The One-Dimensional KDP and Other Ice Rule Models

Abstract

One-dimensional ice, KDP, F, eight-vertex and general six-vertex models are discussed. As these models cannot be defined uniquely in one dimension, various possibilities are presented. For a particular one-dimensional lattice, the KDP, the eight-vertex and the six­ vertex models undergo a first-order phase transition. The KDP model is considered also in the presence of electric fields. Thermodynamic quantities of the model show unusual behavior. dimensional ice models are discussed. In § 3 the one-dimensional KDP and F models are considered. We will see that only the KDP model with a particular boundary condition for the vertical direction undergoes a first-order phase transition, and that the model is equivalent to Nagle's one-dimensional KDP model. Occur­ rence of a phase transition in one-dimensional systems is briefly discussed. In § 4 the KDP model which has a transition is considered in external electric fields. The careful consideration of the possible ways of defining the one-dimensional KDP models is necessary when we introduce the external fields precisely. It is shown that the transition temperature is increased and the order of the trasition is not changed by the introduction of the fields. The latter fact is a remarkable difference from the result of the two-dimensional KDP model. In § 5 the one­ dimensional eight-vertex model is considered and the possible phase transition is also of first order which differs from the two-dimensional case. Condition for the occurrence of the transition in the general six-vertex model which is a generalization