arXiv subjects

J. J. M. Verbaarschot

Publications and source records attributed to J. J. M. Verbaarschot.

At least 55 records · Page 3Linked to original sources

arXiv2006

Phase of the Fermion Determinant at Nonzero Chemical Potential

We show that in the microscopic domain of QCD (also known as the $ε$-domain) at nonzero chemical potential the average phase factor of the fermion determinant is nonzero for $μ< m_π/2$ and is exponentially suppressed for larger values of the chemical potential. This follows from the chiral Lagrangian that describes the low-energy limit of the expectation value of the phase factor. Explicit expressions for the average phase factor are derived using a random matrix formulation of the zero momentum limit of this chiral Lagrangian.

hep-lat↗

arXiv2006

Surprises for QCD at Nonzero Chemical Potential

In this lecture we compare different QCD-like partition functions with bosonic quarks and fermionic quarks at nonzero chemical potential. Although it is not a surprise that the ground state properties of a fermionic quantum system and a bosonic quantum system are completely different, the behavior of partition functions with bosonic quarks does not follow our naive expectation. Among other surprises, we find that the partition function with one bosonic quark only exists at nonzero chemical potential if a conjugate bosonic quark and a conjugate fermionic quark are added to the partition function.

hep-ph↗

arXiv2005

The Sign Problem is the Solution

The unquenched spectral density of the Dirac operator at $μ\neq0$ is complex and has oscillations with a period inversely proportional to the volume and an amplitude that grows exponentially with the volume. Here we show how the oscillations lead to the discontinuity of the chiral condensate.

hep-lat↗

arXiv2005

Random Matrices

We review elementary properties of random matrices and discuss widely used mathematical methods for both hermitian and nonhermitian random matrix ensembles. Applications to a wide range of physics problems are summarized. This paper originally appeared as an article in the Wiley Encyclopedia of Electrical and Electronics Engineering.

hep-ph↗

arXiv2005

Chiral Symmetry Breaking at Nonzero Chemical Potential

We consider chiral symmetry breaking at nonzero chemical potential and discuss the relation with the spectrum of the Dirac operator. We solve the so called Silver Blaze Problem that the chiral condensate at zero temperature does not depend on the chemical potential while this is not the case for the Dirac spectrum and the weight of the partition function.

hep-ph↗

arXiv2005

Sum Rules for the Dirac Spectrum of the Schwinger Model

The inverse eigenvalues of the Dirac operator in the Schwinger model satisfy the same Leutwyler-Smilga sum rules as in the case of QCD with one flavor. In this paper we give a microscopic derivation of these sum rules in the sector of arbitrary topological charge. We show that the sum rules can be obtained from the clustering property of the scalar correlation functions. This argument also holds for other theories with a mass gap and broken chiral symmetry such as QCD with one flavor. For QCD with several flavors a modified clustering property is derived from the low energy chiral Lagrangian. We also obtain sum rules for a fixed external gauge field and show their relation with the bosonized version of the Schwinger model. In the sector of topological charge $ν$ the sum rules are consistent with a shift of the Dirac spectrum away from zero by $ν/2$ average level spacings. This shift is also required to obtain a nonzero chiral condensate in the massless limit. Finally, we discuss the Dirac spectrum for a closely related two-dimensional theory for which the gauge field action is quadratic in the the gauge fields. This theory of so called random Dirac fermions has been discussed extensively in the context of the quantum Hall effect and d-wave super-conductors.

hep-th↗

arXiv2005

QCD, Chiral Random Matrix Theory and Integrability

Random Matrix Theory has been a unifying approach in physics and mathematics.In these lectures we discuss applications of Random Matrix Theory to QCD and emphasize underlying integrable structures. In the first lecture we give an overview of QCD, its low-energy limit and the microscopic limit of the Dirac spectrum which, as we will see in the second lecture, can be described by chiral Random Matrix Theory. The main topic of the third lecture is the recent developments on the relation between the QCD partition function and integrable hierarchies (in our case the Toda lattice hierarchy). This is an efficient way to obtain the QCD Dirac spectrum from the low energy limit of the QCD partition function. Finally, we will discuss the QCD Dirac spectrum at nonzero chemical potential. We will show that the microscopic spectral density is given by the replica limit of the Toda lattice equation. Recent results by Osborn on the Dirac spectrum of full QCD will be discussed.

