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International Journal of Modern Physics B

TOWARDS A COUPLED-CLUSTER TREATMENT OF SU (N) LATTICE GAUGE FIELD THEORY

R. F. Bishop, N.E. Ligterink, Niels R. Walet

Published Dec 15, 2006• 5 citations
Publisher

Abstract

A consistent approach to Hamiltonian SU (N) lattice gauge field theory is developed using the maximal-tree gauge and an appropriately chosen set of angular variables. The various constraints are carefully discussed, as is a practical means for their implementation. A complete set of variables for the colourless sector is thereby determined. We show that the one-plaquette problem in SU (N) gauge theory can be mapped onto a problem of N fermions on a torus, which is solved numerically for the low-lying energy spectra for N ≤ 5. We end with a brief discussion of how to extend the approach to include the spatial (inter-plaquette) correlations of the full theory, by using a coupled-cluster method parametrisation of the full wave functional.

DOI
10.1142/s021797920603603x

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An ab initio formulation of microscopic quantum many-body theory, namely, the coupled-cluster method (CCM), is applied to the lattice gauge models, Z(2)Z(2)Z(2) in 2+1 dimensions, and U(1) in 1+1 and 2+1 dimensions. Both mode-mode couplings and plaquette-plaquette correlations are considered. In particular, within the one-plaquette approximation for the U(1) model, the CCM is able to include mode couplings of arbitrarily high order. It therefore reproduces essentially the exact numerical results for the equivalent Mathieu problem. These include the ground-state, plaquette, and excitation energies as functions of the coupling constant. Two-plaquette equations are solved by local approximations for both the U(1) and Z(2)Z(2)Z(2) models, and the results are impressive. Detailed comparisons with other methods, particularly perturbation theory, are made and discussed, with emphasis on the nonperturbative nature of the problem.
We apply the coupled cluster method (CCM) to the U(1) lattice gauge model in 2+1 dimensions, basing our analysis on a reference state which already contains a certain class of single-plaquette correlations, the so-called "mean-field" reference state. We consider the simplest possible approximation scheme for the coupled cluster calculation, in which only single-plaquette correlations are included. The use of the "mean-field" reference state allows us to reproduce, within this level of approximation, the correct leading-order behavior of the ground-state energy of the equivalent Mathieu problem in both the strong- and weak-coupling limits. In addition, we have calculated the mean plaquette and the odd- and even-parity mass gaps.
What has since become known as the coupled cluster method (CCM) was invented some 35 years ago to calculate ground-state energies of closed-shell atomic nuclei.The method has since been extended in many different ways to the point where it is nowadays recognized as providing one of the most universally applicable, most powerful, and numerically most accurate of all ab initio methods in the microscopic quantum many-body theory of finite or infinite systems.We review here the applications within nuclear physics, and thereby show how the CCM provides a unifying approach of almost unique stature.The early calculations on both closed-and open-shell nuclei and on infinite nuclear matter are thus related to the recent extensions which provide both extremely accurate calculations for light nuclei and a framework for handling various problems in nuclear field theory.The latter include pion-nucleon field theory and such Hamiltonian lattice gauge field theories as the Abelian U(l) model and the non-Abelian SU(2) model.In all cases we stress the underlying concepts and the high quality of the results obtained.