1. Mott-Berezinsky formula, instantons, and integrability
Ilya A. Gruzberg
In collaboration with Adam Nahum (Oxford University)
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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. Euler Symposium, Saint Petersburg, Russia, July 8th, 2011
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

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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
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
N. F. Mott ‘68
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

9. 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

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

11. 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

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. Gor’kov, O. N. Dorokhov, F. V. Prigara ‘83
D. A. Ivanov, P. M. Ostrovsky, M. A. Skvortsov ‘09
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

13. 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

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

15. 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

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)
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

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
Euler Symposium, Saint Petersburg, Russia, July 8th, 2011

21. 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