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Progress of Theoretical Physics

Landau Theory and the Inverse Temperature Series Expansion in the Method of Symmetry Breaking Potential

M. Yamashita

Published Feb 1, 1982• 4 citations
Publisher

Abstract

The Landau theory on the phase transition derived statistical-mechanically on the basis of the symmetry breaking potential method with the use of inverse temperature series expansion is examined. In that method an estimate of the mean field is improved systematically by taking account of short range correlations. The results at low temperatures in relation to the expansion parameter are investigated. The critical temperature obtained is equal to that in the [n, 0] Padé approximation applied to the inverse of susceptibility. By making use of many results obtained up to now in the calculation of susceptibility with the use of inverse temperature series expansion, the critical temperatures of the Ising, Heisenberg and XY models are determined. The possibility of long range ordering in low dimensional systems and the case of the symmetry breaking potential incommutative with the Hamiltonian are also investigated.

DOI
10.1143/ptp.67.375

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The Landau theory on the phase transition derived statistical-mechanically on the basis of the symmetry breaking potential method with the use of inverse temperature series expansion is examined. In that method an estimate of the mean field is improved systematically by taking account of short range correlations. The results at low temperatures in relation to the expansion parameter are investigated. The critical temperature obtained is equal to that in the [n, 0] Padé approximation applied to the inverse of susceptibility. By making use of many results obtained up to now in the calculation of susceptibility with the use of inverse temperature series expansion, the critical temperatures of the Ising, Heisenberg and XY models are determined. The possibility of long range ordering in low dimensional systems and the case of the symmetry breaking potential incommutative with the Hamiltonian are also investigated.
To avoid the difficulties with the usual type of (canonical) high-temperature series which arise when T<Tc and H=O, it is suggested that a high-temperature expansion be accomplished using the microcanonical ensembles of systems defined as involving only those states having values of the order parameter equal to the temperature-averaged one. The method is applied for the Ising and Heisenberg model utilising the molecular-field picture, and for the Heisenberg model using the spin-wave picture. In the latter case, the spin-wave energy is expanded according to the high-temperature parameter. It is shown that (m,m+1)-type Pade approximants are good at all temperatures. It is found that the longitudinal susceptibility chi L, diverges like H-1/2 at T<Tc; consequently, the critical exponent gamma ' no longer holds. A new critical exponent, gamma is defined as scaling the reduced susceptibility, chi 0=H1/2chi L. Deriving the scaling-law relating gamma with beta and delta , gamma is estimated as 0.18.
We show that an high temperature expansion at fixed order parameter can be derived for the quantum Ising model. The basic point is to consider a statistical generating functional associated to the local spin state. The probability at thermal equilibrium of this state reflects directly the occurrence of a spontaneous symmetry breaking. It is possible to recover the expansion around the mean field in the system dimensionality if the ``direction'' in the Hilbert space of local spin states is suitably chosen. Results for the free energy at the critical temperature, as a function of the transverse field, in first order approximation in the inverse system dimensionality are compared with those of the standard approach.
A new method of deriving high temperature series expansions for the susceptibility of the Ising and classical vector spin models is presented. The method involves generating the expansion for the inverse of the susceptibility, in which case only contributions from star graphs need to be considered. An additional term (the thirteenth) has been added to the Ising high temperature susceptibility series for the FCC lattice. Analysis of the extended series does not indicate that revision of previous critical estimates is required. In a test of the capabilities of the method for other systems, seven terms of the susceptibility series of the FCC Heisenberg and planar vector models have been reproduced. Techniques used in generating the graph data are discussed.
We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.
A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.
The self-consistency conditions inherent in the mean-field approximation are extended to take into account some pair correlations.Ii is shown that this extension is systematically improvable and applicable to the spin systems with short-range interactions in the entire temperature region.With two correlations included and the renormalization-group ideas implemented, the critical properties of the Ising model are satisfactorily revealed.Moreover, the improved estimates of the thermodynamical quantities for the quantum linear XY model from the finite-size calculations are found down to very low temperatures.