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Journalof Southwest China Normal University

Study of variational-cumulant expansion of the quantum Heisenberg hyperfilm

Xing Yang

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The quantum Heisenberg hyperfilm is studied by the variational\|cumulant expansion method. From the analytic properties of free energy, the critical reduced temperature θ\-c(l) can in principle be calculated to any order cumulant expansion as a function of the number l of hyperlayers in hyperfilm. Explicit expressions for θ\-c(l) up to the third order are giren for various lattices and spins. The numberical results are given and discussed.

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The quantum Heisenberg hyperfilm is studied by the variational|cumulant expansion method. From the analytic properties of free energy, the critical reduced temperature θ-c(l) can in principle be calculated to any order cumulant expansion as a function of the number l of hyperlayers in hyperfilm. Explicit expressions for θ-c(l) up to the third order are giren for various lattices and spins. The numberical results are given and discussed.
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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.
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.