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A systematic coupled-cluster calculations of the ground state of U(1) lattice gauge models in (1+1) and (2+1) dimensions

R.F. Bishop, Yang Xian

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

We have applied the microscopic coupled-cluster method (CCM) of many-body quantum theory to the U(1) lattice gauge models in (1+1) and (2+1) dimensions. The mode couplings and plaquette correlations are studied in detail by means of a hierarchical truncation scheme. Good numerical results for the ground-state energy have been obtained for a range of coupling constants varying from strong to weak. The systematic and nonperturbative natures of the CCM are emphasized.

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We have applied the microscopic coupled-cluster method (CCM) of many-body quantum theory to the U(1) lattice gauge models in (1+1) and (2+1) dimensions. The mode couplings and plaquette correlations are studied in detail by means of a hierarchical truncation scheme. Good numerical results for the ground-state energy have been obtained for a range of coupling constants varying from strong to weak. The systematic and nonperturbative natures of the CCM are emphasized.
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.
A review is given of our recent application of a systematic microscopic formulation of quantum many-body theory, namely the coupled-cluster method (CCM), to Hamiltonian U(1)U(1)U(1) lattice gauge models in the pure gauge sector. It is emphasized that our CCM results represent a natural extension or resummation of the results from the strong-coupling perturbation theory.
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.
The coupled-cluster method (CCM) is applied to anisotropic-quantum-antiferromagnetic models in one and two dimensions (1D and 2D). Several hierarchical approximation schemes of the CCM are defined, which are specialized for the spin-lattice models. Good results are obtained for both ground-state and excited-state energies. The spin-spin correlation functions and the staggered magnetizations are also calculated. We have found that the CCM gives a qualitatively correct description of the entire Heisenberg-Ising phase, including the critical point where a phase transition occurs in 1D. Some interesting differences between 1D and 2D models near the critical point are discussed.
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.