Random matrix model of QCD at finite density and the nature of the quenched limit

We use a random matrix model to study chiral symmetry breaking in QCD at finite chemical potential $\mu$. We solve the model and compute the eigenvalue density of the Dirac matrix on a complex plane. A naive ``replica trick'' fails for $\mu\neq0$: we find that quenched QCD is not a simple $n\to0$ limit of QCD with $n$ quarks. It is the limit of a theory with $2n$ quarks: $n$ quarks with original action and $n$ quarks with conjugate action. The results agree with earlier studies of lattice QCD at $\mu\neq0$ and provide a simple analytical explanation of a long-standing puzzle.Comment: 9 pages, revtex3.0, 4 epsf figures. Packed by "uufiles" script. Revised version (PRL): minor change

Random Matrices and the Convergence of Partition Function Zeros in Finite Density QCD

We apply the Glasgow method for lattice QCD at finite chemical potential to a schematic random matrix model (RMM). In this method the zeros of the partition function are obtained by averaging the coefficients of its expansion in powers of the chemical potential. In this paper we investigate the phase structure by means of Glasgow averaging and demonstrate that the method converges to the correct analytically known result. We conclude that the statistics needed for complete convergence grows exponentially with the size of the system, in our case, the dimension of the Dirac matrix. The use of an unquenched ensemble at $\mu=0$ does not give an improvement over a quenched ensemble. We elucidate the phenomenon of a faster convergence of certain zeros of the partition function. The imprecision affecting the coefficients of the polynomial in the chemical potential can be interpeted as the appearance of a spurious phase. This phase dominates in the regions where the exact partition function is exponentially small, introducing additional phase boundaries, and hiding part of the true ones. The zeros along the surviving parts of the true boundaries remain unaffected.Comment: 17 pages, 14 figures, typos correcte

Fermion determinants in matrix models of QCD at nonzero chemical potential

The presence of a chemical potential completely changes the analytical structure of the QCD partition function. In particular, the eigenvalues of the Dirac operator are distributed over a finite area in the complex plane, whereas the zeros of the partition function in the complex mass plane remain on a curve. In this paper we study the effects of the fermion determinant at nonzero chemical potential on the Dirac spectrum by means of the resolvent, G(z), of the QCD Dirac operator. The resolvent is studied both in a one-dimensional U(1) model (Gibbs model) and in a random matrix model with the global symmetries of the QCD partition function. In both cases we find that, if the argument z of the resolvent is not equal to the mass m in the fermion determinant, the resolvent diverges in the thermodynamic limit. However, for z =m the resolvent in both models is well defined. In particular, the nature of the limit $z \rightarrow m$ is illuminated in the Gibbs model. The phase structure of the random matrix model in the complex m and \mu-planes is investigated both by a saddle point approximation and via the distribution of Yang-Lee zeros. Both methods are in complete agreement and lead to a well-defined chiral condensate and quark number density.Comment: 27 pages, 6 figures, Late

Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential

In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.Comment: 4 pages, 3 figures, to appear in Phys. Rev. Let

Instantons and the Chiral Phase Transition at non-zero Baryon Density

We study an interacting ensemble of instantons at finite baryon chemical potential. We emphasize the importance of fermionic zero modes and calculate the fermion induced interaction between instantons at non-zero chemical potential. We show that unquenched simulations of the instanton ensemble are feasible in two regimes, for sufficiently small and for very large chemical potential. At very large chemical potential chiral symmetry is restored and the instanton ensemble is dominated by strongly correlated chain-like configurations.Comment: 37 pages, 10 figures, uses revtex, revised version (minor corrections and expanded dicussion