On Hubbard-Stratonovich Transformations over Hyperbolic Domains

TL;DR

This paper proves the validity of Hubbard-Stratonovich identities over hyperbolic domains and introduces a new identity for systems with chiral symmetry, offering alternative derivation methods for nonlinear sigma models.

Contribution

It extends the mathematical foundation of Hubbard-Stratonovich transformations to hyperbolic domains and presents a novel identity for chiral symmetric disordered systems.

Findings

Validated Hubbard-Stratonovich identities over hyperbolic domains

Introduced a new HS identity for chiral symmetric systems

Outlined an alternative derivation of the nonlinear sigma model

Abstract

We discuss and prove validity of the Hubbard-Stratonovich (HS) identities over hyperbolic domains which are used frequently in the studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with "chiral" symmetry. Apart from this we outline a way of deriving the nonlinear $\sigma$-model from the gauge-invariant Wegner $k-$orbital model avoiding the use of the HS transformations.