The Six-Vertex Model
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In this chapter we consider exact solutions for a class of two-dimensional models which have completely different behaviour from the Ising model. These models are based on a square lattice version of a completely hydrogen-bonded network, like that of ice I (see Appendix A.3). With all bond configurations having equal energy, the free energy is related to the ground-state (zero-point) entropy of a d = 2 ice analogue (square ice), which was evaluated by Lieb (1967a). As well as water, certain other substances, such as KH2PO4 (potassium dihydrogen phosphate), crystallize as four-coordinated hydrogen-bonded networks. KH2PO4 is a ferroelectric with a spontaneous electrical polarization in zero field below a critical temperature. Square lattice analogues of ferroelectrics and antiferroelectrics can be constructed by appropriate assignment of energies to the various bond configurations. Lieb (1967b, 1967c) obtained expressions for the free energy of these models by an extension of the method used for square ice. In contrast to the Ising model, some results can be obtained for non-zero fields (electric fields in this case) (Sutherland 1967, Lieb 1969). The development in this chapter is based on the extensive review article of Lieb and Wu (1972).
KeywordsPartition Function Critical Temperature Ising Model Vertical Edge Equilibrium Free Energy
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