The Replica Limit of Unitary Matrix Integrals
We investigate the replica trick for the microscopic spectral density, $ρ_s(x)$, 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 $ν$. 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 $ρ_s(x)$ 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 $ν=0$ we recover the small-$x$ logarithmic singularity of the resolvent by means of the replica trick. For half integer $ν$, 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.