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
Glasses
=
Systems defying thermodynamic
equilibration
• Breaking of ergodicity (τrel → ∞)
• 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 τrel ~ exp[Lθ]
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 τrel → ∞, 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:
• τrel = ∞ (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
Anderson:
Many-body quantum glass:
H0 : non-ergodic
T : potentially
restores ergodicity
Inhibited hopping model
Eigenstate at λ = 0
Aim: show that for λ ≤ λc 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
H0 is fully localized: has an extensive set of local, conserved
operators → “ integrable system”
γ
Simple arguments for localization
• Hybridization between eigenstates
with different τi,l;m is suppressed,
since energies El;m~ J >> λ
→ Expect: “Hopping ~λ in the lattice
labeled with τi,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 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 H0
→ Initial inhomogeneities remain frozen in dynamics!
Eigenstate perturbation theory
Perturb localized eigenstates of H0, 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 li = lj ±1):
Degeneracy is lifted by hybridization at order O(λ)
But: Most configurations remain degenerate at first order.
Lifting degeneracies at order O(λ2)
Generic lifting mechanism: Virtual barrier hops: ΔΕ ~λ2/J
→ In general two eigenstates don’t hybridize unless they can be
connected by only two barrier hops (matrix element ~ λ2/J).
Lifting degeneracies at order O(λ2)
But: some eigenstates remain exactly degenerate at order O(λ2):
Lifting degeneracies at order O(λ2)
But: some eigenstates remain exactly degenerate at order O(λ2):
In random eigenstates their density is small ~ 0.034 (ρbarrier=1/2).
→ Eigenstates with same shift ΔΕ ~λ2/J are connected by matrix
elements ~ λn~30 <<< λ4. 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 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 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 ρbarrierL 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
L
Linear response
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 < λc !
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?
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The ideal quantum glass transition

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