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Progress of Theoretical Physics

Perturbational Approach to the Two-Dimensional Next Nearest Neighbor Ising Problem

Akira Kawakami, Takeo Osawa

Published Aug 1, 1969• 2 citations
Publisher

Abstract

The Feynman diagram technique is applied to the Ising model with the next nearest neighbor interaction according to the suggestions by Hurst and Gibberd. The partition function in the lowest order approximation is evaluated. The critical temperature is evaluated in this approximation, and is shown to be exact when either the nearest or the next nearest neighbor interaction is very small. The rules for perturbation calculation are discussed and some simple diagrams are investigated.

DOI
10.1143/ptp.42.196

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The Feynman diagram technique is applied to the Ising model with the next nearest neighbor interaction according to the suggestions by Hurst and Gibberd. The partition function in the lowest order approximation is evaluated. The critical temperature is evaluated in this approximation, and is shown to be exact when either the nearest or the next nearest neighbor interaction is very small. The rules for perturbation calculation are discussed and some simple diagrams are investigated.
We obtain an approximation of the phase diagram of an Ising model with both nearest-neighbor and next-nearest-neighbor interactions on the square lattice. We use the Fisher zeros of the partition functions of a sequence of finite-sized systems along with various extrapolation methods to obtain phase transition points. In addition, we obtain an approximate value of the correlation length critical exponent nu. Our results are compared to previous results for this system using a wide variety of other approaches.
Thermodynamic perturbation theory is applied to systems that undergo a phase transition. It is shown that certain conditions are necessary in order that the application of the perturbation does not change the qualitative features of the phase transition, e.g., the critical indices. The method is then applied to the two-dimensional-square Ising lattice with nearest-and next-nearest-neighbor interactions. The shift in the critical temperature is determined to two orders in the peturbation parameter, viz., the ratio of the strength of the next-nearest-neighbor force to the strength of the nearest-neighbor force. A similar calculation is also made for a two-dimensional-lattice potential of Kac and Helfand.
The next-nearest-neighbor two-dimensional Ising model is cast into a form which resembles a one-dimensional interacting many-fermion system. An approximation, which has previously been shown to be successful in giving the critical properties of the two-dimensional ferroelectric problem, is used. From the approximate expression obtained, the critical indices are found to be α = α′ = 0, β = ⅛, which agrees with the results obtained from series expansions and plausible physical arguments. The critical temperature obtained agrees to within 6% of the series expansion results.
We investigate the effect of including doman-wall interactions in the two-dimensional axial next-nearest-neighbor Ising model. At low temperatures this problem is reduced to a one-dimensional system of interacting fermions which can be treated exactly. It is found that the critical boundaries of the low-temperature phases are in good agreement with those obtained using a free-fermion approximation. In contrast with the monotonic behavior derived from the free-fermion approach, the wall density or wave number displays reentrant phenomena when the ratio of the next-nearest-neighbor and nearest-neighbor interactions is greater than 1/2.
The critical values of the ratio of interaction energies at which the ordering of spins in the ground state is indeterminate (and probably Tc=0{T}_{c}=0Tc​=0) are calculated exactly for a variety of Ising lattices with antiferromagnetic next-nearest-neighbor coupling, and for anisotropic nearest-neighbor coupling. Using these results and the exact two-dimensional solutions, the general dependence of the critical point on interaction ratio is sketched. The complex case of the antiferromagnetic fcc is reconsidered.