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Condensed Matter Physics

GAUSSIAN APPROXIMATION FOR ISING MODEL. VARIATIONAL APPROACH

Yanishevsky

Published Jan 1, 2000• 0 citations
Publisher

Abstract

A variant of a Gaussian approximation in Ising model of phase transition is suggested.The method of functional integration with the application of a variational method is put in the basis of the approach.The Hamiltonian (the functional) is presented which in the method of functional integration produces the known results of a two-tails approximation obtained by a diagram method.The application of a variational method eliminates a principal demerit of the two-tails approximation -incorrect description of the phase transition order.The results of numerical calculations for two and threedimensional lattices are presented.

DOI
10.5488/cmp.3.1.201

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A variant of a Gaussian approximation in Ising model of phase transition is suggested.The method of functional integration with the application of a variational method is put in the basis of the approach.The Hamiltonian (the functional) is presented which in the method of functional integration produces the known results of a two-tails approximation obtained by a diagram method.The application of a variational method eliminates a principal demerit of the two-tails approximation -incorrect description of the phase transition order.The results of numerical calculations for two and threedimensional lattices are presented.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.
To the lowest order approximation, the method of variational cumulant expansion has been successful in the determination of the critical point of a spin system on the Ising model. In the second order calculation, unphysical phase transition arises. In this letter, we first analyze the origin of unphysical phase transitions in high-order variational cumulant expansion calculations. We then explain the basis on which a conjecture is proposed that to any order m of the variational cumulant expansion, the critical point is given by the bifurcation point of the free energy. The theory is finally verified numerically for the two-dimensional case.
We investigate the Kramers-Wannier approximation for the three-dimensional (3D) Ising model. The variational state is represented by an effective 2D Ising model, which contains two variational parameters. We numerically calculate the variational partition function using the corner transfer matrix renormalization group (CTMRG) method, and find its maximum with respect to the variational parameters. The value of the calculated transition point, Kc = 0.2184, is only 1.5% less than the true Kc. This result is better than that obtained using the corner transfer tensor renormalization group (CTTRG) approach. The calculated phase transition is mean-field like.
To study the variational-cumulant expansion (VCE) method, the 2-dimensional Ising model has been selected. The model is evaluated to the sixth order approximation with a scheme for the enumeration of connected diagrams. The internal energy , specific heat and spontaneous magnetization have been calculated. A comparison with the exact results shows that thermodynamic and magnetic quantities converge to the exact values rather quickly and the position of phase transition point approaches to the exact pointstand order by order.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.
The standard variational-cumulant expansion (VC) combined with a scheme for the enumeration of connected diagrams is employed to study the simple Ising model with the second-order phase transition. This model is evaluated to the 5th order approximations. A comparison with the 2-dimensional exact results shows that thermodynamic quantities converge rather slowly to the exact values in the intermediate region. A strict analytical calculation of the critical temperature with the 5th order approximation both for and cumulant expansion suggests that VC can achieve rather high standard accuracy if the variational parameters could be determined up to higher degrees of corrections rather than the first one. Anomalies are found for and at extremely low temperature. A detailed study of implies that thermodynamic formulae should be carefully chosen in VC approach so as to avoid these anomalies.