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Physical review. B, Condensed matter

Comparison between linked-cluster expansion methods for the Heisenberg antiferromagnet on the square lattice

Zheng Weihong, J. Oitmaa, Chris J. Hamer

Published Oct 1, 1995• 13 citations
Publisher

Abstract

Comparisons are made between the results of three different linked cluster expansion methods, as applied to the S=1/2 Heisenberg antiferromagnet on a square lattice at zero temperature. Connected moments of the Hamiltonian are calculated up to 15th order, extending previous results of Lee and Lo, and are used to form a t expansion. The t expansion is found to be less accurate than the coupled cluster expansion, which in turn is less accurate than the standard perturbation expansion for this system.

DOI
10.1103/physrevb.52.10278

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Comparisons are made between the results of three different linked cluster expansion methods, as applied to the S=1/2 Heisenberg antiferromagnet on a square lattice at zero temperature. Connected moments of the Hamiltonian are calculated up to 15th order, extending previous results of Lee and Lo, and are used to form a t expansion. The t expansion is found to be less accurate than the coupled cluster expansion, which in turn is less accurate than the standard perturbation expansion for this system.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
The spin-half Heisenberg antiferromagnet (HAF) on the square and triangular lattices is studied using the coupled-cluster-method (CCM) technique of quantum many-body theory. The phase relations between different expansion coefficients of the ground-state wave function in an Ising basis for the square lattice HAF is exactly known via the Marshall-Peierls sign rule, although no equivalent sign rule has yet been obtained for the triangular-lattice HAF. Here the CCM is used to give accurate estimates for the Ising-expansion coefficients for these systems, and CCM results are noted to be fully consistent with the Marshall-Peierls sign rule for the square-lattice case. For the triangular-lattice HAF, a heuristic rule is presented which fits our CCM results for the Ising-expansion coefficients of states which correspond to two-body excitations with respect to the reference state. It is also seen that Ising-expansion coefficients which describe localized, m-body excitations with respect to the reference state are found to be highly converged, and from this result we infer that the nodal surface of the triangular lattice HAF is being accurately modeled. Using these results, we are able to make suggestions regarding possible extensions of existing quantum Monte Carlo simulations for the triangular-lattice HAF.
We present a method for the direct calculation of the spin stiffness by means of the coupled-cluster method. For the spin-12\frac{1}{2}21​ Heisenberg antiferromagnet on square, triangular, and cubic lattices, we calculate the stiffness in high orders of approximation. For square and cubic lattices, our results are in very good agreement with the best results available in the literature. For the triangular lattice, our result is more precise than any other result obtained thus far by other approximate method.
A coupled-cluster formalism similar to that used in electronic-structure theory is developed for quantum-mechanical systems localized to sites on a lattice. It is shown how diagrammatic methods may be used to represent the cluster contributions in a convenient and systematic manner. The formalism is applied to the square S=1/2 Heisenberg antiferromagnet, yielding with minor computational effort results of quality comparable to those from computationally intensive quantum Monte Carlo calculations.