Many-body dynamics and localizations V. Oganesyan D. Huse A. Pal and S. Mukerjee

outline G. Feher: hole burning, ENDOR and spin-localization Thermodynamics vs. localization Many-body localization – phase diagram DOS, levels and statistics, entropy From quantum to classical – “localized” chaos Insights and plans

Thermodynamics from dynamics: closed systems Dynamics  temperature,entropy,etc Measure the Boltzmann distribution by “watching” a single many-body trajectory Keywords: ergodicity, mixing, diffusion, entanglement

Some not so ergodic closed systems Anderson insulators: coherent trapping of waves (= quantum particle) Fermi-Pasta-Ulam anharmonic chains: apparent “trapping” of energy by nonlinear resonances in clasical many-particle dynamics Are these cases generic? – probably not…

“Locator” expansion: coherent hopping in a disordered environment can converge  particles stay put, no classical “hopping” transport If disorder is strong ALL wave-functions are localized,  – localization length Conductivity at ANY temperature Also takes place with classical waves Rigorous proof(s) – 80’s Original motivation: undetectable spin diffusion

Hole burning -- long lived localized excitations of donor spins ENDOR method for MEASURING the Hamiltonian (G.Feher, 50s)

Variable range hopping conduction energy  E real space Assume  E can be absorbed/emitted by the “bath” 

VRH from interactions? ? energy  E real space Can nearby particles act as an “efficient bath” for the a carrier trying to hop? “Efficient bath” should absorb and emit energy in arbitrary increments. The issue never settled…

Many-body localization? Locator expansion about many-body Slater determinant states made from localized single particle orbitals Interactions induce hopping on the (vast) network of many-body tab= <a| interaction |b> locator states: Still a linear hopping problem, although on a very complicated Fock “lattice” Basko etal,Annals of Physics, Gornyi etal, PRL: there is a many-body “mobility edge” The perfect insulator ( = 0) is stable at sufficiently low temperature. a = b =

MIT phase diagram PRB 75, 155111 (2007) Vc Tc(V) T insulator =0 The infinite T trajectory Metal,  finite Basko et. al. Gornyi et. al. V Many-body localization can survive at infinite (!) temperature, there is a critical interaction/disorder strength

No thermodynamic signatures near infinite temperature Vc =0 V Free energy is analytic V=0 – PWA(58)  The many-body localization transition is a purely dynamic phenomenon Infinite T regime is advantageous for investigating transport numerically in the absence of disorder: simple statistical description of spectra and transition matrix elements applies (Mukerjee etal PRB 2006)

Statistical theory of quantum transport Evaluate (,T) numerically (and exactly) for a finite system understand finite size effects, e.g. separately in distributions of j’s and E’s extrapolate to thermodynamic limit Obstacles: discrete spectra --- need a dense “forest” of delta functions to insure accurate extrapolation to continuum; at low T or weak interactions --- very few dominant transitions, poor stats typical culprit – long-lived “quasiparticle” states with regularly spaced levels, degeneracies and non-trivial matrix elements Solution: destroy quasiparticles with strong interactions + high temperature Bonus: systematic and simple (looking) high temperature expansion This talk PRB 73, 035113 (2006)

Example: conductivity of a clean metal  Size effects and universal features: Statistical noise Wigner deficit hydrodynamic singularity: “universal” Urbach tails at high (!) frequencies: T  PRB 73, 035113 (2006)

Metal-insulator transition at infinite T : conductivity 1/ 1 interaction Disorder 1 2 3 clean dirty insulator metal Anderson Metal =0 =0 V  T 2  T Very strong finite-size effects persist over a wide frequency range (colors--sizes) – difficult to be quantitative 3 

Spectral statistics (3D Anderson) localized and extended states “look” different … Exact energies  r vs. |n> |n+1> |n> En+3 En+2 En+1 En |n+1> En-1 … easy to find degeneracies in a localized spectrum Insulators: Poisson level statistics Metals: level repulsion  Wigner-Dyson statistics But there is a third possibility (e.g. in the 3D Anderson model): critical “intermediate” statistics is a universal property of the mobility edge and can be used to “detect” it.

