The 2-d Ising model in a magnetic field iπ/2 as a soluble case of the sixteen vertex model

De-Zhang Li, Yu‐Jun Zhao, Yao Yao +1 more

We study the residual entropy of a two-dimensional Ising model with crossing and four-spin interactions, both in the case of a zero magnetic field and in an imaginary magnetic field iπ/2kBT. The spin configurations of this Ising model can be mapped into the hydrogen configurations of square ice with the defined standard direction of the hydrogen bonds. Making use of the equivalence of this Ising system with the exactly solved eight-vertex model and taking the low temperature limit, we obtain the residual entropy. Two soluble cases in the zero field and one soluble case in the imaginary field are examined. In the case that the free-fermion condition holds in zero field, we find that the ground states in the low temperature limit include the configurations disobeying the ice rules. In another case in zero field where the four-spin interactions are −∞ and another case in imaginary field where the four-spin interactions are 0, the residual entropy exactly agrees with the result of square ice determined by Lieb in 1967. In the solutions to the latter two cases, we have shown alternative approaches to the residual entropy problem of square ice.

Fa Yueh Wu

T. D. Schultz, Daniel Charles Mattis, Élliott H. Lieb

C.S. Hsue, Ken‐Yu Lin, Fa Yueh Wu

An eight-vertex model with staggered (site-dependent) vertex weights is considered. The model is an extension of the usual one with translationally invariant weights and contains sixteen independent vertex weights. From its Ising representation it is seen that there are actually only eleven independent parameters. After discussing some general symmetry properties of this model, we consider in detail the soluble case of a free-fermion model. We find that the staggered free-fermion model may exhibit up to three phase transitions. Generally the specific heat has logarithmic singularities, expect in some special cases it has an exponent $\ensuremath{\alpha}=\frac{1}{2}$ and the system is frozen below a unique transition point. Conditions for these special cases are given.

De-Zhang Li, Xin Wang, Xiao‐Bao Yang

The Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on honeycomb, triangular and Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field i(π/2)kBT. We employ the decorated lattice technique, star-triangle transformation, and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero-field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model, while the models on the triangular and Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field i(π/2)kBT, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagomé lattices in the imaginary field still exhibit a non-zero residual entropy.

之康 尾関, Yukiyasu Ozeki, 秀稔 西森 +1 more