Mott-Berezinsky formula, instantons, and integrability

1
Mott-Berezinsky formula, instantons, and
integrability

• Ilya A. Gruzberg
• In collaboration with Adam Nahum (Oxford
  University)

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Anderson localization

• Single electron in a random potential (no
  interactions)
• Ensemble of disorder realizations statistical
  treatment
• Possibility of a metal-insulator transition
  (MIT) driven by disorder
• Nature and correlations of wave functions
• Transport properties in the localized phase
• - DC conductivity versus AC
  conductivity
• - Zero versus finite temperatures

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Weak localization

• Qualitative semi-classical picture
• Superposition add probability amplitudes, then
  square
• Interference term vanishes for most pairs of
  paths

D. Khmelnitskii 82 G. Bergmann 84
R. P. Feynman 48
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Weak localization

• Paths with self-intersections
• - Probability amplitudes
• - Return probability
• - Enhanced backscattering
• Reduction of conductivity

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Strong localization
P. W. Anderson 58

• As quantum corrections may reduce
  conductivity to zero!
• Depends on nature of states at Fermi energy
• - Extended, like plane waves
• - Localized, with
• - localization length

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Localization in one dimension

• All states are localized in 1D by arbitrarily
  weak disorder
• Localization length mean free path
• All states are localized in a quasi-1D wire with
  channels
• with localization length
• Large diffusive regime for
  allows to map the problem
• to a 1D supersymmetric sigma model (not
  specific to 1D)
• Deep in the localized phase one can use the
  optimal fluctuation
• method or instantons (not specific to 1D)

N. F. Mott, W. D. Twose 61
D. J. Thouless 73
D. J. Thouless 77
K. B. Efetov 83
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Optimal fluctuation method for DOS
I. Lifshitz, B. Halperin and M. Lax, J. Zittartz
and J. S. Langer

• Tail states exist due to rare fluctuations
• of disorder
• Optimize to get
• DOS in the tails
• Prefactor is given by fluctuation integrals near
• the optimal fluctuation

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Mott argument for AC conductivity
N. F. Mott 68

• Apply an AC electric field to an Anderson
  insulator
• Rate of energy absorption due to transitions
  between states (in 1D)
• Need to estimate the matrix element

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Mott argument

• Consider two potential wells that support
  states at
• The states are localized, and their overlap
  provides mixing between
• the states
• Diagonalize
• Minimal distance

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Mott-Berezinsky formula

• Finally
• In dimensions the wells can be separated in
  any direction which gives
• another factor of the area
• First rigorous derivation has been obtained only
  in 1D
• For large positive energies (so that
  ) Berezinsky invented a
• diagrammatic technique (special for 1D) and
  derived Mott formula in the
• limit of weak disorder

V. L. Berezinsky 73
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Supersymmetry and instantons
R. Hayn, W. John 90

• Write average DOS and AC conductivity in terms
  of Greens functions,
• represent them as functional integrals in a
  field theory with a quartic action
• For large negative energies (deep in the
  localized regime) the action is
• large, can use instanton techniques saddle
  point plus fluctuations near it
• Many degenerate saddle points zero modes
• Saddle point equation is integrable, related to
  a stationary Manakov
• system (vector nonlinear Schroedinger equation)
• Integrability is crucial to find exact
  two-instanton saddle points,
• to control integrals over zero modes, and
  Gaussian fluctuations
• near the saddle point manifold
• Reproduced Mott formula in the weak disorder
  limit

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Other results in 1D and quasi 1D

• Other correlators involving different wave
  functions
• Correlation function of local DOS in 1D
• Correlation function of local DOS in quasi1D
  from sigma model
• Something else?

L. P. Gorkov, O. N. Dorokhov, F. V. Prigara 83
D. A. Ivanov, P. M. Ostrovsky, M. A. Skvortsov 09
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Our model

• Hamiltonian (in units )
• Disorder
• Same model as used for derivation of DMPK
  equation
• Assumptions
• - saddle point technique requires
• - small frequency
• - weak disorder

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Some features and results

• Saddle point equations remain integrable,
  related to stationary
• matrix NLS system
• Two-soliton solutions are known exactly
• (Two-instanton solutions that we need can also
  be found by an ansatz)
• The two instantons may be in different
  directions in the channel space,
• hence there is no minimal distance between
  them!
• Nevertheless, for
  we reproduce Mott-Berezinsky result
• Specifically, we show

F. Demontis, C. van der Mee 08
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Calculation of DOS setup

• Average DOS
• Greens functions as functional integrals over
  superfields
• is a vector (in channel space) of
  supervectors

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Calculation of DOS disorder average

• After a rescaling
• (In the diffusive case (positive energies) one
  proceeds by decoupling the
• quartic term by Hubbard-Stratonovich
  transformation, integrating out the
• superfields, and deriving a sigma model)

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Calculation of DOS saddle point

• Combine bosons
  into
• Rotate integration contour
• The saddle point equation
• Saddle point solutions (instantons)
• The centers and the directions of
  the instantons are
• collective coordinates (corresponding to zero
  modes)
• The classical action
  does not depend on them

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Calculation of DOS fluctuations

• Expand around a classical configuration
• has a zero mode corresponding
  to rotations of
• has a zero mode
  corresponding to translations of ,
• and a negative mode with eigenvalue

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Calculation of DOS fluctuation integrals

• Integrals over collective variables
• Integrals over modes with positive eigenvalues
  give scattering determinants
• Grassmann integrals give the square of the zero
  mode of
• Integral over the negative mode of gives
• Collecting everything together gives
  given above

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Calculation of the AC conductivity

• is much more involved due to appearance of
  nearly zero modes
• Need to use the integrability to determine exact
  two-instanton solutions
• and zero modes
• Surprising cancelation between fluctuation
  integrals over nearly zero modes
• and the integral over the saddle point manifold
• In the end get the Mott-Berezinsky formula plus
  ( -dependent) corrections
• with lower powers of

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Conclusions

• We present a rigorous and conreolled derivation
  of Mott-Berezinsky formula
• for the AC conductivity of a disordered
  quasi-1D wire in the localized tails
• Generalizations to higher dimensions
• Generalizations to other types of disorder
  (non-Gaussian)
• Relation to sigma model