What to calculate

1
Theory of many-body localization
D. M. Basko I. L. Aleiner B. L.
Altshuler Columbia University in the City of New
York
Annals of Physics 321, 1126 (2006)
2
Outline

• 1. Introduction
• models of disorder
• single-particle eigenstates
• Anderson localization
• mobility edge
• hopping conduction
• Effect of electron-electron interaction
• energy conservation problem
• role of the temperature T
• finite-temperature metal-insulator transition
• 3. Effect on nonlinear conduction
• 4. Technical details

3
Single-electron disorder
1.
random potential
dispersion in the clean system
2. Anderson model
dispersion in the clean system
(one can consider an arbitrary lattice)
4
Single-particle states
extended
localized
dc conductivity
thermal population of extended states
all eigenstates are localized
5
d1 all states are localized d2 the same
(?) dgt2 mobility edge
Anderson model (tight-binding, dgt2)
disorder
all eigenstates are localized
DoS
Our main assumption all states are localized
6
Inelastic processes ) transitions between
localized states
?
energy mismatch
?
Level spacing in the localization volume
DoS per unit volume
localization volume
inelastic rate
(any mechanism)
7
Phonon-assisted hopping
?
?
energy difference can always be matched by a
phonon
without Coulomb gap
Mott formula
any bath with a continuous spectrum of
delocalized excitations down to E 0
mechanism- dependent prefactor
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Let us turn off electron-phonon coupling
Can electron-electron interaction produce hopping
conductivity?
Do electrons themselves make a suitable bath?

• Fleishman, Anderson, PRB 21, 2366 (1980)
• Shahbazyan, Raikh, PRB 53, 7299 (1996)
• Kozub, Baranovskii, Shlimak, Solid State Comm.
  113, 587 (2000)
• Nattermann, Giamarchi, Le Doussal, PRL 91,
  056603 (2003)
• Gornyi, Mirlin, Polyakov, cond-mat/0407305 v1

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No phonons, e-e interaction
weak short-range
dimensionless interaction strength (1)
localization volume
DoS per unit volume
level spacing
Matrix elements between localized wave functions
spatial cutoff
energy cutoff
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Energy conservation problem
emission of electron-hole pair ? pair collision
triple collision

• Lower temperatures ? harder to conserve energy
• Need to consider ALL many-electron processes
• to ALL orders of perturbation theory

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The answer is
e-e interaction is sufficient to cause real
transitions
the system is softened, but not
sufficiently NO RELAXATION !!!
finite-temperature metal-insulator transition
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Anderson localization in the many-body Fock space
(A., G., K., L., 1997)
many-body Fock states e-e interaction metal-insula
tor transition temperature
sites with random energies coupling between
sites Anderson transition coordination number
Systematic treatment of many-electron
transitions D. B., I. Aleiner, and B.
Altshuler, Ann. Phys. 321, 11261205 (2006)
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Many-body (de)localization
one-body many-body
localized in space whole volume
eigenstates
Electron-hole excitation
spectral function
0 extended
IPR
gt 0 localized
Developed metallic phase
microcanonical , canonical ) temperature
Insulating phase total energy Ek
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Many-body mobility edge
Large Ek ) high T extended states
interaction ! dephasing ! cutoff of WL (good
metal)
Fermi Golden Rule ! hopping (bad metal)
transition ! mobility edge
(volume)
No activation!
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Open questions
2. Will the transition occur in a boson system?
Strong e-e interaction ? Wigner crystallization
(elementary
excitations are phonons) Impurity potential ?
random pinning (phonon localization and loss
of the long-range order)
3. What is the relation to glasses?
Ergodicity breaking, freezing of relaxation
but Conventional glass transition ? statistics
of potential barriers Many-body localization ?
statistics of random forces How to count
many-electron excitations in a Coulomb system?
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Summary
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Possible experimental signatures (turn the
phonons back on!)
metallic conduction Joule heating, phonon cooling
critical enhancement one phonon ? many e-h pairs
phonon-assisted conduction
Signatures in nonlinear transport
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Electronic temperature from thermal balance
variable-range hopping
usual metal
nearest- -neighbor hopping
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Electronic temperature from thermal balance
weaker phonons
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Electronic temperature from thermal balance
weaker phonons
stronger field
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Electronic temperature from thermal balance
weaker phonons
stronger field
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Bistable I-V curve
necessary
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Exact spectral function
The spectral function shows how a single-particle
excitation is spread over exact eigenstates
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(?0)
one-particle spectral weight
exact spectral function
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(?2)
three-particle spectral weight
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(?4)
five-particle spectral weight
more dense
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(?6)
seven-particle spectral weight
n!1 continuous background?
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What to calculate?
(Anderson, 1958)
random quantity
metal
insulator
insulator
h!0
metal
h
????-??
h?ih?i
behavior for a given realization
probability distribution for a fixed energy
gt 0
metal
working criterion
0
insulator
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Two essential facts
spatial cutoff
energy cutoff
These facts can be modeled in different ways
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Self-consistent Born approximation
?
?
iterations of SCBA max number of particles in
the final state
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Stability of the metallic phase finite
broadening is self-consistent at high temperatures

as long as

(levels well resolved)

quantum kinetic equation for transitions
between localized states
(model-dependent)
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Stability of the insulating phase NO spontaneous
generation of broadening

• is always a solution
• linear stability analysis
• after iterations of SCBA equations

first
, then
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Stability of the insulating phase NO spontaneous
generation of broadening

• is always a solution
• linear stability analysis
• after iterations of SCBA equations

first
, then
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Instead of conclusion FAQ

• Q What if one takes Coulomb interaction?
• A
• (dipole
  transitions!)
• nothing changes,
• ask Levitov
• Q Can we have ?
• A Yes, if the range of is large.
• e-e VRH

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Our model localization volumes ! discrete grains