Two- and three-spin triangular ising model: Variational approximations

Hadey K. Mohamad

Amir Nourhani, Vincent H. Crespi, Paul E. Lammert

Abstract We present a treatment of the triangular lattice antiferromagnetic Ising model (TAFIM) based on a small number of elementary ideas common to statistical and solid-state physics. The TAFIM is represented as a reduced BCS model in one space, one (imaginary) time dimension. The representation is approximate for nonzero temperature, but allows quick derivation of asymptotically exact thermodynamic functions, and the divergence of the spin–spin correlation length. The fermionic representation is exact at zero temperature. We demonstrate the existence of a two-dimensional continuum of zero-temperature equilibrium macrostates characterized by satisfied bond fractions of the three different orientations, and calculate their entropy densities.

R. J. Baxter, M F Sykes, M G Watts

Below its critical temperature, the triangular Ising model with pure three-spin interactions has a spontaneous magnetization M. The first 13 terms of a series expansion for M are known. Here these are re-arranged into a form which can be extrapolated to all terms in a natural way. Thus a conjecture is obtained for M. It fits the scaling prediction beta =1/12. Another order parameter is similarly discussed, together with the difficulties encountered in applying the technique to the two-spin Ising model susceptibility.

Viktor Urumov

Fa Yueh Wu

R. J. Baxter, Fa Yueh Wu

It is shown that the Ising model with three-spin interactions on a triangular lattice is equivalent to a site-colouring problem on a hexagonal lattice. The transfer matrix method is then used to solve the colouring problem. The colouring of two neighbouri