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Mott-Berezinsky formula, instantons, and integrability

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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


1
Mott-Berezinsky formula, instantons, and
integrability
  • Ilya A. Gruzberg
  • In collaboration with Adam Nahum (Oxford
    University)

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2
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

3
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
4
Weak localization
  • Paths with self-intersections
  • - Probability amplitudes
  • - Return probability
  • - Enhanced backscattering
  • Reduction of conductivity

5
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

6
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
7
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

8
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

9
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

10
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
11
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

12
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
13
Our model
  • Hamiltonian (in units )
  • Disorder
  • Same model as used for derivation of DMPK
    equation
  • Assumptions
  • - saddle point technique requires
  • - small frequency
  • - weak disorder

14
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
15
Calculation of DOS setup
  • Average DOS
  • Greens functions as functional integrals over
    superfields
  • is a vector (in channel space) of
    supervectors

16
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)

17
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

18
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

19
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

20
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

21
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
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