A high-order coupled cluster method (CCM) for the zero-temperature ground states of quantum lattice spin systems of general spin quantum number
(s> 1/2) is described and illustrated via the spin-1 anisotropic Heisenberg antiferromagnet on the square lattice. Results for the ground-state energy and sublattice magnetization are seen to be in excellent agreement with the best of other approximate methods. Furthermore, we are able to follow the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter $\ensuremath{\Delta}$ from the trivial Ising limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty})$ down to a critical value ${\ensuremath{\Delta}}_{c},$ at which point the physical solution terminates. The results for the ground-state energy, sublattice magnetization, and phase transition point thus support our choice of this model as a good nontrivial test case of our CCM formalism.
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.
We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly correlated spin systems. In cluster mean-field, the ground state wave function is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allow for a straightforward improvement which uses perturbation theory and coupled-cluster to account for intercluster correlations. We present benchmark calculations on the 1D chain and 2D square J 1 – J 2 Heisenberg model, using cluster mean-field, perturbation theory, and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4, and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly correlated systems. Some ways to improve upon the methods presented in this work are discussed.
Using the perturbative expansion of the wave function from a single determinantal reference, it is possible to define a genealogical hierarchy of model spaces. The corresponding configuration-interaction matrices are dressed in the frame of intermediate Hamiltonian theory from estimates of the coefficients of the outer space determinants, which may take a coupled cluster (CC) or a perturbative form. The deviations from the additivity of transition energies are taken into account as a modification of coupled-cluster amplitudes, and higher-order exclusion-principle-violating corrections are introduced. The resulting equations relative to the first and second generations are applied to anisotropic Heisenberg Hamiltonians for one-dimensional and square two-dimensional spin lattices. The results show the efficiency of the method, which gives as accurate results with adjacent two-body operators as the traditional CC expansion with local four-body operators. With local four-body operators we obtain results comparable to those obtained with local six-body operators in normal CC treatments. \textcopyright{} 1996 The American Physical Society.
In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used
GW approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function
G and screened Coulomb interaction
W.
W is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The
Na9+ cluster and the dissociation of the
C2 molecule are discussed as examples. We find that the
GW approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as
Na9+ clusters. The
C2 molecule is a pathological system with multideterminantal ground state where the
GW approach ceases to be valid.
A modified coupled cluster method is applied to the research of the ground-state energy of periodic systems described by model Hamiltonians. The reference function is always a strongly localized function. The method is applied first to Heisenberg Hamiltonians and spin-frustrated one-dimensional (1-D) chain and square lattices, starting from Neel functions or from products of bond singlets. The same method is then applied to Hubbard Hamiltonian for 1-D chain and two-dimensional (2-D) square lattices starting from Neel function or products of bond molecular orbitals (MOs). In both cases the wave operators involve a very limited number of local operators. Despite its simplicity, the method is able to treat quite satisfactorily highly degenerate situations, approaching correctly the highly delocalized regime from the Neel function or the highly correlated regime from a product of bond MOs. However, the method is not precise enough to treat the subtle phenomenon of bond alternation of polyacetylene. The coupled cluster method from strongly localized reference functions represents an elegant and quite efficient exploratory tool, but its accuracy is limited by the poor treatment of collective effects. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 115–132, 1998