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Journal of Mathematical Physics

Next-Nearest-Neighbor Ising Model

R. W. Gibberd

Published Jun 1, 1969• 30 citations
Publisher

Abstract

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.

DOI
10.1063/1.1664929

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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.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor ( J x and J y ) and isotropic next-nearest-neighbor ( J d ) interactions has an order-disorder phase transition at a temperature T = T c given by the condition e −2 J x / k b T c + e − 2 J y / k b T c + e −2( J x + J y )/ k b T c (2 − e −4 J d / k b T c ) = e 4 J d / k b T c . The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh (2 J x / k b T c )sinh (2 J y / k b T c ) = 1, is recaptured.