hep-th↗

arXiv2005

Chiral Symmetry Breaking and the Dirac Spectrum at Nonzero Chemical Potential

The relation between the spectral density of the QCD Dirac operator at nonzero baryon chemical potential and the chiral condensate is investigated. We use the analytical result for the eigenvalue density in the microscopic regime which shows oscillations with a period that scales as 1/V and an amplitude that diverges exponentially with the volume $V=L^4$. We find that the discontinuity of the chiral condensate is due to the whole oscillating region rather than to an accumulation of eigenvalues at the origin. These results also extend beyond the microscopic regime to chemical potentials $μ\sim 1/L$.

hep-th↗

arXiv2004

Unquenched QCD Dirac Operator Spectra at Nonzero Baryon Chemical Potential

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.

hep-th↗

arXiv2004

The Supersymmetric Method in Random Matrix Theory and Applications to QCD

The supersymmetric method is a powerful method for the evaluation of quenched averages in disordered systems. Among others, this method has been applied to the theory of S-matrix fluctuations, the theory of universal conductance fluctuations and the microscopic spectral density of the QCD Dirac operator. We start this series of lectures with a general review of Random Matrix Theory and the statistical theory of spectra. An elementary introduction of the supersymmetric method in Random Matrix Theory is given in the second and third lecture. We will show that a Random Matrix Theory can be rewritten as an integral over a supermanifold. This integral will be worked out for the Gaussian Unitary Ensemble that describes systems with broken time reversal invariance. We especially emphasize the role of symmetries. As a second example of the application of the supersymemtric method we discuss the calculation of the microscopic spectral density of the QCD Dirac operator. which is known to be given by chiral Random Matrix Theory. Also in this case we use symmetry considerations to rewrite the generating function for the resolvent as an integral over a supermanifold. The main topic of the second last lecture is the recent developments on the relation between the supersymmetric partition function and the Toda lattice hierarchy. We will show that this relation is an efficient way to calculate superintegrals. Finally, we will discuss the quenched QCD Dirac spectrum at nonzero chemical potential. Because of the nonhermiticity of the Dirac operator the usual supersymmetric method has not been successful in this case. However, we will show that the supersymmetric partition function can be evaluated by means of the replica limit of the Toda lattice equation.

hep-th↗

arXiv2004

The QCD Phase Diagram at Non-zero Baryon and Isospin Chemical Potentials

In heavy ion collision experiments as well as in neutron stars, both baryon and isospin chemical potentials are different from zero. In particular, the regime of small isospin chemical potential is phenomenologically important. Using a random matrix model, we find that the phase diagram at non-zero temperature and baryon chemical potential is greatly altered by an arbitrarily small isospin chemical potential: There are two first order phase transitions at low temperature, two critical endpoints, and two crossovers at high temperature. As a consequence, in the region of the phase diagram explored by RHIC experiments, there are two crossovers that separate the hadronic phase from the quark-gluon plasma phase at high temperature.

hep-lat↗

arXiv2004

QCD Dirac Spectra and the Toda Lattice

We discuss the spectrum of the QCD Dirac operator both at zero and at nonzero baryon chemical potential. We show that, in the ergodic domain of QCD, the Dirac spectrum can be obtained from the replica limit of a Toda lattice equation. At zero chemical potential this method explains the factorization of known results into compact and noncompact integrals, and at nonzero chemical potential it allows us to derive the previously unknown microscopic spectral density.

