The replica limit of unitary matrix integrals

Abstract

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 → ∞ is obtained from the n → 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.

Are Bosonic Replicas Faulty?

“…Having formulated a general nonperturbative theory of both fermionic and bosonic replicas, we further concentrate on the chGUE matrix model and develop an integrable theory of the corresponding nonlinear bosonic replica σ model. Contrary to the claims made in the literature [18], the latter is shown to produce the exact expression for the chGUE density of eigenlevels in the physi- …”

“…Having formulated a general nonperturbative theory of both fermionic and bosonic replicas, we further concentrate on the chGUE matrix model and develop an integrable theory of the corresponding nonlinear bosonic replica σ model. Contrary to the claims made in the literature [18], the latter is shown to produce the exact expression for the chGUE density of eigenlevels in the physically relevant limit of infinite-dimensional matrices. This achievement, representing the main outcome of our study, provides strong evidence that the bosonic replicas are as good and reliable as the fermionic ones.…”

“…The present Letter, prompted by the ongoing discussion [6,18,19,20] about a seeming asymmetry in the performance of fermionic and bosonic replicas [21] (fermionic-bosonic dichotomy), addresses the problem of integrability of zero-dimensional bosonic replica field theories much in line with the ideas of Refs. [8,9].…”

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

“…The form of the Gaussian weight w G (Φ) respects the flavor symmetries, but these symmetries do not exclude traces of higher powers of Φ † Φ in the exponent as well as other non-invariant terms. At zero chemical potential it was shown [50,51,52,53,54,55,56,57] that partition functions and eigenvalue correlations in the microscopic limit are independent of the form of this weight, which, in random matrix theory, is known as universality. The only condition for the probability density is that the spectral density near the origin is nonvanishing in the thermodynamic limit.…”

Qcd, Chiral Random Matrix Theoryand Integrability

“…We will consider the microscopic limit where π(x − y)N ρ(x) ≡ r is kept fixed for N → ∞. In that case the two-point function in the center of the spectrum only depends on r and is given by 73) both in the fermionic and the bosonic case. In an eigenvalue representation of the Goldstone fields the microscopic limit of the generating function Z n (r) can be written as [21,37] Z n (r) =…”