Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

Abstract

The microscopic spectral density of the QCD Dirac operator at nonzero baryon chemical potential for an arbitrary number of quark flavors was derived recently from a random matrix model with the global symmetries of QCD. In this paper we show that these results and extensions thereof can be obtained from the replica limit of a Toda lattice equation. This naturally leads to a factorized form into bosonic and fermionic QCD-like partition functions. In the microscopic limit these partition functions are given by the static limit of a chiral Lagrangian that follows from the symmetry breaking pattern. In particular, we elucidate the role of the singularity of the bosonic partition function in the orthogonal polynomials approach. A detailed discussion of the spectral density for one and two flavors is given.

Qcd, Chiral Random Matrix Theoryand Integrability

“…In that case the Dirac operator is non-Hermitian and its eigenvalues are scattered in the complex plane. We will show that also in this case the spectral density can be obtained from the Toda lattice equation [25,26]. Recent analytical results by Osborn [27] for the Dirac spectrum of QCD with dynamical quarks at nonzero chemical potential will be discussed.…”

“…We have oscillations on the scale 1/ΣV with an amplitude that diverges exponentially with the volume [26]. In the thermodynamic limit, these oscillations are visible in a domain where the real part of the exponent in (9.3) is positive.…”

“…In the book by Efetov [32] it is shown that the diffusion of electrons in a disordered medium can be described by the effective action 26) where Q are the Goldstone fields, ν is the density of states, D is the diffusion constant and ω is the energy difference between the advanced and the retarded Green's functions. The matrix Λ is a diagonal matrix with matrix elements ±1 corresponding to the causal character of the Green's functions.…”

“…In this domain we will calculate the resolvent and the spectral density from the replica limit of the Toda lattice equation [25,26].…”

“…The low-energy limit of the generating function for the spectral density can again be written as a τ -function [26]. The spectral density is then obtained from the replica limit of the corresponding Toda lattice equation.…”