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From Interacting Nucleons to Hamiltonian Lattice Gauge Theories: A Unified Many-Body Treatment

R. F. Bishop

Published Jan 1, 1994• 0 citations
OpenAlex

Abstract

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.

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Journal of Physics A Mathematical and General•Aug 20, 2003•0 citations
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.
The coupled cluster method is widely recognized nowadays as providing an ab initio method of great versatility, power, and accuracy for handling in a fully microscopic and systematic way the correlations between particles in quantum many-body systems. The number of successful applications made to date within both chemistry and physics is impressive. In this article we review our recent extensions of the method which now provide a unifying framework for also dealing with strongly interacting infinite quantum lattice systems described by a Hamiltonian. Such systems include both spin-lattice models (such as the anisotropic Heisenberg or XXZ model) exhibiting interesting magnetic properties, and electron lattice models (such as the tJ and Hubbard models), where the spins or fermions are localized on the sites of a regular lattice; as well as lattice gauge theories [such as the Abelian U(1) model of quantum electrodynamics and non-Abelian SU(n) models]. Illustrative results are given for both the XXZ spin lattice model and U(1) lattice gauge theory. © 1994 John Wiley & Sons, Inc.
The coupled cluster method (CCM) is one of the most powerful and most successful fully microscopic, ab initio formulations available for quantum N-body theory, with N finite or infinite. It has probably been applied to more systems in quantum chemistry, nuclear, condensed matter and other areas of physics, and quantum field theory than any other competing method. In nearly all such cases the numerical results are either the best or among the best available. The CCM can deal with ground- and excited-state energies of closed- and open-shell systems, density matrices and hence other properties, sum rules, and the sub-sum-rules that follow from imbedding linear response theory within it. Extensions exist to deal with systems at nonzero temperature and out of equilibrium. At the formal level it provides an exact mapping of the quantum-mechanical problem onto a classical Hamiltonian phase space where the multiconfigurational canonical classical coordinates have specific cluster and locality properties. In this way it can provide exact hierarchical generalizations of mean-field theory and the random phase approximation. We discuss here both the formalism itself and a selection of its applications.
The widely used coupled cluster method (CCM) in quantum many-body theory has recently provided very accurate descriptions of a large number of extended systems. Although its earlier applications to closed-shell and neighboring finite nuclei were also very successful, they have been shrouded in algebraic and technical complexity. Furthermore, they are difficult to compare with more traditional calculations of generalized shell-model theory since, at least at the important level of two-body correlations, they have been largely implemented in relative-coordinate space rather than the more usual oscillator configuration space. The CCM is reviewed here in the precise context of applications to simple finite systems. Special attention is paid to formulate it in such a way that comparison may be made with generalized shell-model or configuration-interaction (CI) theories. Particular regard is paid to an exact incorporation of translational invariance, so that any spuriosity associated with the center-of-mass motion is always avoided. An important side benefit is that the number of many-body configurations in the usual oscillator basis is dramatically reduced. We are thereby able to present both CI and CCM calculations on 4He^{4}\mathrm{He}4He up to the essentially unprecedented level of 60\ensuremath{\Elzxh}\ensuremath{\omega} in oscillator excitation energy, for two popular and quasirealistic choices of the nucleon-nucleon interaction for which exact Monte Carlo results are available for this nucleus. Although even our simplest approximations attain about 95% of the total binding energy, the convergence in the oscillator configuration space is shown to be both very slow and of a complicated nonuniform nature. Strong implications are drawn for standard implementations of generalized shell-model techniques for heavier nuclei.
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.
The phenomenon of pairing in atomic and nuclear many-body systems gives rise to a great number of different physical properties of matter, from areas as seemingly diverse as the shape of stable nuclei to superconductivity in metals and superfluidity in neutron stars. With the experimental realisation of the long sought BCS-BEC crossover observed in trapped atomic gases - where it is possible to fine tune the s-wave scattering length a of a many-fermion system between a dilute, correlated BCS-like superfluid of Cooper pairs and a densely packed BEC of composite bosons - pairing problems in atomic physics have found renewed interest in recent years. Given the high precision techniques involved in producing these trapped gas condensates, we would like to employ a suitably accurate many-body method to study such systems, preferably one which goes beyond the simple mean-field picture.The Coupled Cluster Method (CCM) is a widely applied and highly successful ab initio method in the realm of quantum many-body physics and quantum chemistry, known to be capable of producing extremely accurate results for a wide variety of different many-body systems. It has not found many applications in pairing problems however, at least not in a general sense. Our aim, therefore, is to study various models of pairing using a variety of CCM techniques - we are interested in studying the generic features of pairing problems and in particular, we are especially interested in probing the collective modes of a system which exhibits the BCS-BEC crossover, in either the BCS or BEC limit. The CCM seems a rather good candidate for the job, given the high precision results it can produce.
The coupled cluster method (CCM) of microscopic quantum many-body theory has become an ab initio method of first choice in quantum chemistry and many fields of nuclear, subnuclear and condensed matter physics, when results of high accuracy are required. In recent years it has begun to be applied with equal success to strongly correlated systems of electrons or quantum spins defined on a regular spatial lattice. One regularly finds that the CCM is able to describe accurately the various zero-temperature phases and the quantum phase transitions between them, even when frustration is present and other methods such as quantum Monte Carlo often fail. We illustrate the use and powerfulness of the method here by applying it to a square-lattice spin-half Heisenberg model where frustration is introduced by competing nearest neighbour bonds. The model exhibits the physically interesting phenomenon of competition between magnetic order and dimerization. Results obtained for the model with the CCM are compared with those found from spin-wave theory and from extrapolating the results of exact diagonalizations of small lattices. We show that the CCM is essentially unique among available methods in being able both to describe accurately all phases of this complex model and to provide accurate predictions of the various phase boundaries and the order of the corresponding transitions.