Anderson localization for a supersymmetric sigma model

TL;DR

This paper proves localization at high temperatures in a supersymmetric sigma model related to Anderson localization, using Ward identities from supersymmetry, extending understanding of phase transitions in disordered systems.

Contribution

It establishes high-temperature localization for a 3D supersymmetric sigma model, providing rigorous proof using supersymmetry techniques.

Findings

Localization at high temperatures for all dimensions d≥1

Existence of a diffusive phase at low temperatures in 3D

Use of Ward identities from supersymmetry in analysis

Abstract

We study a lattice sigma model which is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices in 3D. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. The existence of a diffusive phase in 3 dimensions was proved for low temperatures. Here we prove localization at high temperatures for any dimension $d\geq 1$. Our analysis uses Ward identities coming from internal supersymmetry.