Existence of Three Transition Temperatures in Decorated Triangular and Square Ising Lattices with Anisotropic Couplings

on either all longitudinal or all transverse bonds can exhibit three successive phase transitions as the temperature is varied throughout the whole region. The circumstance of the same sort is found in the case of triangular lattice. We investigate in § 2 a partly-decorated triangular lattice as shown in .Fig. 1; viz. the triangular lattice in which no transversely-lying bonds are decorated but all obliquely-drawn bonds are decorated in the same manner throughout them. The partition function of the system of Ising spins distributed on that lattice can be reduced to the partition functi~n of a simple triangular lattice" by a process of iteration. By making use of the dual and the star-triangle transforma tions on that triangular lattice we determine the transition temperatures and show it is possible to select the parameters of exchange couplings so that the system has three successive transition temperatures. We consider in § 3 the triangular and square lattices in which every bond