“One parameter” finite size scaling Universality of gap statistics: consider n=En+1-En and r = <2>/<>2 Can be used like a Binder cumulant e.g. in Monte Carlo of 3D Ising model RMT Poisson r=4/ r=2 W Wc “Data collapse” across 3D Anderson (B. Shklovskii etal PRB ’93) Upshot: No phase transition in finite volume, yet finite size corrections to universal level statistics can be used to find and study the critical point

Many-body spectral statistics Far away from the critical regime it is easy to check that MANY-BODY spectra of clean strongly interacting metal exhibits Wigner-Dyson statistics and that MANY-BODY spectrum of non-interacting insulator is Poisson. Problem: large variations of density-of-states with energy and disorder. MANY-BODY DOS is NOT a smooth function in the thermodynamic limit – not safe to use for normalization

Many-body DOS: universality of a haystack Average level spacing varies rapidly, Hamiltonian is NOT a “random matrix”, it’s local, i.e VERY sparse Complication: ...but a simplification too! 14 sites 12 sites 10 sites 8 sites 6 sites DOS is “thermodynamic” already for very few particles, e.g. it obeys the central limit theorem

Solution: two-gap distribution function Instead consider a dimensionless number constructed from two adjacent gaps: rn=min (n, n+1)/max (n, n+1) Two universal distributions of r can be identified corresponding to Wigner-Dyson and Poisson stats PRB 75, 155111 (2007) Critical distribution? Metal Insulator p(r)=2/(1+r)2 More simply <r> = 2 log 2 -1 in the insulator, while <r>~ 0.53 in the diffusive phase, <r>c=?

L=16 L=8 w=“dirt” Crossover sharpens (but drifts) RMT value ~0.53 Critical stats? Poisson value 2log2-1 w=“dirt” PRB 75, 155111 (2007)

<r> L=16 L=8 “dirt” More on drifts Sizes vs. drifts: L  wc 10 2 10 2 12 1 14 ~.5 So far so good: the drift is slowing but would be nice to get rid of drifts altogether—should be possible if caused by “irrelevant” operators, e.g. by tweaking H and/or r… so far only minor improvement <r> L=16 L=8 “dirt” PRB 75, 155111 (2007)

Possibilities There is no insulating phase  the drift continues, localization only exists at infinite disorder OR The drift continues, but only vertically: the “critical statistics” is Poisson, this is also suggested by other considerations and existing results on Anderson in high dimensions or Bethe lattice (Canopy graphs). Similar to e.g. 5D Ising model and Kosterlitz-Thouless flows – the critical point “belongs” to one of the phases, 1-parameter scaling is violated OR...

Open questions Is there a finite temperature insulating phase? Is there a phase transition at finite disorder strength? Good news: sharpening crossover Bad news: drifts A rigorous proof of many-body localization would help Scaling and field theory at the transition? Eventually need either to go beyond one parameter scaling Or to consider sufficiently different quantities for one parameter scaling analysis, e.g. individual wavefunctions.

Another view of localization: entanglement (entropy) B size strong disorder expected slope (ala Boltzmann) = - log 2 Weak disorder <A log A>

Classical many-body localization? Finite temperature transport in all quantum models can usually be discussed in terms of classical diffusive models (e.g. PRB 73, 035113 (2006) )…. or is there classical localization?! Two extreme regimes: J=0 – each spin precesses in it’s own random field hj (orientation and magnitude) – no transport hj=0 – spin and energy diffusion (e.g. numerics D. Landau et.al.) How are these regimes “connected”? I.e. what is (energy) conductivity/diffusion constant as a function of J/|h|, esp. as J  0? Will keep to infinite T

The answer J D Clean limit good diffusion Surprise – an enormous suppression of diffusion Conjecture #1 -- no transition. Conjecture #2 – phenomena not dimension specific. Justification – qualitative and quantitative analysis of space-time evolution of Kubo

How to measure diffusion Open boundaries – current states; slow density relaxation Will focus on “fast” current relaxation – Kubo Localization -- the integral vanishes! The challenge – accurately compute the long-time tail of <jj> “Exact simulation” – only roundoff errors

Optical conductivity, small J’s uncorrelated spins * ~ J few spin contribution 

Optical conductivity, small J’s Low freq. limit? powerlaws? /J

Diffusive regime: long time tails ( J)0.33 Appear robust w.r.t. to changes in disorder strength or type A very strong violation of mode-coupling prediction – confirmed for a wide range of disordered stochastic systems What’s different here? – nonlinearity, perhaps

Diffusive regime: finite size/time effects J=0.4,0.32,0.17 Red data – finite size effects in small rings (10~20 spins) C=10 throughout

Insulating regime: formalism Sample specific semilocal Kubo conductivity Useful facts: positive definite for all t short times – computable analytically, “follows” disorder (i.i.d. etc) “infinite” times – uniform, the magnitude specifies sample specific DC conductivity

Semilocal Kubo: localized “noise” t=1/J t=0

Longer times: diffusion and bottlenecks t >>1/J 1 / t  0

New (?) classical phenomena Very strong suppression of diffusion already at infinite temperature Strong non-analytic corrections to diffusion at finite frequency – at odds with existing theory A sort of localization of current noise and chaos exists at short times and distances – it is ultimately destabilized, presumably by Arnold like diffusion, but in real space

Lessons for the quantum problem Many phenomena should translate, esp. in the diffusive regime The regime of localized chaos may/should disappear altogether – complete many-body localization?