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Acta Physica Sinica

STUDY OF VARIATIONAL-CUMULANT EXPANSION OF THE QUANTUM HEISENBERG HYPERFILM

H Chen

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 given for various lattices and spins.

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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 given for various lattices and spins.
With the variational cumulant expansion (VCE) method, the thermodynamic behaviors of antiferromagnetic Heisenberg films in simple cubic lattices are studied analytically. From the analytic properties of the free energy, in principle we are able to calculate analytically the critical temperatures and the thermodynamic functions, to any order cumulant as the functions of the number of L (the hyperlayers in the hyperfilm). Explicit expressions for up to the fourth order are given. A comparison with the existing results for 3-dimensional system is given. The effective range of the interaction is obtained from numerical results.
An alternative approach is proposed to treat the critical point for general Ising spin systems. From the analytic properties of the free energy, we are in principle able to calculate analytically the critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​(l) for (d+1)-dimensional Ising films to any order cumulant as a function of the number l of hyperlayers in the hyperfilm. Explicit expressions up to the fourth order are given. It is shown from the general expression for Tc(m){\mathit{T}}_{\mathit{c}}^{(\mathit{m})}Tc(m)​(l) that [Tc{\mathit{T}}_{\mathit{c}}Tc​(l)-Tc{\mathit{T}}_{\mathit{c}}Tc​]/Tc{\mathit{T}}_{\mathit{c}}Tc​=b/(l+a) in the limit l\ensuremath{\rightarrow}\ensuremath{\infty}. Comparisons with existing results for d=2 are discussed.
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.