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Physical review. B, Condensed matter

Linked-cluster expansion for quantum spin systems and the perpendicular susceptibility of the Ising model

Yung‐Li Wang, Christopher D. Wentworth, B. Westwański

Published Aug 1, 1985• 30 citations
Publisher

Abstract

The Wick reduction theorem is used in a linked-cluster-expansion calculation to facilitate the evaluation of the multi-integrals of the ordered products encountered in the expansion. The method resolves a major problem in the general linked-cluster expansion for quantum spin systems. We apply it to the evaluation of the perpendicular susceptibility of the Ising model. An eighth-order linked-cluster series is obtained for a general lattice. The temperature behavior of the perpendicular susceptibility of the three-dimensional Ising model (fcc lattice) is discussed.

DOI
10.1103/physrevb.32.1805

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The Wick reduction theorem is used in a linked-cluster-expansion calculation to facilitate the evaluation of the multi-integrals of the ordered products encountered in the expansion. The method resolves a major problem in the general linked-cluster expansion for quantum spin systems. We apply it to the evaluation of the perpendicular susceptibility of the Ising model. An eighth-order linked-cluster series is obtained for a general lattice. The temperature behavior of the perpendicular susceptibility of the three-dimensional Ising model (fcc lattice) is discussed.
A linked-cluster expansion technique for spin systems has been used to obtain the free energy, the magnetization, and the susceptibility series for three Ising systems: (1) The Ising model; (2) The Blume-Capel model; and (3) The Blume-Emery-Griffiths (BEG) model. Each Hamiltonian is first divided into a single-ion potential and a term describing the interaction of the spin fluctuations. Only the latter is treated as perturbation in the series expansion. In the parametric phase our series reduces to the exact high-temperature series. However, in the ordered phase the same series can be used to study the low temperature behavior, such as the variation of the magnetization with temperature. The analysis of the eighth-order series for systems (1) and (2) show results in excellent agreement with the previous ones obtained from the high-temperature and low-temperature series. The series analysis for the BEG model shows quite different behavior from that predicted by the mean-field approximation.
Using the linked-cluster expansion, we have obtained the free energy, uniform susceptibility, and staggered susceptibility of the s-band Hubbard model to the sixth order in terms of βt, where t is the hopping integral to the nearest neighbors and β is the inverse of temperature. We show that a special technique based on the multiple-site Wick reduction theorem is more convenient than previous perturbation calculations. Our linked-cluster expansion result is for an arbitrary value of the on-site repulsion U. It has made corrections of previous results of several papers (up to fourth order) and achieved the sixth-order term.
The linked-cluster expansion technique for the high-temperature expansion of spin model is reviewed. A new algorithm for the computation of three-point and higher Green’s functions is presented. Series are computed for all components of two-point Green’s functions for a generalized 3D Ising model, to 25th order on the bcc lattice and to 23rd order on the sc lattice. Series for zero-momentum four-, six-, and eight-point functions are computed to 21st, 19th, and 17th order respectively on the bcc lattice. 1
Similar to various series expansions that are used to approximate mathematical func- tions, the linked-cluster expansion is an approximation method that allows us to approach the actual values of a very large physical system’s different physical quan- tities by systematically studying smaller systems embedded in this larger system. The main concept in linked-cluster expansion, weight, represents the additional con- tribution to a certain physical quantity by increasing the system size by one unit. These weights are used to eventually build up the result on a larger system. In our case, we focus on the partition function, a quantity that can be used to calculate several essential thermodynamic aspects of the system such as average energy, spe- cific heat, magnetic susceptibility, etc. We study these weights for the Ising and Potts models on a one dimensional lattice and Bethe lattice, with and without an external magnetic field. Previous studies have shown that in the one dimensional Ising model, adding more than two lattice sites to the system does not create any ad- ditional contribution to the partition function (i.e. their weights are all zero), giving us the result for an infinitely large system with just two lattice sites, a remarkable simplification. In our study, we prove that this property holds not only for the Ising model, but also its generalization, the Potts model, and show that it is a result of a spin flip symmetry inherent in the system. In order to test this, we break this symmetry using an externally applied magnetic field and show that in this case, for magnetic dipole energies comparable with the exchange energy between neighboring spins, this special property vanishes.
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.