## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

3-15-1981

#### Publication Source

Physical Review B

#### Volume

23

#### Issue

6

#### Start Page

2640

#### Last Page

2673

#### DOI

10.1103/PhysRevB.23.2640

#### Abstract

A general field-theoretic formulation of the Anderson model for the localization of wave functions in a random potential is given in terms of n-component replicated fields in the limit n→0, and is analyzed primarily for spatial dimension d≥4. Lengths ξ_{1} and ξ_{2} associated with the spatial decay of correlations in the single-particle and two-particle Green's functions, respectively, are introduced. Two different regimes, the weak coupling and strong coupling, are distinguished depending on whether ξ_{1}^{−1} or ξ_{2}^{−1}, respectively, vanishes as the mobility energy, E_{c}, is approached. The weak-coupling regime vanishes as d→4+. Mean-field theory is developed from the uniform minimum of the Lagrangian for both the strong- and weak-coupling cases. For the strong-coupling case it gives the exponents v_{a}=1/4, γ_{a}=β_{a}=1/2, η=0, and μ=1, where β_{a} is the exponent associated with the density of extended states and μ is that associated with the conductivity. Simple heuristic arguments are used to verify the correctness of these unusual mean-field values. Infrared divergences in perturbation theory for the strong-coupling case occur for d<8, and an ε expansion (ε=8−d) is developed which is found to be identical to that previously analyzed for the statistics of lattice animals and which gives β_{a}=1/2−ε/12, η=−ε/9, v_{a}=1/4+ε/36, and μ=1−5ε/36. The results are consistent with the Ward identity, which in combination with scaling arguments requires that β_{a}+γ_{a}=1. The treatment takes account of the fact that the average of the on-site Green's function [G(x⃗ ,x⃗ ;E)]_{av} is nonzero and is predicated on this quantity being real, i.e., on the density of states vanishing at the mobility edge. We also show that localized states emerge naturally from local minima of finite action in the Lagrangian. These instanton solutions are analyzed on a lattice where the cutoff produced by the lattice constant leads to *lattice instantons* which exist for all d, in contrast to the case for the continuum model where instanton solutions seem not to occur for d>4. This analysis leads to a density of localized states ρ_{loc} satisfying 1nρ_{loc}~−E^{2} at large E and 1nρ_{loc}~−|E−E_{c}|^{−ζ} at the mobility edge, where for the weak-coupling case ζ=(1/2)(d−4) and for the strong-coupling case ζ=(d−2+η)v_{a}−2β_{a}=1/2+ε/18 for d<8 and ζ=(1/4)(d−6) for d>8. A brief discussion of the relationship between this work and the theories of localization below four dimensions is presented.

#### Recommended Citation

Harris, A.,
&
Lubensky, T. C.
(1981).
Mean-Field Theory and ε Expansion for Anderson Localization.
*Physical Review B,*
*23*
(6),
2640-2673.
http://dx.doi.org/10.1103/PhysRevB.23.2640

**Date Posted:** 12 August 2015

This document has been peer reviewed.