1. Many-body dynamics and localizations
V. Oganesyan
D. Huse
A. Pal and S. Mukerjee

2. 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

3. 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

4. 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…

5. “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) – 80’s
Original motivation: undetectable spin diffusion

6. Hole burning -- long lived localized excitations of donor spins
ENDOR method for MEASURING the Hamiltonian (G.Feher, 50s)

7. Variable range hopping conduction
energy
 E
real space
Assume  E can be absorbed/emitted by the “bath” 

8. VRH from interactions? ? energy  E real space
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…

9. 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= <a| interaction |b>
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.
a =
b =

10. MIT phase diagram
PRB 75, (2007) Vc
Tc(V) T
insulator
=0
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

11. No thermodynamic signatures near infinite temperatureVc
=0 V
Free energy is analytic
V=0 – 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)

12. Statistical theory of quantum transport
Evaluate (,T) numerically (and exactly) for a finite system
understand finite size effects, e.g. separately in distributions of j’s and E’s
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, (2006)

13. Example: conductivity of a clean metal
Size effects and universal features:
Statistical noise
Wigner deficit
hydrodynamic singularity:
“universal” Urbach tails
at high (!) frequencies:
T 
PRB 73, (2006)

14. Metal-insulator transition at infinite T : conductivity
1/ 1
interaction
Disorder 1 2 3
clean
dirty
insulator
metal
Anderson
Metal
=0
=0 V  T 2  T
Very strong finite-size effects persist over a wide frequency range (colors--sizes) – difficult to be quantitative 3 

15. Spectral statistics (3D Anderson)
localized and extended states “look” different …
Exact
energies  r
vs.
|n>
|n+1>
|n>
En+3
En+2
En+1 En
|n+1>
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.

16. “One parameter” finite size scaling
Universality of gap statistics:
consider n=En+1-En
and
r = <2>/<>2
Can be used
like a Binder
cumulant
e.g. in
Monte Carlo
of 3D Ising model
RMT
Poisson
r=4/
r=2 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

17. 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

18. Many-body DOS: universality of a haystack
Average level spacing varies rapidly, Hamiltonian is NOT a “random matrix”, it’s local, i.e VERY sparse
Complication:
...but a simplification too!
14 sites
12 sites
10 sites
8 sites
6 sites
DOS is “thermodynamic”
already for very few particles, e.g. it obeys the central limit theorem

19. Solution: two-gap distribution function
Instead consider a dimensionless number constructed from two adjacent gaps:
rn=min (n, n+1)/max (n, n+1)
Two universal distributions of r can be identified corresponding to Wigner-Dyson and Poisson stats
PRB 75, (2007)
Critical
distribution?
Metal
Insulator
p(r)=2/(1+r)2
More simply <r> = 2 log 2 -1 in the insulator,
while <r>~ 0.53 in the diffusive phase, <r>c=?

20. L=16 L=8 w=“dirt” Crossover sharpens (but drifts) RMT value ~0.53
Critical stats?
Poisson value 2log2-1
w=“dirt”
PRB 75, (2007)

21. <r> L=16 L=8 “dirt” More on drifts Sizes vs. drifts: L  wc 10 2
10 2
12 1
14 ~.5
So far so good: the
drift is slowing but
would be nice
to get rid of drifts
altogether—should
be possible if caused by “irrelevant” operators, e.g. by tweaking H and/or r…
so far only minor
improvement
<r>
L=16
L=8
“dirt”
PRB 75, (2007)

22. 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...

23. 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.

24. Another view of localization: entanglement (entropy)B
size
strong
disorder
expected slope (ala Boltzmann) = - log 2
Weak
disorder
<A log A>

25. 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, (2006) )….
or is there classical localization?!
Two extreme regimes:
J=0 – each spin precesses in it’s own random field hj (orientation and magnitude) – no transport
hj=0 – 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

26. The answer J D Clean limit good diffusion
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

27. 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 <jj>
“Exact simulation” – only roundoff errors

28. Optical conductivity, small J’s
uncorrelated spins
* ~ J
few spin contribution 

29. Optical conductivity, small J’s
Low freq. limit?
powerlaws?
/J

30. 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
What’s different here? – nonlinearity, perhaps

31. Diffusive regime: finite size/time effects
J=0.4,0.32,0.17
Red data – finite size effects in small rings (10~20 spins)
C=10 throughout

32. 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

33. Semilocal Kubo: localized “noise”
t=1/J
t=0

34. Longer times: diffusion and bottlenecks
t >>1/J
1 / t  0

35. 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

36. 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?