Ideal Quantum Glass Transitions:Many-body localization without quenched disorderTIDS 15 1-5 September 2013 Sant Feliu de Guixols

Markus Mller

Mauro Schiulaz

MotivationG. Carleo, F. Becca, M. Schiro, M. Fabrizio, Scientific Reports 2, 243 (2012). Dynamics starting from inhomogeneous initial condition At large U: relaxation time grows (diverges?) with L!

Glass transition ?! Why how ??

Glasses =Systems defying thermodynamic equilibration

Breaking of ergodicity (rel ) Absence of full thermalizationHow can this occur in general?

Routes to glassinessSpin glasses + cousins: (Coulomb glasses?)Usual classical ingredients: Disorder + Frustration

Barriers grow with size L rel ~ exp[L]

Structural glasses (viscous, supercooled liquids):Steric frustration + self-generated disorder growing time- and length-scales (Kirkpatrick, Thirumalai, Wolynes)But: Eternal debate without conclusion: Is there an ideal glass transition at finite T: Can rel , before full jamming and incompressibility are reached ???

Ideal glass transition? Classical structural glasses in finite d: ??? How can barriers become infinite without jamming?

BUT

Quantum ideal (disorder free) glasses can exist! Extra ingredient: Anderson localization! - Properties:

rel = (ergodicity broken) Self-generated disorderNo d.c. transport / no diffusion in thermodynamic limit

Classical frustration plays no role!Glass because of quantum effects, NOT despite of them ( quantum spin glass, superglass)

Model: inhibited hopping in 1dH0 : non-ergodic T : potentially restores ergodicity Anderson:Many-body quantum glass:Inhibited hopping modelEigenstate at = 0 Aim: show that for c many-body localized quantum glass!

Signatures of quantum glass, many-body localizationNo thermalization

Persistence of spatial inhomogeneity in long-time average

Spontaneous breaking of translational invariance

No d.c. transport

Essential ingredientsH0 is fully localized: has an extensive set of local, conserved operators integrable system

Simple arguments for localizationHybridization between eigenstates with different i,l;m is suppressed, since energies El;m~ J >>

Expect: Hopping ~ in the lattice labeled with i,l;m is localized

BUT: Caveat: El;m does not depend on site i! Spectrum of H0 is extensively degenerate resonant delocalization?

Summary of the argumentDegeneracies are lifted at low orders in perturbation theory in

Near-degeneracies are much more weakly coupled than their level splitting (typically)

Rare resonances occur locally, but dont percolate

Perturbation theory [in lifted basis] converges for

Eigenstate perturbation theoryPerturb localized eigenstates of H0, expand in That is: Choose basis of fixed barrier positions, do not fix momentum! Lifting of degeneracies?

Resonances and hybridization3 degenerate configurations (with li = lj 1): Degeneracy is lifted by hybridization at order O() But: Most configurations remain degenerate at first order.

Lifting degeneracies at order O(2)Generic lifting mechanism: Virtual barrier hops: ~2/J In general two eigenstates dont hybridize unless they can be connected by only two barrier hops (matrix element ~ 2/J).

Lifting degeneracies at order O(2)But: some eigenstates remain exactly degenerate at order O(2):

Lifting degeneracies at order O(2)But: some eigenstates remain exactly degenerate at order O(2):In random eigenstates their density is small ~ 0.034 (barrier=1/2).

Eigenstates with same shift ~2/J are connected by matrix elements ~ n~30

Dynamic localizationIn typical, random eigenstates, perturbation theory converges at small !

Eigenstates are localized close to inhomogeneous eigenstates of H0

Initial inhomogeneities remain frozen in dynamics! [Expand the initial state in eigenstates and check!]

BUT: highly atypical, nearly periodic eigenstates hybridize over large distances and delocalize! Nevertheless, generic initial conditions have exponentially small weight on such eigenstates, and remain localized.

Independent, direct check?

Numerical verification?Make use of translational invariance!

Spontaneous symmetry breakingBreak translational invariance by very weak disorder WCheck eigenstate inhomogeneitySpontaneous dynamical breaking of translational invariance = self-induced many-body localization;

Susceptibility to disorder?All barrierL barriers must be moved to hybridize the degenerate barrier configurations (rigid rotations)

exponentially large mass, exponentially small splitting of the band, expoentially strong response to disorderAnalytics: Disorder-response dominated by mixing of L nearly degenerate momentum states

Susceptibility to disorder?Linear responseSlope: SusceptibilityLAnalytics: Disorder-response dominated by mixing of L nearly degenerate momentum states

Susceptibility to disorder?Many-body problemNon-intercating particles>>Comparison with free particles

Susceptibility to disorder?Glass transition?

No exponential sensitivity to W!Very rough estimate:Ideal quantum glass exists in a substantial range 0 < < c !

For the expertsRecent conjectures (Huse& Oganesyan; Serbyn&Papic&Abanin 2013)Many-body localization Existence of an extensive set of local conserved quantities, as in integrable models.These conservation laws prohibit thermalization.??In disorder-free quantum glasses, such operators seem not to exist for > 0 .They seem to be inconsistent with rare delocalized states.

When is the above conjecture correct? What does it imply when it does not hold?

ConclusionsSelf-generated disorder [initial conditions] can induce many-body Anderson localization in closed, disorder-free quantum systems

Here: Manybody localization = spontaneous dynamical breaking of translational symmetry

Genuine, ideal dynamic quantum glass Induced by quantum effects BUT: requires coherence = absence of noise/dephasing In reality: ergodicity breaking up to time scale controlled by remaining dissipative processes.

Open questionsNature of the glass / localization transition as function of ?

Temperature dependence? Delocalization due to reduced disorder at low T?

Relation with Anderson orthogonality catastrophy (if any)?Localization and non-thermalization in strongly correlated systems?

Interplay between manybody Anderson localization and classical frustration in phase space? Interacting insulator-to-conductor transitions?