Localization Theory in Zero Dimension and the Structure of Diffusion Poles

TL;DR

This paper examines the diffusion pole structure in localized phases, revealing paradoxes in instanton analysis and emphasizing the need for self-consistent methods to determine the diffusion coefficient.

Contribution

It identifies fundamental paradoxes in the instanton approach to localization theory and clarifies the limitations of this method in zero-dimensional models.

Findings

The 1/f singularity is linked to high-order perturbation theory.

High-order behaviors of ^{RA} and U^{RA} are identical.

The instanton method yields only a 1/( + i) singularity with an undetermined parameter.

Abstract

The 1/[-i\omega + D(\omega, q)q^2] diffusion pole in the localized phase transfers to the 1/\omega Berezinskii-Gorkov singularity, which can be analyzed by the instanton method (M V. Sadovskii, 1982; J. L. Cardy, 1978). Straightforward use of this approach leads to contradictions, which do not disappear even if the problem is extremely simplied by taking zero-dimensional limit. On the contrary, they are extremely sharpened in this case and become paradoxes. The main paradox is specified by the following statements: (i) the 1/\omega singularity is determined by high orders of perturbation theory, (ii) the high-order behaviors for two quantities \Phi^{RA} and U^{RA} are the same, and (iii) \Phi^{RA} has the 1/\omega singularity, whereas U^{RA} does not have it. Solution to the paradox indicates that the instanton method makes it possible to obtain only the 1/(\omega + i\gamma) singularity, where the parameter \gamma remains indefinite and must be determined from additional conditions. This conceptually confirms the necessity of the self-consistent treatment for the diffusion coefficient that is used in the Vollhardt-Wolfle type theories.