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Physical review. B, Condensed matter

Equivalence of the linked-cluster expansion and the variational cumulant expansion in the high-temperature limit

Yu Xia, Hao Che, D. L. Lin, H. M. Wang

Published Aug 1, 1992• 5 citations
Publisher

Abstract

It is shown for the Ising model that the linked-cluster expansion (LCE) is a special case of the variational cumulant expansion (VCE) in the high-temperature limit. The VCE was originally developed and formulated to treat lattice-gauge-field models. The free energy calculated from the VCE with a trial Hamiltonian of mean-field form in the high-temperature limit is in fact identical to that from the LCE. Thus, a more general expansion that is valid for the whole temperature range is established for the treatment of Ising systems.

DOI
10.1103/physrevb.46.3125

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It is shown for the Ising model that the linked-cluster expansion (LCE) is a special case of the variational cumulant expansion (VCE) in the high-temperature limit. The VCE was originally developed and formulated to treat lattice-gauge-field models. The free energy calculated from the VCE with a trial Hamiltonian of mean-field form in the high-temperature limit is in fact identical to that from the LCE. Thus, a more general expansion that is valid for the whole temperature range is established for the treatment of Ising systems.
The variational-cumulant-expansion (VCE) method is adopted to calculate the Heisenberg model. The free energy of the model is expanded to the third order, and the analytical formulas for the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ are given for each order. The first order of VCE corresponds to the mean-field theory, the second order gives the same result for Tc{\mathit{T}}_{\mathit{c}}Tc​ as the 1/z expansion, and the third order gives a more complicated correction. As a comparison, the results of the two-dimensional (2D) Ising model for VCE to the seventh order are shown. We carry out a trial to predict the critical temperature of the infinite order of the VCE, and get a very accurate result for the 2D Ising model. A prediction of Tc{\mathit{T}}_{\mathit{c}}Tc​ for the Heisenberg model is also made.
The modified variational cumulant expansion (VCE) method valid for the whole temperature range and for arbitrary fields is developed to study the critical properties of the general q-state Potts model used to stimulate various magnetic systems. The series expansion is generated directly in terms of the Potts Hamiltonian. The free energy and the internal energy are calculated up to the third-order cumulant terms for the d-dimensional hypercubic q-state Potts model. The expression for the critical temperature Tc is derived in a closed form which allows to determine Tc up to arbitrary order cumulant terms. The modified VCE method eliminates the unphysical first-order phase transition at the mean-field critical point in the internal energy of the Potts model which plagues the conventional VCE method.
The variational-cumulant expansion (VCE) method is used to calculate the quantum XY model, for which the trial action does not commute with the action , with arbitrary magnitude of spin on various one-, two- and three-dimensional lattices. The free energy of the model is expanded to the third order and the analytical formulae of critical temperature are given for each order. To testify this method in a wider range, the Heisenberg model on different lattices is also calculated, which has more comparable results. Weighted fitting method of statistical mathematics is adopted to predict of infinite order.
Following the approach recently proposed by Lin, Che, and Xia [Phys. Rev. A 46, 1805 (1992)], we calculate the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ for Ising films with lattice structures other than simple cubic by means of the variational cumulant expansion (VCE) of the free energy of the spin system. The critical temperature is derived in terms of the ratio of two successive cumulants to an arbitrary order of accuracy. Since the cumulant of any order m can be calculated analytically by a graphic method, analytical expressions of Tc{\mathit{T}}_{\mathit{c}}Tc​ can be derived, in principle, as functions of the number L of spin layers in the film to an arbitrary order of accuracy in the VCE. We describe the graphic method in great detail in this paper, and calculate Tc{\mathit{T}}_{\mathit{c}}Tc​ up to the fourth-order cumulant for films of body-centered-cubic lattice and to the third order for films of face-centered-cubic lattice. Our results show that in the limit of L\ensuremath{\rightarrow}\ensuremath{\infty}, Tc{\mathit{T}}_{\mathit{c}}Tc​ approaches the corresponding bulk value rather quickly with increasing m in both cases.
We consider 1/d1/d1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate 1/d1/d1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in 1/d.1/d.1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in 1/d.1/d.1/d.