Mott-Berezinsky formula, instantons,
and integrability
Ilya A. Gruzberg
In collaboration with Adam Nahum (Oxford University)
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
Localization in one dimension
• All states are localized in 1D by arbitrarily weak disorder N. F. Mott, W. D. Twose ‘61
• Localization length = mean free path
• All states are localized in a quasi-1D wire with
with localization length
D. J. Thouless ‘73
channels
• 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)
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
• Apply an AC electric field to an Anderson insulator
N. F. Mott ‘68
• Rate of energy absorption due to transitions between states (in 1D)
• Need to estimate the matrix element
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
V. L. Berezinsky ‘73
• For large positive energies (so that
) Berezinsky invented a
diagrammatic technique (special for 1D) and derived Mott formula in the
limit of “weak disorder”
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
Supersymmetry and instantons
R. Hayn, W. John ‘90
• Write average DOS and AC conductivity in terms of Green’s 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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
Other results in 1D and quasi 1D
• Other correlators involving different wave functions
• Correlation function of local DOS in 1D
L. P. Gor’kov, O. N. Dorokhov, F. V. Prigara ‘83
• Correlation function of local DOS in quasi1D from sigma model
D. A. Ivanov, P. M. Ostrovsky, M. A. Skvortsov ‘09
• Something else?
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
Our model
• Hamiltonian (in units
)
• Disorder
• Same model as used for derivation of DMPK equation
• Assumptions:
- saddle point technique requires
- small frequency
- “weak disorder”
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
Some features and results
• Saddle point equations remain integrable, related to stationary
matrix NLS system
• Two-soliton solutions are known exactly
F. Demontis, C. van der Mee ‘08
(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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
Calculation of DOS: setup
• Average DOS
• Green’s functions as functional integrals over superfields
•
is a vector (in channel space) of supervectors
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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)
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
,
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
• Collecting everything together gives
gives
given above
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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 (
with lower powers of
-dependent) corrections
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011