Sarang Gopalakrishnan

Response'of'many,body'localized'
systems
Sarang Gopalakrishnan (Caltech)
Michael(Knap((TU(Munich)
Kartiek Agarwal,(Eugene(Demler (Harvard)
Vedika Khemani,(David(Huse (Princeton)
Markus(Mueller((Paul(Scherrer Inst.)
Rahul(Nandkishore (CU(Boulder)
Sid(Parameswaran (UC(Irvine)
How'does'an'isolated'system'equilibrate?
Classical picture: energy gets equipartitioned, system forgets
Quantum picture: unitary time evolution, no forgetting?
Equilibration'='spreading'out'in'Fock'space
ky
In quantum-mechanical language:
kx
Eigenstates are superpositions of all configurations; thus
“delocalized” over energy shell
i
0
0
|n, ni + e |n , n i + e
i
0
|n00 , n00 i + . . .
“Local measurements” insensitive to phases, look like
microcanonical ensemble (“eigenstate thermalization”)
Fock,space'hopping'model
|n1 , n1 i
|n4 , n4 i
|n5 , n5 i
|n3 , n3 i
|n2 , n2 i
|n0 , n0 i
Collisions(of(particles(correspond(to(FockKspace(hopping
Q:(Are(there(variants(of(this(hopping(model(with(localization(transitions?
A'simple'disordered'Fock space
H=
X
hi
z
i
+J
z z
i i+1
i
Eigenstates with no transverse field: | "##"" . . .i
+✏
x
i
| """"" . . .i
Energies are random numbers (but correlated)
Transverse field causes “hopping” on this
network
However, hopping might be too weak to
overcome randomness: “many-body localization”
| ""#"" . . .i
| "#""" . . .i
| "##"" . . .i
Eigenstate (de)localization'transition
Thermal/ergodic phase
Many-body localized phase
Eigenstates are spread out over
configurations:
Eigenstates are “dressed”
classical strings
Local measurements see thermal
behavior
Highly entangled (volume law)
Local observables vary strongly
Weak entanglement (area law)
MBL1transition1=1breakdown1of1
statistical1mechanics
disorder
strength
Eigenstate'perspective'vs.'experiment
Experiments do not look at eigenstates,
only dynamics
Eigenstates are sensitive to
infinitesimal coupling to bath
What is actually measured in expt?
• Far-from-eqm dynamics
• On finite timescales
• In imperfectly isolated systems
Are we doomed?
temperature
This'is'a'standard'theme'in'QPTs…
critical(fan
Nearly(FM(regime
Nearly(PM(regime
Transverse(field
max(frequency,(coupling(to(bath)
This'is'a'standard'theme'in'QPTs…
critical(fan
thermal(regime
nearly(localized(regime
disorder
Kubo'ac'conductivity
energy
sample
time
parallel plate capacitors
create oscillatory field
Mott, Pollak, Efros, Geballe...
Isolated'resonances'in'insulators
x
Splitting of resonance ~ hybridization matrix element ~ exp(-x/ξ)
Resonance “responds” only at frequencies ω ~ exp(-x/ξ): ω specifies x
Current matrix element ~ distance / transit time across resonance ~ xω
Resonance,counting
Collective resonances are “feline”
. . . ""## . . . $ . . . ##"" . . .
Small/local resonances have large splittings,
large/collective resonances have small splittings
MBL adaptation of Mott’s (1968) argument for a.c. conductivity:
• Look for many-body resonances with splitting ω
• These rearrange n nearby spins such that ! ⇠ W exp( n/⇣)
• # of n-spin rearrangements at infinite temperature ~ 2n [cf. polynomial growth usually]
• # of n-spin rearrangements at frequency ω: N (!) ⇠ ! ⇣ log 2
• AC conductivity: T (!) ⇠ ! 2 N (!) ⇠ ! 2
QPT
!1
deep MBL
!0
From'conductivity'to'quench'dynamics
FT of spectral function is autocorrelator
Meaning of autocorrelator:
Given particle at site i at time zero
How likely is it to be there at t?
Quench dynamics
Put a particle in at time zero
Ask if it is still there at time t
Late-time decay of contrast due to slow
resonances decaying
hO(t)i
O1 ⇠ t
1
QPT
!1
deep MBL
!0
Does'linear'response'even'work?
Linear-response picture:
Response dominated by resonant pairs
This depends on rotating-wave approx.
Δ
Δ
ω
A
A
ω
Response'crossover:'absence'of'dc'limit
Δ
Δ
ω
A
A
ω
Power'laws'in'nonlinear'response
Heating at time t:
LZ transitions with rate 1/t
Rate of nth-order rearrangement
n
⇠ W 2 exp( 2n/⇣)/A
Phase space: (A/W) 2n
Result: # of rearrangements at time t:
A1
/2
t
/2
max(frequency,(coupling(to(bath)
What'if'bath'sets'the'cutoff?
critical(fan
thermal(regime
nearly(localized(regime
disorder
MBL'+'baths
Two energy scales of bath:
• System-bath coupling g
• Bath “bandwidth” or
correlation time Δ
Case of slowly fluctuating baths
(e.g. nuclear spins, electron spins…)
• Bath only gives energy in small increments
• To exchange energy with bath, must go to
high order
2 2 1
• System-bath decay rate ~ g
Summary
Response functions in the MBL phase behave as continuously varying power
laws of time, frequency, drive amplitude, bath bandwidth…
All these exponents are simple functions of the one parameter φ
φ -> 0 deep in the MBL phase
φ -> 1 at MBL-thermal transition
No clean dc limit for response functions: linear response breaks down as drive
frequency goes to zero
Crossover between resonances and adiabatic rearrangements