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

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 order of the quantum chromodynamics transition predicted by the standard model of particle physics

We determine the nature of the QCD transition using lattice calculations for physical quark masses. Susceptibilities are extrapolated to vanishing lattice spacing for three physical volumes, the smallest and largest of which differ by a factor of five. This ensures that a true transition should result in a dramatic increase of the susceptibilities.No such behaviour is observed: our finite-size scaling analysis shows that the finite-temperature QCD transition in the hot early Universe was not a real phase transition, but an analytic crossover (involving a rapid change, as opposed to a jump, as the temperature varied). As such, it will be difficult to find experimental evidence of this transition from astronomical observations.Comment: 7 pages, 4 figure

Character Expansions for the Orthogonal and Symplectic Groups

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the U(N) group. All three expansions have been checked for all N by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups.Comment: 20 pages, in REVTE

Effective Lagrangians and Chiral Random Matrix Theory

Recently, sum rules were derived for the inverse eigenvalues of the Dirac operator. They were obtained in two different ways: i) starting from the low-energy effective Lagrangian and ii) starting from a random matrix theory with the symmetries of the Dirac operator. This suggests that the effective theory can be obtained directly from the random matrix theory. Previously, this was shown for three or more colors with fundamental fermions. In this paper we construct the effective theory from a random matrix theory for two colors in the fundamental representation and for an arbitrary number of colors in the adjoint representation. We construct a fermionic partition function for Majorana fermions in Euclidean space time. Their reality condition is formulated in terms of complex conjugation of the second kind.Comment: 27 page

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

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

Spatial structure of quark Cooper pairs in a color superconductor

Spatial structure of Cooper pairs with quantum numbers color 3^*, I=J=L=S=0 in ud 2 flavor quark matter is studied by solving the gap equation and calculating the coherence length in full momentum range without the weak coupling approximation. Although the gap at the Fermi surface and the coherence length depend on density weakly, the shape of the r-space pair wave function varies strongly with density. This result indicates that quark Cooper pairs become more bosonic at higher densities.Comment: 10 pages, 3 figures. The frequency dependence of the gap and the limitation on the type I/type II discussion are mentioned briefly. To appear in Phys. Rev.

Slowing Out of Equilibrium Near the QCD Critical Point

The QCD phase diagram may feature a critical end point at a temperature T and baryon chemical potential $\mu$ which is accessible in heavy ion collisions. The universal long wavelength fluctuations which develop near this Ising critical point result in experimental signatures which can be used to find the critical point. The magnitude of the observed effects depends on how large the correlation length $\xi$ becomes. Because the matter created in a heavy ion collision cools through the critical region of the phase diagram in a finite time, critical slowing down limits the growth of $\xi$, preventing it from staying in equilibrium. This is the fundamental nonequilibrium effect which must be calculated in order to make quantitative predictions for experiment. We use universal nonequilibrium dynamics and phenomenologically motivated values for the necessary nonuniversal quantities to estimate how much the growth of $\xi$ is slowed.Comment: 21 pages, 5 figures, reference added, typo corrected, to appear in Phys. Rev.

Universal Scaling of the Chiral Condensate in Finite-Volume Gauge Theories

We confront exact analytical predictions for the finite-volume scaling of the chiral condensate with data from quenched lattice gauge theory simulations. Using staggered fermions in both the fundamental and adjoint representations, and gauge groups SU(2) and SU(3), we are able to test simultaneously all of the three chiral universality classes. With overlap fermions we also test the predictions for gauge field sectors of non-zero topological charge. Excellent agreement is found in most cases, and the deviations are understood in the others.Comment: Expanded discussion of overlap fermion results. 17 pages revtex, 7 postscript figure