probes boundary Quantum Hall wave function: ky to organize ES [Li & Haldane] ky kx MBL phase: conserved quantities label ES """"i = c 0 " ""i ""i + + e 4apple e apple " "#i ""i r=1 ##i ##i r=4 + e 2apple " "#i #"i + r=2 +.. Coefficients decay as C "..."##"""# {z } r "..." / e appler

Power-law entanglement spectrum Hierarchical structure of h (r) (r) i/e 2appler L = P L r=0 (r) ih (r) but non-orthogonal Orthogonalize perturbatively (r) / e 4appler multiplicity is 2 r (0) (1) (1) (2) (2) (2) (2) Power-law entanglement spectrum k / 1 k 4apple ln 2

Numerics for XXZ spin chain Numerical studies for XXZ spin chain, J =J z =1 H = X i (h i S z i + J?S + i S i+1 + h.c.) + X i J z S z i S z i+1 Power law entanglement spectrum: disorder W = 5 k / k 1 more details in: [arxiv:1605.05737]

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

Summary and outlook MBL: new universality class of non-thermalizing systems Properties: dynamics, matrix elements, entanglement hô(t)i Questions: MBL in d>1, symmetries, MPS/MPO description breakdown of MBL, mobility edge ho(1)i 1 t a hi S z ji exp( apple 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

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