The Replica Limit of Unitary Matrix Integrals

TL;DR

This paper examines the use of the replica trick in calculating the microscopic spectral density of the Euclidean QCD Dirac operator, revealing its limitations and advantages compared to supersymmetric methods.

Contribution

It demonstrates how the replica trick can recover the spectral density in certain cases and highlights its failure in others, advocating for supersymmetric methods as more reliable.

Findings

Replica trick reproduces exact results for half-integer topological charge.

Replica symmetric saddle-point expansion captures smooth spectral contributions.

Replica trick fails to produce unique spectral densities in general cases.

Abstract

We investigate the replica trick for the microscopic spectral density, $\rho_s(x)$, of the Euclidean QCD Dirac operator. Our starting point is the low-energy limit of the QCD partition function for $n$ fermionic flavors (or replicas) in the sector of topological charge $\nu$. In the domain of the smallest eigenvalues, this partition function is simply given by a U(n) unitary matrix integral. We show that the asymptotic behavior of $\rho_s(x)$ for $x \to \infty$ is obtained from the $n\to 0$ limit of this integral. The smooth contributions to this series are obtained from an expansion about the replica symmetric saddle-point, whereas the oscillatory terms follow from an expansion about a saddle-point that breaks the replica symmetry. For $\nu =0$ we recover the small-$x$ logarithmic singularity of the resolvent by means of the replica trick. For half integer $\nu$, when the saddle point expansion of the U(n) integral terminates, the replica trick reproduces the exact analytical result. In all other cases only an asymptotic series that does not uniquely determine the microscopic spectral density is obtained. We argue that bosonic replicas fail to reproduce the microscopic spectral density. In all cases, the exact answer is obtained naturally by means of the supersymmetric method.