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International Journal of Quantum Chemistry

The coupled cluster theory of quantum lattice systems

R. F. Bishop, Yang Xian

Published Feb 12, 1994• 1 citation
Publisher

Abstract

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.

DOI
10.1002/qua.560520817

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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.
Strongly interacting quantum spin–lattice models exhibit a wide variety of phases with diverse and subtle magnetic ordering properties. Their detailed description within a unified microscopic framework poses a real challenge for the many-body theorist. By specific application to the spin-½ anisotropic Heisenberg model on a square lattice, we show how the ab initio coupled cluster method, which has already been very successfully applied to a wide variety of quantum many-body and field-theoretic systems, may be very efficiently and systematically implemented for spin–lattice models. Results for such local properties as the ground-state energy and sublattice magnetization are thereby obtained which are on a par with those from the best of the available alternative methods. Furthermore, we demonstrate explicitly how the coupled cluster method now also provides an effective and fully microscopic tool to yield systematic and accurate estimates of the zero-temperature quantum phase transition boundaries between states of different quantum order, as well as of the critical behaviour of the system in the vicinity of the transition points.
Quantum spins arranged on regular geometrical lattices, and interacting through a wide diversity of specified model Hamiltonians, provide a rich variety of magnetically ordered phases with many subtle properties. The zero-temperature quantum phase transitions between states of different order can be explored by the variation of internal control parameters within the interaction Hamiltonian. We illustrate how the systematic inclusion of multi-spin correlations for such lattice systems may be efficiently and effectively implemented via the coupled cluster method (CCM), which has been widely demonstrated on many previous occasions to be one of the most versatile methods of microscopic quantum many-body theory. Its power is illustrated by application to the spin-½ anisotropic Heisenberg (or XXZ) model on an infinite square lattice. The results obtained for such properties as the ground-state energy and sublattice magnetization are found to be as accurate as those obtained by any available alternative method. We show also how the CCM can now provide accurate ab initio estimates for the critical values of the parameters which mark the positions of phase boundaries between states of different quantum order, as well as for the critical indices associated with the singular behaviour near these transition points of various physical quantities.
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 (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.
We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spinlattice models of interest in quantum magnetism, including their quantum phase transitions.The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (socalled LSUBm) approximations, by the use of computer-algebraic techniques.The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUBm results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, m → ∞, of the truncation index m, which denotes the only approximation made.All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half J XXZ 1 -J XXZ 2 model on a honeycomb lattice with nearestneighbor and next-nearest-neighbor interactions with exchange couplings J 1 > 0 and J 2 ≡ κJ 1 > 0, respectively, where both interactions are of the same anisotropic XXZ type.We show how the method can be used to determine the entire zerotemperature ground-state phase diagram of the model in the range 0 ≤ κ ≤ 1 of the frustration parameter and 0 ≤ Δ ≤ 1 of the spin-space anisotropy parameter.In particular, we identify a candidate quantum spin-liquid region in the phase space.
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.