The anisotropic 3D Ising model

Abstract

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order-disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. J x and J y ) are small compared to the third exchange energy (J z ), the following asymptotically exact equation for the critical temperature is derived, sinh(2J z /k B T c )sinh(2(J x þ J y )/k B T c )) ¼ 1. This expression is in perfect agreement with a mathematically rigorous result Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from $0.84 at ((H x þ H y )/H z ) ¼ 10 À2 to $0.76 at ((H x þ H y )/H z ) ¼ 10 À20 .

Response to ‘Comment on a recent conjectured solution of the three-dimensional Ising model’

“…The inequality (equation 2B.8) (or other similar ones) of [10], which is important for proving rigorous results, is valid only for β > 0. Actually, if we plotted [12], which agree with Fisher's rigorous formulae in this limit [13].…”

“…From another angle, we could think that the analytic nature of the expansions for the conjectured solution is supported by these mathematical theorems [3,[8][9][10]. In addition, the conjectured solution reduces to Zandvliet et al's results of the anisotropic 3D Ising model where two of the three exchange energies are small compared to the third one [12], which agree with Fisher's rigorous formulae in this limit [13].…”

Reply to comment on "Exact Solution for Three-Dimensional Ising Model" added by Reply to the Reply by Jacques H.H. Perk

“…(32) in Ref. [1] coincide completely with the results found in the domain wall analysis [5] and the asymptotically exact values in the anisotropic limit [6,7].…”

Spontaneous magnetization of the square 2D Ising lattice with nearest- and weak next-nearest-neighbour interactions

“…In the literature, several routes to determine the boundary tension of a number of 2D and 3D lattices without crossing bonds have been put forward [7][8][9][10]. Recently, we derived an expression for the boundary tension (or boundary free energy) along the high symmetry (10) direction, F (10) , of a square 2D Ising model with crossing bonds [11].…”