Quasi-diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

TL;DR

This paper investigates a 3D supersymmetric hyperbolic sigma model, revealing a diffusive phase at low temperatures through advanced Green's function estimates and supersymmetry Ward identities, shedding light on localization phenomena.

Contribution

It introduces a novel lattice field model reflecting Anderson localization and delocalization, employing supersymmetry and Green's function techniques to analyze phase behavior.

Findings

Existence of a diffusive phase in 3D at low temperatures

Use of supersymmetry Ward identities for analysis

Green's function estimates underpin phase transition insights

Abstract

We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a `diffusive' phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green's functions and a family of Ward identities coming from internal supersymmetry.