Polychromatic Potts model: a new lattice-statistical problem and some exact results

We propose a generalisation of the q-state Potts model in which two neighbouring spins in states s and s' have a coupling -J$( s, s'), and develop an exact graphical expansion of the partition function where each cluster in the graph can be coloured in one of C 6 q colours. This spin model reduces to the standard Potts model (one colour) when J , = J ; it also encompasses many new applications including correlared polychromatic bond percolation, the dilute branched polymer problem, and weighted clusters in the standard Potts model. The Potts model (for a review see Wu 1982) is a natural generalisation of the lattice-gas or Ising model in which each spin can exist in more (and fewer!) than two states. This model has attracted considerable recent attention, because the extra degree of freedom exhibited by q, the number of states, permits the model to encompass a wide range of physical phenomena of recent interest. These range from surface adsorption (Alexander 1975) and structural phase transitions (Aharony et a1 1977) to percolation (Kasteleyn and Fortuin 1969), biophysics, and diffusion in porous media (Stephen 1983). The purpose of this letter is to propose, and present exact results for, a generalisation of the standard Potts model; we call this the polychromatic Potts model. Special cases of this model are of considerable current interest. These include the problem of the dilute brunched polymers, a correlated polychromatic percolation processll, and weighted clusters in the standard Potts model. Consider a lattice of N sites and E edges. Associate with the ith site a spin variable si = 1 , 2 , , . , , q such that two neighbouring spins in spin states s and s' interact with a Potts interaction -J,S(s, s'). There is also an external field H, applied to spins in state s. Thus the reduced Hamiltonian is where K, = J,/ kT, L, = H,/ kT and the first summation is over all nearest-neighbour pairs. This defines a polychromatic Potts model which reduces to the standard 8 Present address: Division of Materials Research, National Science Foundation, Washington DC 20550, USA. (1 Supported in part by grants from NSF, ARO and ONR. ll This is not to be confused with random site polychromatic percolation, introduced by Zallen (1977). @ 1983 The Institute of Physics L751 L752 Letter to the Editor ‘monochromatic’ Potts model upon taking J, = J for all s and H , = 0, s > 1. Following Baxter (1973) we write eKsS(s, s’) = 1 + u,S(s, s’), (2) where U, =eKs1. Then the partition function defined by ( l ) , &(q; {K,}; IL,)) = f . . f n [I + vsS(si, sj)In exp(L,,), (3) can be conveniently expressed in terms of a graphical expansion. There are 2“ terms that arise when the product in (3) is expanded. Each term of the expansion is placed in a 1 : 1 correspondence with a graph G embedded on the lattice if we draw a bond on the edge of the lattice for each factor v,S(s, s’). After carrying out the sums over the 4 N states of the system, we are left with a single sum over the 2€ graphs G: s l = l s N = l ( i j ) z N ( q ; { K } ; W,)) = C ~ [ U % e x p ( ~ s , ) + & exp(L2sc)+. . .+U> exp(~,s,)l. (4) G c Here the product is taken over all connected clusters in G, including isolated points; b, = 0, 1 ,2 , . . . and s, = 1 , 2 , 3 , . . . are, respectively, the numbers of bonds and sites in each cluster. When K , = K, and L, = LS(s, l) , equation (4) reduces to the known expansion for the monochromatic Potts model (Wu 1978),