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arXiv:1010.2159 · Physical Review A

Coupled-cluster method: A lattice-path-based subsystem approximation scheme for quantum lattice models

R. F. Bishop, P. H. Y. Li

Published Apr 29, 2011• 16 citations
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Abstract

An approximation hierarchy, called the lattice-path-based subsystem (LPSUB$m$) approximation scheme, is described for the coupled-cluster method (CCM). It is applicable to systems defined on a regular spatial lattice. We then apply it to two well-studied prototypical (spin-$\frac{1}{2}$ Heisenberg antiferromagnetic) spin-lattice models, namely, the $\mathit{XXZ}$ and the $\mathit{XY}$ models on the square lattice in two dimensions. Results are obtained in each case for the ground-state energy, the ground-state sublattice magnetization, and the quantum critical point. They are all in good agreement with those from such alternative methods as spin-wave theory, series expansions, quantum Monte Carlo methods, and the CCM using the alternative lattice-animal-based subsystem (LSUB$m$) and the distance-based subsystem (DSUB$m$) schemes. Each of the three CCM schemes (LSUB$m$, DSUB$m$, and LPSUB$m$) for use with systems defined on a regular spatial lattice is shown to have its own advantages in particular applications.

DOI
10.1103/physreva.83.042111

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An approximation hierarchy, called the lattice-path-based subsystem (LPSUBmmm) approximation scheme, is described for the coupled-cluster method (CCM). It is applicable to systems defined on a regular spatial lattice. We then apply it to two well-studied prototypical (spin-12\frac{1}{2}21​ Heisenberg antiferromagnetic) spin-lattice models, namely, the XXZ\mathit{XXZ}XXZ and the XY\mathit{XY}XY models on the square lattice in two dimensions. Results are obtained in each case for the ground-state energy, the ground-state sublattice magnetization, and the quantum critical point. They are all in good agreement with those from such alternative methods as spin-wave theory, series expansions, quantum Monte Carlo methods, and the CCM using the alternative lattice-animal-based subsystem (LSUBmmm) and the distance-based subsystem (DSUBmmm) schemes. Each of the three CCM schemes (LSUBmmm, DSUBmmm, and LPSUBmmm) for use with systems defined on a regular spatial lattice is shown to have its own advantages in particular applications.
A new approximate scheme, DSUBm, is described for the coupled cluster method.We apply it to two wellstudied (spin-1/2 Heisenberg antiferromagnet) spin-lattice models, namely: the XXZ and the XY models on the square lattice in two dimensions.Results are obtained in each case for the ground-state energy, the sublattice magnetization and the quantum critical point.They are in good agreement with those from such alternative methods as spin-wave theory, series expansions, exact diagonalization techniques, quantum Monte Carlo methods and those from the CCM using the LSUBm scheme.
A review of the coupled cluster method (CCM) applied to lattice quantum spin systems is presented here. The CCM formalism is explained and an application to the spin-half {\it XXZ} model on the square lattice is presented. Low orders of approximation are carried out analytically and a high-order CCM formulation is presented. Results for the SUB2 approximation are carried out numerically for SUB2-mmm and analytically for SUB2. It is found that SUB2-mmm results converge rapidly to the full SUB2 solution, including new results for the SUB2-mmm limiting points compared to the SUB2 critical point. Results for the ground-state energy and the sublattice magnetisation are presented. A study of the excitation spectrum of this model at the SUB2 critical point is given. Indeed, the shape of the excitation spectrum at the SUB2 critical point is identical to that predicted by spin-wave theory for the isotropic model, albeit with a multiplicative factor of 1.1672. This result compares very well to results of cumulant series expansions and Monte Carlo simulation that again predict a similar shape for the excitation spectrum, but with multiplicative factors of 1.18 and 1.21$\pm$0.03, respectively. Results for the isotropic Heisenberg model on the square lattice for the spin-one antiferromagnet and the spin-one/spin-half ferrimagnet are also given.
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)(s>~1/2)(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.
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.
The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is largely due to the application of computational techniques that allow the method to be applied to high orders of approximation using a localized scheme known as the LSUBm scheme. A hitherto unreported aspect of this scheme is that results for LSUBm expectation values behave in distinctly different ways with odd and even values of m. Here, we consider the behavior of ground-state expectation values of odd and even orders of the CCM LSUBm approximation for unfrustrated spin-half Heisenberg antiferromagnets on the square and honeycomb lattice and the frustrated spin-half Heisenberg antiferromagnet on the triangular lattice. We demonstrate that results for odd and even orders of approximation show qualitatively different behavior for both the ground-state energy and the sublattice magnetization. Indeed, the odd series consistently forms an upper branch of results, and the even series a lower branch with respect to both ground-state energy and sublattice magnetization, for all of the models considered here.
High-Order Coupled Cluster Method (CCM) Formalism 1: Ground- and Excited-State Properties of Lattice Quantum Spin Systems with s>=1/2s >= 1/2s>=1/2
The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is due largely to the application of computational techniques that allow the method to be applied to high orders of approximation using localised approximation schemes, e.g., such as the LSUBmmm scheme. In this article, the high-order CCM formalism for the ground and excited states of quantum magnetic systems are extended to those with spin quantum number s≥12s \ge \frac 12s≥21​. Solution strategies for the ket- and bra-state equations are also considered. Aspects of extrapolation of CCM expectation values are discussed and future topics regarding extrapolations are presented.