Michael Creutz

Monte Carlo computations in lattice gauge theories

Abstract The formulation of gauge theories on Euclidean space-time lattices and the application of the Monte Carlo computational technique to the ensuing systems are reviewed. A variety of numerical results obtained for lattice gauge theories are presented and discussed.

Deterministic Ising dynamics

A deterministic cellular automaton rule is presented which simulates the Ising model. On each cell in addition to an Ising spin is a space-time parity bit and a variable playing the role of a momentum conjugate to the spin. The procedure permits study of nonequilibrium phenomena, heat flow, mixing, and time correlations. The algorithm can make full use of multispin coding, thus permitting fast programs involving parallel processing on serial machines.

A Statistical Approach to Quantum Mechanics

A Monte Carlo method is used to evaluate the Euclidean version of Feynman’s sum over particle histories. Following Feynman’s treatment, individual paths are defined on a discrete (imaginary) time lattice with periodic boundary conditions. On each lattice site, a continuous position variable xi specifies the spatial location of the particle. Using a modified Metropolis algorithm, the low-lying energy eigenvalues, / &,(x)jZ, the propagator, and the effective potential for the anharmonic oscillator are computed, in good agreement with theory. For a deep double-well potential, instantons were found in our computer simulations appearing as multi-kink configurations on the lattice.

On invariant integration over SU(N)

We give a graphical algorithm for evaluation of invariant integrals of polynomials in SU(N) group elements. Such integrals occur in strongly coupled lattice gauge theory. The results are expressed in terms of totally antisymmetric tensors and Kronecker delta symbols.

Quantum Fields on the Computer

This book provides an overview of recent progress in computer simulations of non-perturbative phenomena in quantum field theory, particularly in the context of the lattice approach. It is a collection of extensive self-contained reviews of various subtopics, including algorithms, spectroscopy, finite temperature physics, Yukawa and chiral theories, bounds on the Higgs meson mass, the renormalization group, and weak decays of hadrons. Physicists with some knowledge of lattice gauge ideas should find this book a useful and interesting source of information on the recent developments in the field.

Microcanonical simulation of Ising systems

Abstract Numerical simulations of the microcanonical ensemble for Ising systems are described. We explain how to write very fast algorithms for such simulations, relate correlations measured in the microcanonical ensemble to those in the canonical ensemble and discuss criteria for convergence and ergodicity.

Quark masses and chiral symmetry

I discuss the global structure of the strongly interacting gauge theory of quarks and gluons as a function of the quark masses and the {ital CP}-violating parameter {theta}. I concentration on whether a first order phase transition occurs at {theta}={pi}. I show why this is expected when multiple flavors have a small degenerate mass. This transition can be removed by sufficient flavor breaking. I speculate on the implications of this structure for Wilson`s lattice fermions.

Aspects of chiral symmetry and the lattice

This article is a pedagogical exploration of the nonperturbative issues entwining lattice gauge theory, anomalies, and chiral symmetry. After briefly reviewing the importance of chiral symmetry in particle physics, the author discusses how anomalies complicate lattice formulations. Considerable information can be deduced from effective chiral Lagrangians, helping interpret the expectations for lattice models and elucidating the role of the CP-violating parameter {Theta}. One particularly elegant scheme for exploring this physics on the lattice is presented in some detail. This uses an auxiliary extra space-time dimension, with the physical world being a four-dimensional interface.

Low temperature expansion for the Ising model.

On simple cubic lattices, we compute the low-temperature expansion for the energy of the Ising model through 50 excited bonds in three dimensions and 44 excited bonds in four dimensions. We also give the magnetization through 42 excited bonds. Our method is a recursive enumeration of states with given energies on a set of finite lattices with generalized helical boundary conditions. A linear combination of such lattices cancels finite volume effects.

Spontaneous violation of CP symmetry in the strong interactions

Some time ago Dashen [Phys. Rev. D 3, 1879 (1971)] pointed out that spontaneous CP violation can occur in the strong interactions. I show how a simple effective Lagrangian exposes the remarkably large domain of quark mass parameters for which this occurs. I close with some warnings for lattice simulations.