Development of (4-epsilon)-dimensional theory for the density of states near the Anderson transition

TL;DR

This paper develops a (4-epsilon)-dimensional theoretical framework to analyze the density of states near the Anderson transition, addressing renormalization issues and eliminating spurious poles to improve understanding of the transition's properties.

Contribution

It introduces an epsilon-expansion approach for the Anderson transition problem, systematically handling different dimensional theories and resolving the spurious pole issue.

Findings

Elimination of the spurious pole in the density of states

Asymptotically exact approximations for different dimensions

Shift of the phase transition point in the complex plane

Abstract

The density of states for the Schroedinger equation with a Gaussian random potential is determined by the functional integral corresponding to the phi^4 theory with a `wrong' sign of the interaction constant. The special role of the dimension d=4 for such a problem can be seen from different viewpoints but is fundamentally determined by the renormalizability of the theory. The construction of an epsilon-expansion in direct analogy with the phase-transition theory gives rise to the problem of a `spurious' pole. To solve this problem, a proper treatment of the factorial divergency of the perturbation series is necessary. Simplifications arising in high dimensions can be used for the development of a (4-epsilon)-dimensional theory, but this requires successive consideration of four types of theories: a nonrenormalizable theory for d>4, nonrenormalizable and renormalizable theories in the logarithmic situation (d=4), and a super-renormalizable theory for d<4. An approximation is found for each type of theory giving asymptotically exact results. The qualitative effect is the same in all four cases and consists in a shifting of the phase transition point in the complex plane. This results in the elimination of the `spurious' pole and in regularity of the density of states for all energies. A discussion is given of the calculation of high orders of perturbation theory and a perspective of the epsilon-expansion for the problem of conductivity near the Anderson transition.