Critical point of QCD at finite T and \mu, lattice results for physical quark masses

A critical point (E) is expected in QCD on the temperature (T) versus baryonic chemical potential (\mu) plane. Using a recently proposed lattice method for \mu \neq 0 we study dynamical QCD with n_f=2+1 staggered quarks of physical masses on L_t=4 lattices. Our result for the critical point is T_E=162 \pm 2 MeV and \mu_E= 360 \pm 40 MeV. For the critical temperature at \mu=0 we obtained T_c=164 \pm 2 MeV. This work extends our previous study [Z. Fodor and S.D.Katz, JHEP 0203 (2002) 014] by two means. It decreases the light quark masses (m_{u,d}) by a factor of three down to their physical values. Furthermore, in order to approach the thermodynamical limit we increase our largest volume by a factor of three. As expected, decreasing m_{u,d} decreased \mu_E. Note, that the continuum extrapolation is still missingComment: 10 pages, 2 figure

Color Superconductivity in Asymmetric Matter

The influence of different chemical potential for different flavors on color superconductivity is analyzed. It is found that there is a first order transition as the asymmetry grows. This transition proceeds through the formation of bubbles of low density, flavor asymmetric normal phase inside a high density, superconducting phase with a gap {\it larger} than the one found in the symmetric case. For small fixed asymmetries the system is normal at low densities and superconducting only above some critical density. For larger asymmetries the two massless quarks system stays in the mixed state for arbitrarily high densities.Comment: 8 pages, 2 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

Imaginary chemical potential and finite fermion density on the lattice

Standard lattice fermion algorithms run into the well-known sign problem at real chemical potential. In this paper we investigate the possibility of using imaginary chemical potential, and argue that it has advantages over other methods, particularly for probing the physics at finite temperature as well as density. As a feasibility study, we present numerical results for the partition function of the two-dimensional Hubbard model with imaginary chemical potential. We also note that systems with a net imbalance of isospin may be simulated using a real chemical potential that couples to I_3 without suffering from the sign problem.Comment: 9 pages, LaTe

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

Lattice determination of the critical point of QCD at finite T and \mu

Based on universal arguments it is believed that there is a critical point (E) in QCD on the temperature (T) versus chemical potential (\mu) plane, which is of extreme importance for heavy-ion experiments. Using finite size scaling and a recently proposed lattice method to study QCD at finite \mu we determine the location of E in QCD with n_f=2+1 dynamical staggered quarks with semi-realistic masses on $L_t=4$ lattices. Our result is T_E=160 \pm 3.5 MeV and \mu_E= 725 \pm 35 MeV. For the critical temperature at \mu=0 we obtained T_c=172 \pm 3 MeV.Comment: misprints corrected, version to appear in JHE

Self-consistent parametrization of the two-flavor isotropic color-superconducting ground state

Lack of Lorentz invariance of QCD at finite quark chemical potential in general implies the need of Lorentz non-invariant condensates for the self-consistent description of the color-superconducting ground state. Moreover, the spontaneous breakdown of color SU(3) in this state naturally leads to the existence of SU(3) non-invariant non-superconducting expectation values. We illustrate these observations by analyzing the properties of an effective 2-flavor Nambu-Jona-Lasinio type Lagrangian and discuss the possibility of color-superconducting states with effectively gapless fermionic excitations. It turns out that the effect of condensates so far neglected can yield new interesting phenomena.Comment: 16 pages, 3 figure

Quantum Chaos in the Yang-Mills-Higgs System at Finite Temperature

The quantum chaos in the finite-temperature Yang-Mills-Higgs system is studied. The energy spectrum of a spatially homogeneous SU(2) Yang-Mills-Higgs is calculated within thermofield dynamics. Level statistics of the spectra is studied by plotting nearest-level spacing distribution histograms. It is found that finite temperature effects lead to a strengthening of chaotic effects, i.e. spectrum which has Poissonian distribution at zero temperature has Gaussian distribution at finite-temperature.Comment: 6 pages, 5 figures, Revte

Statistical analysis and the equivalent of a Thouless energy in lattice QCD Dirac spectra

Random Matrix Theory (RMT) is a powerful statistical tool to model spectral fluctuations. This approach has also found fruitful application in Quantum Chromodynamics (QCD). Importantly, RMT provides very efficient means to separate different scales in the spectral fluctuations. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator for staggered fermions from SU(2) lattice gauge theory for different lattice size and gauge couplings. In disordered systems, the Thouless energy sets the universal scale for which RMT applies. This relates to recent theoretical studies which suggest a strong analogy between QCD and disordered systems. The wealth of data allows us to analyze several statistical measures in the bulk of the spectrum with high quality. We find deviations which allows us to give an estimate for this universal scale. Other deviations than these are seen whose possible origin is discussed. Moreover, we work out higher order correlators as well, in particular three--point correlation functions.Comment: 24 pages, 24 figures, all included except one figure, missing eps file available at http://pluto.mpi-hd.mpg.de/~wilke/diff3.eps.gz, revised version, to appear in PRD, minor modifications and corrected typos, Fig.4 revise

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