hep-th↗

arXiv2004

Diquark and Pion Condensation in Random Matrix Models for two-color QCD

We introduce a random matrix model with the symmetries of QCD with two colors at nonzero isospin and baryon chemical potentials and temperature. We analyze its phase diagram and find phases with condensation of pion and diquark states in addition to the phases with spontaneously broken chiral symmetries. In the limit of small chemical potentials and quark masses, we reproduce the mean field results obtained from chiral Lagrangians. As in the case of QCD with three colors, the presence of two chemical potentials breaks the flavor symmetry and leads to phases that are characterized by different behaviors of the chiral condensates for each flavor. In particular, the phase diagram we obtain is similar to QCD with three colors and three flavors of quarks of equal masses at zero baryon chemical potential and nonzero isospin and strange chemical potentials. A tricritical point of the superfluid transitions found in lattice calculations and from an analysis in terms of chiral Lagrangians does not appear in the random matrix model. Remarkably, at fixed isospin chemical potential, for the regions outside of the superfluid phases, the phase diagram in the temperature - baryon chemical potential plane for two colors and three colors are qualitatively the same.

hep-ph↗

arXiv2004

Supersymmetric Quenching of the Toda Lattice Equation

The average of the ratio of powers of the spectral determinants of the Dirac operator in the $ε$-regime of QCD is shown to satisfy a Toda lattice equation. The quenched limit of this Toda lattice equation is obtained using the supersymmetric method. This super symmetric approach is then shown to be equivalent to taking the replica limit of the Toda lattice equation. Among other, the factorization of the microscopic spectral correlation functions of the QCD Dirac operator into fermionic and bosonic partition functions follows naturally from both approaches. While the replica approach relies on an analytic continuation in the number of flavors no such assumptions are made in the present approach where the numbers of flavors in the Toda lattice equation are strictly integer.

hep-th↗

arXiv2004

Factorization of Correlation Functions and the Replica Limit of the Toda Lattice Equation

Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and non-Hermitian theories in the Wigner-Dyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive two-point correlation functions for class A and class AIII as well as several one-point correlation functions in class AIII. In the non-Hermitian case the spectral density of non-Hermitian complex random matrices in the weak non-Hermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the Hatano-Nelson model) which is known from earlier work. New results are obtained for the spectral density in the weak non-Hermiticity limit of a quenched chiral random matrix model at nonzero chemical potential. These results apply to the ergodic or $ε$ domain of quenched QCD at nonzero chemical potential. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one- and two-point functions into a product of a bosonic (noncompact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic, the bosonic and the supersymmetric partition functions are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation.

hep-th↗

arXiv2003

The Factorization Method for Monte Carlo Simulations of Systems With a Complex Action

We propose a method for Monte Carlo simulations of systems with a complex action. The method has the advantages of being in principle applicable to any such system and provides a solution to the overlap problem. In some cases, like in the IKKT matrix model, a finite size scaling extrapolation can provide results for systems whose size would make it prohibitive to simulate directly.

hep-lat↗

arXiv2003

Developments in Random Matrix Theory

In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given.

cond-mat.dis-nn↗

arXiv2003

The QCD Phase Diagram at Nonzero Temperature, Baryon and Isospin Chemical Potentials in Random Matrix Theory

We introduce a random matrix model with the symmetries of QCD at finite temperature and chemical potentials for baryon number and isospin. We analyze the phase diagram of this model in the chemical potential plane for different temperatures and quark masses. We find a rich phase structure with five different phases separated by both first and second order lines. The phases are characterized by the pion condensate and the chiral condensate for each of the flavors. In agreement with lattice simulations, we find that in the phase with zero pion condensate the critical temperature depends in the same way on the baryon number chemical potential and on the isospin chemical potential. At nonzero quark mass, we remarkably find that the critical end point at nonzero temperature and baryon chemical potential is split in two by an arbitrarily small isospin chemical potential. As a consequence, there are two crossovers that separate the hadronic phase from the quark-gluon plasma phase at high temperature. Detailed analytical results are obtained at zero temperature and in the chiral limit.

hep-ph↗