Equivalence of domains for hyperbolic Hubbard-Stratonovich transformations

TL;DR

This paper proves the validity of hyperbolic Hubbard-Stratonovich transformations for general non-compact symmetry groups, clarifying their relation to other variants and explaining sign factors in orthogonal cases.

Contribution

It extends the proof of hyperbolic HS transformations to all non-compact symmetry groups and relates these transformations to other variants through domain deformations.

Findings

Validated hyperbolic HS transformations for general non-compact groups.

Connected Pruisken-Schaefer transformations to other HS variants.

Explained origin of sign factors in orthogonal symmetry cases.

Abstract

We settle a long standing issue concerning the traditional derivation of non-compact non-linear sigma models in the theory of disordered electron systems: the hyperbolic Hubbard-Stratonovich (HS) transformation of Pruisken-Schaefer type. Only recently the validity of such transformations was proved in the case of U(p,q) (non-compact unitary) and O(p,q) (non-compact orthogonal) symmetry. In this article we give a proof for general non-compact symmetry groups. Moreover we show that the Pruisken-Schaefer type transformations are related to other variants of the HS transformation by deformation of the domain of integration. In particular we clarify the origin of surprising sign factors which were recently discovered in the case of orthogonal symmetry.