Ideal Quantum Glass Transitions:
Many-body localization without quenched disorder
Markus Müller
Mauro Schiulaz
TIDS 15 1-5 September 2013 Sant Feliu de Guixols

Motivation
G. 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 ??
L= 8 , 10 , 12

=
Systems defying thermodynamic equilibration
• Breaking of ergodicity (τ rel
→
•
Absence of full thermalization
∞)
How can this occur in general?

Routes to glassiness
Spin glasses + cousins: (Coulomb glasses?)
Usual classical ingredients: Disorder + Frustration
→ Barriers grow with size L τ rel
~ exp[L
θ
]
Structural glasses (viscous, supercooled liquids):
Steric frustration + self-generated disorder
→ growing timeand length-scales (Kirkpatrick, Thirumalai, Wolynes)
But: Eternal debate without conclusion:
Is there an
“ideal” glass transition at finite T:
Can τ 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:
• τ rel
= ∞ (ergodicity broken)
•
Self-generated disorder
•
No 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 1d
Many-body quantum glass:
H
0
: non-ergodic
T : potentially restores ergodicity
Anderson:
Inhibited hopping model Eigenstate at λ = 0
Aim: show that for λ ≤ λ c many-body localized quantum glass!

Signatures of quantum glass, many-body localization
•
No thermalization
•
Persistence of spatial inhomogeneity in longtime average
•
Spontaneous breaking of translational invariance
•
No d.c. transport

Essential ingredients
H
0 is fully localized: has an extensive set of local, conserved operators → “ integrable system”
γ

Simple arguments for localization
•
Hybridization between eigenstates with different τ i,l;m since energies E l;m is suppressed,
~ J >> λ
→
Expect: “Hopping ~
λ in the lattice labeled with τ i,l;m is localized”
•
BUT: Caveat: E l;m does not depend on site i!
→ Spectrum of H
0 is extensively degenerate → resonant delocalization?

Summary of the argument
•
Degeneracies are lifted at low orders in perturbation theory in λ
•
Near-degeneracies are much more weakly coupled than their level splitting (typically)
→ Rare resonances occur locally, but don’t percolate
→ Perturbation theory [in lifted basis] converges for λ << J
(same reasons as in systems with quenched disorder,
[Basko, Aleiner, Altshuler 2006, Imbrie&Spencer, unpub.])
→ Eigenstates are localized close to inhomogeneous eigenstates of H
0
→ Initial inhomogeneities remain frozen in dynamics!
→ Eigenstate thermalization hypothesis is not obeyed!

Eigenstate perturbation theory
Perturb localized eigenstates of H
0
, expand in λ
That is: Choose basis of fixed barrier positions, do not fix momentum!
Lifting of degeneracies?

Resonances and hybridization
3 degenerate configurations (with l i
= l j
± 1):
Degeneracy is lifted by hybridization at order O(λ)
But: Most configurations remain degenerate at first order.

Lifting degeneracies at order O(λ 2 )
Generic lifting mechanism: Virtual barrier hops: ΔΕ ~λ 2 /J
→ In general two eigenstates don’t hybridize unless they can be connected by only two barrier hops (matrix element ~ λ 2 /J).

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

Lifting degeneracies at order O(λ 2 )
But: some eigenstates remain exactly degenerate at order O(λ 2 ):
In random eigenstates their density is small ~ 0.034 (ρ barrier
=1/2).
→ Eigenstates with same shift ΔΕ ~λ 2 /J are connected by matrix elements ~ λ n~30 <<< λ 4 . Resonances are very rare!
→ In typical, random eigenstates, perturbation theory converges!

Dynamic localization
In typical, random eigenstates, perturbation theory converges at small λ!
→ Eigenstates are localized close to inhomogeneous eigenstates of H
0
→ 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 breaking
Break translational invariance by very weak disorder W
Check eigenstate inhomogeneity
Spontaneous dynamical breaking of translational invariance = self-induced many-body localization
;

Susceptibility to disorder
?
Analytics : Disorder-response dominated by mixing of L nearly degenerate momentum states
All ρ barrier
L 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 disorder

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

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

Susceptibility to disorder
?
Glass transition?
No exponential sensitivity to W!
Very rough estimate:
Ideal quantum glass exists in a substantial range 0 < λ < λ c
!

For the experts
Recent 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 λ > 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?

Conclusions
•
Self-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 questions
• Nature of the glass / localization transition as function of λ?
•
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?