Spectrum of the SU(3) Dirac operator on the lattice: Transition from random matrix theory to chiral perturbation theory

We calculate complete spectra of the Kogut-Susskind Dirac operator on the lattice in quenched SU(3) gauge theory for various values of coupling constant and lattice size. From these spectra we compute the connected and disconnected scalar susceptibilities and find agreement with chiral random matrix theory up to a certain energy scale, the Thouless energy. The dependence of this scale on the lattice volume is analyzed. In the case of the connected susceptibility this dependence is anomalous, and we explain the reason for this. We present a model of chiral perturbation theory that is capable of describing the data beyond the Thouless energy and that has a common range of applicability with chiral random matrix theory.Comment: 8 pages, RevTeX, 15 .eps figure

Small eigenvalues of the SU(3) Dirac operator on the lattice and in Random Matrix Theory

We have calculated complete spectra of the staggered Dirac operator on the lattice in quenched SU(3) gauge theory for \beta = 5.4 and various lattice sizes. The microscopic spectral density, the distribution of the smallest eigenvalue, and the two-point spectral correlation function are analyzed. We find the expected agreement of the lattice data with universal predictions of the chiral unitary ensemble of random matrix theory up to a certain energy scale, the Thouless energy. The deviations from the universal predictions are determined using the disconnected scalar susceptibility. We find that the Thouless energy scales with the lattice size as expected from theoretical arguments making use of the Gell-Mann--Oakes--Renner relation.Comment: REVTeX, 5 pages, 4 figure

Simulation of a modified Hubbard model with a chemical potential using a meron-cluster algorithm

We show how a variant of the Hubbard model can be simulated using a meron-cluster algorithm. This provides a major improvement in our ability to determine the behavior of these types of models. We also present some results that clearly demonstrate the existence of a superconducting state in this model.Comment: 9 pages, Lattice2002(plenary

A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential

We solve a new chiral Random Two-Matrix Theory by means of biorthogonal polynomials for any matrix size $N$. By deriving the relevant kernels we find explicit formulas for all $(n,k)$-point spectral (mixed or unmixed) correlation functions. In the microscopic limit we find the corresponding scaling functions, and thus derive all spectral correlators in this limit as well. We extend these results to the ordinary (non-chiral) ensembles, and also there provide explicit solutions for any finite size $N$, and in the microscopic scaling limit. Our results give the general analytical expressions for the microscopic correlation functions of the Dirac operator eigenvalues in theories with imaginary baryon and isospin chemical potential, and can be used to extract the tree-level pion decay constant from lattice gauge theory configurations. We find exact agreement with previous computations based on the low-energy effective field theory in the two special cases where comparisons are possible.Comment: 31 pages 2 figures, v2 missing term in partially quenched results inserted, fig 2 update

A new chiral two-matrix theory for dirac spectra with imaginary chemical potential

We solve a new chiral Random Two-Matrix Theory by means of biorthogonal polynomials for any matrix size $N$. By deriving the relevant kernels we find explicit formulas for all $(n,k)$-point spectral (mixed or unmixed) correlation functions. In the microscopic limit we find the corresponding scaling functions, and thus derive all spectral correlators in this limit as well. We extend these results to the ordinary (non-chiral) ensembles, and also there provide explicit solutions for any finite size $N$, and in the microscopic scaling limit. Our results give the general analytical expressions for the microscopic correlation functions of the Dirac operator eigenvalues in theories with imaginary baryon and isospin chemical potential, and can be used to extract the tree-level pion decay constant from lattice gauge theory configurations. We find exact agreement with previous computations based on the low-energy effective field theory in the two special cases where comparisons are possible

Meron-Cluster Approach to Systems of Strongly Correlated Electrons

Numerical simulations of strongly correlated electron systems suffer from the notorious fermion sign problem which has prevented progress in understanding if systems like the Hubbard model display high-temperature superconductivity. Here we show how the fermion sign problem can be solved completely with meron-cluster methods in a large class of models of strongly correlated electron systems, some of which are in the extended Hubbard model family and show s-wave superconductivity. In these models we also find that on-site repulsion can even coexist with a weak chemical potential without introducing sign problems. We argue that since these models can be simulated efficiently using cluster algorithms they are ideal for studying many of the interesting phenomena in strongly correlated electron systems.Comment: 36 Pages, 13 figures, plain Late

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

Finite size scaling of meson propagators with isospin chemical potential

We determine the volume and mass dependence of scalar and pseudoscalar two-point functions in N_f-flavour QCD, in the presence of an isospin chemical potential and at fixed gauge-field topology. We obtain these results at second order in the \epsilon-expansion of Chiral Perturbation Theory and evaluate all relevant zero-mode group integrals analytically. The virtue of working with a non-vanishing chemical potential is that it provides the correlation functions with a dependence on both the chiral condensate, \Sigma, and the pion decay constant, F, already at leading order. Our results may therefore be useful for improving the determination of these constants from lattice QCD calculations. As a side product, we rectify an earlier calculation of the O(\epsilon^2) finite-volume correction to the decay constant appearing in the partition function. We also compute a generalised partition function which is useful for evaluating U(N_f) group integrals

Small eigenvalues of the staggered Dirac operator in the adjoint representation and Random Matrix Theory

The low-lying spectrum of the Dirac operator is predicted to be universal, within three classes, depending on symmetry properties specified according to random matrix theory. The three universal classes are the orthogonal, unitary and symplectic ensemble. Lattice gauge theory with staggered fermions has verified two of the cases so far, unitary and symplectic, with staggered fermions in the fundamental representation of SU(3) and SU(2). We verify the missing case here, namely orthogonal, with staggered fermions in the adjoint representation of SU(N_c), N_c=2, 3.Comment: 3 pages, revtex, 2 postscript figure

Calculation of Effective Coulomb Interaction for Pr3+Pr^{3+}, U4+U^{4+}, and UPt3UPt_3

In this paper, the Slater integrals for a screened Coulomb interaction of the the Yukawa form are calculated and by fitting the Thomas-Fermi wavevector, good agreement is obtained with experiment for the multiplet spectra of $Pr^{3+}$ and $U^{4+}$ ions. Moreover, a predicted multiplet spectrum for the heavy fermion superconductor $UPt_3$ is shown with a calculated Coulomb U of 1.6 eV. These effective Coulomb interactions, which are quite simple to calculate, should be useful inputs to further many-body calculations in correlated electron metals.Comment: 8 pages, revtex, 3 uuencoded postscript figure