Europhysics Letters (EPL)

The Coupled-Cluster Approximation: A Simple Approach to Spin-Lattice and Hubbard Models

M. Röger, J. H. Hetherington

Published Feb 1, 1990• 47 citations

Abstract

The coupled-cluster approximation (CCA) is applied to the ground state of strongly interacting fermions or spins on a regular lattice. A new scheme of successive CCAs, suitable for the lattice problem is proposed. Excellent approximations of the ground-state energies are obtained at the lowest levels in a straightforward way. Two examples are given: the antiferromagnetic Heisenberg linear chain and the half-filled Hubbard model on a two-dimensional square lattice, with comparison to exact results and/or other approximations.

DOI: 10.1209/0295-5075/11/3/012

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Europhysics Letters (EPL)

The Coupled-Cluster Approximation: A Simple Approach to Spin-Lattice and Hubbard Models

M. Röger, J. H. Hetherington

Published Feb 1, 1990• 47 citations

Publisher

Abstract

DOI: 10.1209/0295-5075/11/3/012

Related papers

Coupled-cluster method applied to the motion of a single hole in a Hubbard antiferromagnet

F. Petit, M. Röger

The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U ${)}^{0.70}$ , in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole ${\mathit{m}}^{\mathrm{*}}$ =8t/W (W represents the bandwidth) is proportional to U/T ( ${\mathit{m}}^{\mathrm{*}}$ \ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value ${\mathit{m}}^{\mathrm{*}}$ \ensuremath{\approxeq}84 for large U/t.The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U${)}^{0.70}$, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole ${\mathit{m}}^{\mathrm{*}}$=8t/W (W represents the bandwidth) is proportional to U/T (${\mathit{m}}^{\mathrm{*}}$\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value ${\mathit{m}}^{\mathrm{*}}$\ensuremath{\approxeq}84 for large U/t.

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The Coupled Cluster Method Applied to the {\it XXZ} Model on the Square Lattice

DAMIAN J. J. FARNELL, R. F. Bishop

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- $m$ and analytically for SUB2. It is found that SUB2- $m$ results converge rapidly to the full SUB2 solution, including new results for the SUB2- $m$ 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$0.03, respectively. Results for the isotropic Heisenberg model on the square lattice for the spin-one antiferromagnet and the spin-one/spin-half ferrimagnet are also given.A review of the coupled cluster method (CCM) applied to lattice quantum spin systems is presented here. The CCM formalism is explained and an application to the spin-half {\it XXZ} model on the square lattice is presented. Low orders of approximation are carried out analytically and a high-order CCM formulation is presented. Results for the SUB2 approximation are carried out numerically for SUB2-$m$ and analytically for SUB2. It is found that SUB2-$m$ results converge rapidly to the full SUB2 solution, including new results for the SUB2-$m$ 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.

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Coupled-cluster approximation for a spin-1 Heisenberg antiferromagnet with anisotropic exchange interaction and easy-plane single-ion anisotropy

W. H. Wong, Chi‐Fai Lo, Y. L. Wang

In this paper we have applied the coupled-cluster method to investigate the ground state of the spin-1 Heisenberg antiferromagnet with anisotropic exchange interaction and easy-plane single-ion anisotropy. A coupled-cluster approximation up to the second level has been performed to evaluate the ground-state energy and staggered magnetization for various lattices such as linear chain, honeycomb, square, simple cubic, and body-centered-cubic lattice. The convergence of our results is fairly rapid, and thus the coupled-cluster method seems to be a practical tool for calculating ground-state energy of a spin system since only the first few levels of approximation will be needed in actual practice.In this paper we have applied the coupled-cluster method to investigate the ground state of the spin-1 Heisenberg antiferromagnet with anisotropic exchange interaction and easy-plane single-ion anisotropy. A coupled-cluster approximation up to the second level has been performed to evaluate the ground-state energy and staggered magnetization for various lattices such as linear chain, honeycomb, square, simple cubic, and body-centered-cubic lattice. The convergence of our results is fairly rapid, and thus the coupled-cluster method seems to be a practical tool for calculating ground-state energy of a spin system since only the first few levels of approximation will be needed in actual practice.

Physical review. B, Condensed matterSep 1, 199425 citations

{)}^{0.70}

, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole

{\mathit{m}}^{\mathrm{*}}

=8t/W (W represents the bandwidth) is proportional to U/T (

{\mathit{m}}^{\mathrm{*}}

\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value

{\mathit{m}}^{\mathrm{*}}

\ensuremath{\approxeq}84 for large U/t.

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.

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.

m

and analytically for SUB2. It is found that SUB2-

m

results converge rapidly to the full SUB2 solution, including new results for the SUB2-

m

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.