Non-Commutativity of the Zero Chemical Potential Limit and the Thermodynamic Limit in Finite Density Systems

Monte Carlo simulations of finite density systems are often plagued by the complex action problem. We point out that there exists certain non-commutativity in the zero chemical potential limit and the thermodynamic limit when one tries to study such systems by reweighting techniques. This is demonstrated by explicit calculations in a Random Matrix Theory, which is thought to be a simple qualitative model for finite density QCD. The factorization method allows us to understand how the non-commutativity, which appears at the intermediate steps, cancels in the end results for physical observables.Comment: 7 pages, 9 figure

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

The Fractal Geometry of Critical Systems

We investigate the geometry of a critical system undergoing a second order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point T=Tc, we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite size effects we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.Comment: 1 LaTeX file, 4 figures in ps-files. Accepted for publication in Physical Review

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

Three-dimensional QCD in the adjoint representation and random matrix theory

In this paper we complete the derivations of finite volume partition functions for QCD using random matrix theories by calculating the effective low-energy partition function for three-dimensional QCD in the adjoint representation from a random matrix theory with the same global symmetries. As expected, this case corresponds to Dyson index $\beta =4$, that is, the Dirac operator can be written in terms of real quaternions. After discussing the issue of defining Majorana fermions in Euclidean space, the actual matrix model calculation turns out to be simple. We find that the symmetry breaking pattern is $O(2N_f) \to O(N_f) \times O(N_f)$, as expected from the correspondence between symmetric (super)spaces and random matrix universality classes found by Zirnbauer. We also derive the first Leutwyler--Smilga sum rule.Comment: LaTeX, 19 pages. Minor corrections, added comments, to appear on Phys. Rev.

Hatano-Nelson model with a periodic potential

We study a generalisation of the Hatano-Nelson Hamiltonian in which a periodic modulation of the site energies is present in addition to the usual random distribution. The system can then become localized by disorder or develop a band gap, and the eigenspectrum shows a wide variety of topologies. We determine the phase diagram, and perform a finite size scaling analysis of the localization transition.Comment: 7 pages, 10 figure

Chiral Phase Transition within Effective Models with Constituent Quarks

We investigate the chiral phase transition at nonzero temperature $T$ and baryon-chemical potential $\mu_B$ within the framework of the linear sigma model and the Nambu-Jona-Lasinio model. For small bare quark masses we find in both models a smooth crossover transition for nonzero $T$ and $\mu_B=0$ and a first order transition for T=0 and nonzero $\mu_B$. We calculate explicitly the first order phase transition line and spinodal lines in the $(T,\mu_B)$ plane. As expected they all end in a critical point. We find that, in the linear sigma model, the sigma mass goes to zero at the critical point. This is in contrast to the NJL model, where the sigma mass, as defined in the random phase approximation, does not vanish. We also compute the adiabatic lines in the $(T,\mu_B)$ plane. Within the models studied here, the critical point does not serve as a ``focusing'' point in the adiabatic expansion.Comment: 22 pages, 18 figure

Statistics of S-matrix poles for chaotic systems with broken time reversal invariance: a conjecture

In the framework of a random matrix description of chaotic quantum scattering the positions of $S-$matrix poles are given by complex eigenvalues $Z_i$ of an effective non-Hermitian random-matrix Hamiltonian. We put forward a conjecture on statistics of $Z_i$ for systems with broken time-reversal invariance and verify that it allows to reproduce statistical characteristics of Wigner time delays known from independent calculations. We analyze the ensuing two-point statistical measures as e.g. spectral form factor and the number variance. In addition we find the density of complex eigenvalues of real asymmetric matrices generalizing the recent result by Efetov\cite{Efnh}.Comment: 4 page

Correlation functions of the BC Calogero-Sutherland model

The BC-type Calogero-Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica sigma-model formalism suitable to treat all chiral RMEs. By exploiting ''generalized color-flavor transformation'' we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c=1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point.Comment: 12 pages, no figure, REVTeX4. sect.V updated, references added (v3

Random Matrices close to Hermitian or unitary: overview of methods and results

The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical contexts, most importantly in random matrix description of quantum chaotic scattering as well as in the context of QCD-inspired random matrix models.Comment: Published version, with a few more misprints correcte