Exact solution for a layered two-dimensional Ising lattice

L.V. Mikheev, et al

R. Shankar

Theo M. Nieuwenhuizen, Henri Orland

Two-dimensional Ising models are studied on square and triangular lattices with layered disordered bonds (McCoy-Wu-type models). It is shown that exact solutions exist for diluted exponential distributions of couplings. In the ferromagnetic case the free energy exhibits an essential singularity (slightly different from the result of McCoy and Wu) at the transition point. If frustration is present the transitions occurs at T=0, and the specific heat is linear.

Tohru Morita

The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.

Degang Zhang

J. Wosiek

We have found a simple criterion which allows for the straightforward determination of the order-disorder critical temperatures. The method reproduces exactly known results for the two-dimensional Ising, Potts, and Z(N5) models. It also works for the Ising model on the triangular lattice. For systems that are not self-dual, our proposition remains an unproven conjecture. It predicts ${\mathrm{\ensuremath{\beta}}}_{\mathit{c}}$=0.2656. . . for the two coupled layers of Ising spins. The critical temperature of the three-dimensional Ising model is related to the free energy of the two-layer Ising system.