Exact generating function of a zero-dimensional supersymmetric non-linear sigma model

TL;DR

This paper derives an exact generating function for a supersymmetric non-linear sigma model related to random matrices, revealing its dependence on only three invariants and impacting the study of Anderson transitions.

Contribution

It provides an exact analytical expression for the generating function, clarifying its invariance properties and implications for renormalization group analyses.

Findings

Dependence on only three invariant functions of the source

Recovery of known results in the literature

Implications for the functional renormalization group approach

Abstract

We compute exactly the generating function of a supersymmetric non-linear sigma model describ-ing random matrices belonging to the unitary class. Although an arbitrary source explicitly breaksthe supersymmetry, a careful analysis of the invariance of the generating function allows us to showthat it depends on only three invariant functions of the source. This generating function allows usto recover various results found in the literature. It also questions the possibility of a functionalrenormalization group study of the three-dimensional Anderson transition.