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Mathematical Physics Seminar

Ilya Gruzberg - Disordered wires, integrability, instantons, and zero modes

Date & time: Wednesday, 07 October 2020 at 10:45AM - 11:45AM

Ilya Gruzberg – Ohio State University

Wednesday, October 7, 10:45AM


"Disordered wires, integrability, instantons, and zero modes"

 Non-interacting electrons in disordered wires are Anderson-localized, and the DC conductivity of a wire vanishes at T=0. However, the AC conductivity is non-vanishing, and its asymptotic form at low frequencies was obtained in a qualitative way by Mott. This form was later derived by Berezinsky for a strictly one-dimensional (1D) disordered wire. Then Hayn and John re-derived the Mott-Berezinsky formula applying instanton techniques to deeply localized states in the Lifshits tails. We extend the instanton approach to the case of quasi-1D wires with arbitrary number of channels. The extension requires us to find exact two-instanton solutions of an integrable time-independent matrix nonlinear Schrodigner equation, as well as to exactly evaluate the fluctuation determinant around the two-instanton solutions. The determinant contains multiple zero modes, and we develop techniques for dealing with them. We demonstrate quite generally that the contribution of the zero modes to the fluctuation determinant exactly cancels the Jacobian that appears when the collective variables are introduced. These results greatly simplify the derivation of the Mott-Berezinsky formula for quasi-1D wires, as well as corrections to it, and other physical correlation functions.

G. M. Falco, A. A. Fedorenko, and I. A. Gruzberg, On functional determinants of matrix differential operators with multiple zero modes, J. Phys. A. 50, 485201 (2017).
 G. M. Falco, A. A. Fedorenko, and I. A. Gruzberg, Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law, Europhysics Letters 120, 37004 (2017).
A. Nahum and I. A. Gruzberg, Wave function correlations and low-frequency Mott-Berezinskii conductivity of quasi-one dimensional wires, in preparation.

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