Order-disorder in hexagonal lattices

Abstract

Some relations for the partition functions of two-dimensional infinite triangular and honeycomb lattices with I s i n g interaction between neighbours without magnetic field are given in the general case of different interactions in the three directions. Exact evaluation of these partition functions is obtained with the aid of the method of B r u r i a K a u f m a n. The theory of this method is simplified by avoiding the use of abstract group-properties. Some thermodynamic properties of the isotropic hexagonal lattices are given. These are compared with the corresponding properties of the quadratic lattice. All honeycomb and nearly all triangular lattices have a transition temperature. At this temperature the energy is continuous and the specific heat becomes infinite as — In | T — T c |, as in the rectangular case, solved by O n s a g e r and K a u f m a n. There are exceptional triangular lattices without transition temperature: for example those with equal negative interactions.