Tilo Wettig

RANDOM MATRIX THEORY AND CHIRAL SYMMETRY IN QCD

▪ Abstract Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues increases with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) also follows from the same low-energy effective partition function. The valid...

MICROSCOPIC UNIVERSALITY IN THE SPECTRUM OF THE LATTICE DIRAC OPERATOR

Large ensembles of complete spectra of the Euclidean Dirac operator for staggered fermions are calculated for SU(2) lattice gauge theory. The accumulation of eigenvalues near zero is analyzed as a signal of chiral symmetry breaking and compared with parameter-free predictions from chiral random-matrix theory. Excellent agreement for the distribution of the smallest eigenvalue and the microscopic spectral density is found. This provides direct evidence for the conjecture that these quantities are universal functions.

Overview of the QCDSP and QCDOC computers

The QCDSP and QCDOC computers are two generations of multithousand-node multidimensional mesh-based computers designed to study quantum chromodynamics (QCD), the theory of the strong nuclear force. QCDSP (QCD on digital signal processors), a four-dimensional mesh machine, was completed in 1998; in that year, it won the Gordon Bell Prize in the price/performance category. Two large installations--of 8,192 and 12,288 nodes, with a combined peak speed of one teraflops--have been in operation since. QCD-on-a-chip (QCDOC) utilizes a sixdimensional mesh and compute nodes fabricated with IBM systemon-a-chip technology. It offers a tenfold improvement in price/ performance. Currently, 100-node versions are operating, and there are plans to build three 12,288-node, 10-teraflops machines. In this paper, we describe the architecture of both the QCDSP and QCDOC machines, the operating systems employed, the user software environment, and the performance of our application-- lattice QCD.

Smallest Dirac eigenvalue distribution from random matrix theory

Institut fu¨r Theoretische Physik, Technische Universit¨at Mu¨nchen, D-85747 Garching, Germany(March 2, 1998)We derive the hole probability and the distribution of the smallest eigenvalue of chiral hermitianrandom matrices corresponding to Dirac operators coupled to massive quarks in QCD. They areexpressed in terms of the QCD partition function in the mesoscopic regime. Their universality isexplicitly related to that of the microscopic massive Bessel kernel.PACS number(s): 05.45.+b, 12.38.Aw, 12.38.Lg

An Itzykson–Zuber‐like integral and diffusion for complex ordinary and supermatrices

We compute an analogue of the Itzykson–Zuber integral for the case of arbitrary complex matrices. The calculation is done for both ordinary and supermatrices by transferring the Itzykson–Zuber diffusion equation method to the space of arbitrary complex matrices. The integral is of interest for applications in quantum chromodynamics and the theory of two‐dimensional quantum gravity.

Microscopic spectrum of the QCD Dirac operator with finite quark masses

We compute the microscopic spectrum of the QCD Dirac operator in the presence of dynamical fermions in the framework of random-matrix theory for the chiral Gaussian unitary ensemble. We obtain results for the microscopic spectral correlators, the microscopic spectral density, and the distribution of the smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses, and arbitrary topological charge.

Universal spectral correlations of the Dirac operator at finite temperature

Abstract Using the graded eigenvalue method a rea recently computed extension of the Itzykysn-Zuber integral to complex matrices, we compute the k-point spectral correlation functions of the Dirac operator in a chiral random matrix model with a deterministic diagonal matrix added. We obtain results both on the scale of the mean level spacing and on the microscopic scale. We find that the microscopic spectral correlations have the same functional form as at zero temperature, provided that the microscopic variables are rescaled by the temperature-dependent chiral condensate.

Singular values of the Dirac operator in dense QCD-like theories

A bstractWe study the singular values of the Dirac operator in dense QCD-like theories at zero temperature. The Dirac singular values are real and nonnegative at any nonzero quark density. The scale of their spectrum is set by the diquark condensate, in contrast to the complex Dirac eigenvalues whose scale is set by the chiral condensate at low density and by the BCS gap at high density. We identify three different low-energy effective theories with diquark sources applicable at low, intermediate, and high density, together with their overlapping domains of validity. We derive a number of exact formulas for the Dirac singular values, including Banks-Casher-type relations for the diquark condensate, Smilga-Stern-type relations for the slope of the singular value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We construct random matrix theories and determine the form of the microscopic spectral correlation functions of the singular values for all nonzero quark densities. We also derive a rigorous index theorem for non-Hermitian Diracoperators. Our results can in principle be tested in lattice simulations.

Overlap Dirac operator at nonzero chemical potential and random matrix theory

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma5 Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.

QPACE: Quantum Chromodynamics Parallel Computing on the Cell Broadband Engine

Application-driven computers for lattice gauge theory simulations have often been based on system-on-chip designs, but the development costs can be prohibitive for academic project budgets. An alternative approach uses compute nodes based on a commercial processor tightly coupled to a custom-designed network processor. Preliminary analysis shows that this solution offers good performance, but it also entails several challenges, including those arising from the processors multicore structure and from implementing the network processor on a field-programmable gate array.