Mott-Berezinsky formula, instantons, and integrability

Ilya A. Gruzberg

In collaboration with Adam Nahum (Oxford University) Euler Symposium, Saint Petersburg, Russia, July 8 th , 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 D. Khmelnitskii ‘82 G. Bergmann ‘84 • Superposition: add probability amplitudes, then square R. P. Feynman ‘48 • Interference term vanishes for most pairs of paths

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 D. J. Thouless ‘73 • All states are localized in a quasi-1D wire with channels with localization length D. J. Thouless ‘77 • Large diffusive regime for allows to map the problem to a 1D supersymmetric sigma model (not specific to 1D) K. B. Efetov ‘83 • 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

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 gives • Collecting everything together 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 ( -dependent) corrections with lower powers of 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