Showing papers on "Potts model published in 1976"

Equivalence of the Potts model or Whitney polynomial with an ice-type model

Abstract: The partition function of the Potts model (1952) on any lattice can readily be written as a Whitney polynomial (1932). Temperley and Lieb (Proc. R. Soc., vol.A322, p.251 of 1971) have used operator methods to show that, for a square lattice, this problem is in turn equivalent to a staggered ice-type model. Here the authors rederive this equivalence by a graphical method, which they believe to be simpler, and which applies to any planar lattice. For instance, they also show that the Potts model on the triangular or honeycomb lattice is equivalent to an ice-type model on a Kagome lattice.

Renormalization of the Potts model

Abstract: The author studies the Potts model with a general number of states. First, he discusses the situation in which the Landau theory leads to a first order transition, but which does show fixed points of the renormalization group. Here, there are many questions which need further clarification. Then he describes, rather pedagogically, the logic behind the application of dimensional regularization to critical phenomena. He argues that this is a particularly natural approach. This technique is then applied to the Potts model, for which the critical exponents are computed to O((6-d)2), when there is a fixed point. The one state results, which correspond to the percolation problem, are compared with other calculations and with numerical simulation.

Renormalisation-group results for the three-state Potts model

Abstract: A lower-bound renormalisation transformation of the type introduced by Kadanoff (see J. Statist. Phys., vol.14, p.171, 1976) is applied to the three-state Potts model. For both d=2 and d=3 the Kadanoff transformation predicts a continuous transition. Values for the critical exponents and the critical temperature are reported. The consistency of the d=2 results with series estimates gives the authors some confidence in the predictions for d=3.

The three-spin Ising model as an eight-vertex model

Abstract: The three-spin Ising model is related to a special case of the eight-vertex model by a spin transformation. It follows that the models have the same free energy magnetization and polarization.

Model fluid mixture which exhibits tricritical points.—Part 1

Abstract: A ternary fluid mixture, in which molecular pairs of like species have no interaction and molecular pairs of unlike species interact as hard spheres, is studied in a mean-field approximation., As the pressure is raised the system passes from a state of complete miscibility to states with 3 bicritical points, to a state with 3 tricritical points, to states with 6 bicritical points, to a state with a quadruple point, and finally to states of almost total mutual immiscibility. The behaviour near a tricritical point is discussed in detail, and the system compared briefly with a lattice model—the 3-states Potts model.

Estimates of the critical exponent β for the Potts model using a variational approximation

Abstract: The q-state Potts model on a square lattice is transformed into a staggered ice model. A variational principle for the largest eigenvalue of the transfer matrix of this model is used to develop a set of matrix equations. In the limit of these matrices becoming infinite, the equations determine the partition function and order parameter of the Potts model exactly. We have solved the equations numerically for finite matrices, obtaining estimates of these quantities and, as a result, the critical exponent β.

Two-layered Ising model with short-range two-spin interactions showing the critical behaviour of the Ashkin-Teller model

Abstract: A two-layered Ising model is presented which maps to the Ashkin-Teller (AT) model (1943) for certain ratios of the two-spin couplings. Thus the model is able to exhibit the same critical exponents as the AT model, which are in general different from the usual Ising exponents; even continuous dependence of the critical exponents on the energy parameters is to be expected.

wenee of the Potts model or Whitney polynomial no b-t9pe model

Abstract: The partition function of the Potts model on any lattice can readily be written as a Whitney polynomial. Temperley and Lieb have used operator methods to show that, for a square lattice, this problem is in turn equivalent to a staggered ice-type model. Here we rederive this equivalence by a graphical method, which we believe to be simpler, and which applies to any planar lattice. For instance, we also show that the Potts model on the triangular or honeycomb lattice is equivalent to an ice-type model on a Kagomt lattice. There is at present considerable interest amongst statistical mechanics and com- binatoridits in the evaluation of the Whitney polynomial of a graph. This is because thisproblem is the same as obtaining the partition function of the Potts (1952) model on hegraph (Kasteleyn and Fortuin 1969, Fortuin and Kasteleyn 1972, Baxter 1973, lahiein addition it contains the percolation and colouring problem as special cases. Some exact results are available for the square lattice graph. In particular, when the arsociated Potts model has two states per spin, it becomes the king model and the ppblem is soluble. Also, Temperley and Lieb (1971) have established a remarkable esuivalence between the Whitney polynomial €or a square lattice 2 and the partition hKtbn Of a staggered ice-type model on a related square lattice 2". . Although the polynomial has not yet been evaluated exactly for the square lattice, it to think that it may be. The problem has therefore attracted attention $"W theoreticians, to the extent that we feel it worthwhile presenting a re- htiOn of the equivalence established by Temperley and Lieb. We use graphical which we believe to be simpler than the operator method of Temperley and 'b. they apply to any planar lattice, regular or not.