Another critique of the replica trick

TL;DR

This paper critically reviews the use of the replica trick in disordered electron systems, highlighting its limitations to the perturbative regime and explaining the special case of beta=2 using the Duistermaat-Heckman theorem.

Contribution

It provides a critical analysis of the replica trick's validity and clarifies the exactness of saddle-point results for unitary symmetry using mathematical theorems.

Findings

Replica trick validity is limited to perturbative regimes

Exact saddle-point results for beta=2 are due to the Duistermaat-Heckman theorem

The paper clarifies the scope and limitations of previous approaches

Abstract

Kamenev and Mezard, and Yurkevich and Lerner, have recently shown how to reproduce the large-frequency asymptotics of the energy level correlations for disordered electron systems, by doing perturbation theory around the saddles of the compact nonlinear sigma model derived from fermionic replicas. We present a critical review of their procedure and argue that its validity is limited to the perturbative regime of large frequency. The miraculous exactness of the saddle-point answer for beta = 2 (unitary symmetry) in the universal limit, is shown to be a special feature due to the Duistermaat-Heckman theorem.