Manybody dynamics and localizations

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Many-body dynamics and localizations

• V. Oganesyan
• D. Huse
• A. Pal and S. Mukerjee

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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

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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
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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

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• 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) 80s
• Original motivation undetectable spin diffusion

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Hole burning -- long lived localized excitations
of donor spins ENDOR method for MEASURING the
Hamiltonian (G.Feher, 50s)
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Variable range hopping conduction
energy
real space
? E
Assume ? E can be absorbed/emitted by the bath ?
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VRH from interactions?
energy
?
real space
? E
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
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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 lta interaction bgt 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.
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MIT phase diagram
PRB 75, 155111 (2007)
Vc
Tc(V)
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
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No thermodynamic signatures near infinite
temperature
Vc
?0
V
Free energy is analytic
V0 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)
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Statistical theory of quantum transport

• Evaluate ?(?,T) numerically (and exactly) for a
  finite system
• understand finite size effects, e.g. separately
  in distributions of js and Es
• 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)
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Example conductivity of a clean metal
?
Size effects and universal features Statistical
noise Wigner deficit hydrodynamic
singularity universal Urbach tails at high
(!) frequencies
PRB 73, 035113 (2006)
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Metal-insulator transition at infinite T
conductivity
T?
1/?
1
Metal
?0
?0
V
?
T?
2
?
T?
3
Very strong finite-size effects persist over a
wide frequency range (colors--sizes) difficult
to be quantitative
?
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Spectral statistics (3D Anderson)

• localized and extended states look different

Exact energies
n1gt
ngt
ngt
En3
En2
En1
En
n1gt
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.
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One parameter finite size scaling
1/L

• Universality of gap statistics
• consider ?nEn1-En
• and
• r lt?2gt/lt?gt2

Can be used like a Binder cumulant e.g. in Monte
Carlo of 3D Ising model
RMT
Poisson
r4/?
r2
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
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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

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Many-body DOS universality of a haystack
Average level spacing varies rapidly, Hamiltonian
is NOT a random matrix, its local, i.e VERY
sparse
Complication
...but a simplification too!
DOS is thermodynamic already for very few
particles, e.g. it obeys the central limit theorem
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Solution two-gap distribution function

• Instead consider a dimensionless number
  constructed from two adjacent gaps
• rnmin (?n, ?n1)/max (?n, ?n1)
• 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/(1r)2

• More simply ltrgt 2 log 2 -1 in the insulator,
• while ltrgt 0.53 in the diffusive phase, ltrgtc?

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Crossover sharpens (but drifts)
RMT value 0.53
L16
L8
Critical stats?
Poisson value 2log2-1
wdirt
PRB 75, 155111 (2007)
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More on drifts

• Sizes vs. drifts
• L ? wc
• 10 2
• 12 1
• 14 .5
• So far so good the
• drift is slowing but
• would be nice
• to get rid of drifts
• altogethershould
• be possible if caused by irrelevant operators,
  e.g. by tweaking H and/or r
• so far only minor
• improvement

ltrgt
L16
L8
dirt
PRB 75, 155111 (2007)
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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...

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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.

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Another view of localization entanglement
(entropy)
B
A
size
strong disorder
expected slope (ala Boltzmann) - log 2
Weak disorder
lt?A log ?Agt
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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
• J0 each spin precesses in its own random
  field hj (orientation and magnitude) no
  transport
• hj0 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

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The answer
Clean limit good diffusion
J
D
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
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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 ltjjgt
• Exact simulation only roundoff errors

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Optical conductivity, small Js
?/J2
uncorrelated spins
? J few spin contribution
?
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Optical conductivity, small Js
?/J2
Low freq. limit? powerlaws?
?/J
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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 Whats
different here? nonlinearity, perhaps
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Diffusive regime finite size/time effects
J0.4,0.32,0.17
Red data finite size effects in small rings
(1020 spins) C10 throughout
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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

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Semilocal Kubo localized noise
t1/J
t0
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Longer times diffusion and bottlenecks
t gtgt1/J
1 / t ? 0
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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

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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?