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Mott-Berezinsky formula, instantons, and integrability


Mott-Berezinsky formula, instantons, and integrability Ilya A. Gruzberg In collaboration with Adam Nahum (Oxford University) Euler Symposium, Saint Petersburg, Russia ... – PowerPoint PPT presentation

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Title: Mott-Berezinsky formula, instantons, and integrability

Mott-Berezinsky formula, instantons, and
  • Ilya A. Gruzberg
  • In collaboration with Adam Nahum (Oxford

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Anderson localization
  • Single electron in a random potential (no
  • Ensemble of disorder realizations statistical
  • 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
  • - Zero versus finite temperatures

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

D. Khmelnitskii 82 G. Bergmann 84
R. P. Feynman 48
Weak localization
  • Paths with self-intersections
  • - Probability amplitudes
  • - Return probability
  • - Enhanced backscattering
  • Reduction of conductivity

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

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
  • 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
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

Mott argument for AC conductivity
N. F. Mott 68
  • Apply an AC electric field to an Anderson
  • Rate of energy absorption due to transitions
    between states (in 1D)
  • Need to estimate the matrix element

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

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
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

Other results in 1D and quasi 1D
  • Other correlators involving different wave
  • 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
Our model
  • Hamiltonian (in units )
  • Disorder
  • Same model as used for derivation of DMPK
  • Assumptions
  • - saddle point technique requires
  • - small frequency
  • - weak disorder

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
  • Nevertheless, for
    we reproduce Mott-Berezinsky result
  • Specifically, we show

F. Demontis, C. van der Mee 08
Calculation of DOS setup
  • Average DOS
  • Greens functions as functional integrals over
  • is a vector (in channel space) of

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)

Calculation of DOS saddle point
  • Combine bosons
  • 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
  • The classical action
    does not depend on them

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

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

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

  • 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
  • Relation to sigma model
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