Hyperbolic Hubbard-Stratonovich transformation made rigorous

TL;DR

This paper rigorously formulates the hyperbolic Hubbard-Stratonovich transformation for disordered electron systems with non-compact orthogonal symmetry, clarifying its mathematical structure and addressing longstanding issues.

Contribution

It provides a precise mathematical formulation of the hyperbolic HS transformation in O(p,q) symmetry, confirming the sign-alternating sum structure of the integral.

Findings

The HS integral is a sign-alternating sum of integrals over disjoint domains.

The formulation confirms the proposal by Pruisken and Schaefer.

Addresses a long-standing issue in the mathematical foundation of disordered systems.

Abstract

We revisit a long standing issue in the theory of disordered electron systems and their effective description by a non-linear sigma model: the hyperbolic Hubbard-Stratonovich (HS) transformation in the bosonic sector. For time-reversal invariant systems without spin this sector is known to have a non-compact orthogonal symmetry O(p,q). There exists an old proposal by Pruisken and Schaefer how to do the HS transformation in an O(p,q) invariant way. Giving a precise formulation of this proposal, we show that the HS integral is a sign-alternating sum of integrals over disjoint domains.