The Potts model on a Bethe lattice with nonmagnetic impurities
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DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
The Potts Model on a Bethe Lattice with Nonmagnetic Impurities S. V. Semkin and V. P. Smagin Vladivistok State University of Economics and Service (VSUES), Vladivostok, 690600 Russia e-mail: [email protected] Received April 17, 2015
Abstract—We have obtained a solution for the Potts model on a Bethe lattice with mobile nonmagnetic impurities. A method is proposed for constructing a “pseudochaotic” impurity distribution by a vanishing correlation in the arrangement of impurity atoms for the nearest sites. For a pseudochaotic impurity distribution, we obtained the phase-transition temperature, magnetization, and spontaneous magnetization jumps at the phase-transition temperature. DOI: 10.1134/S1063776115100131
1. INTRODUCTION The Potts model [1] is formulated as follows. Let us consider a certain regular lattice. We put in correspondence to each site a quantity σi (“spin”) that can assume n different values (say, 1, 2, …, n). Two neighboring spins σi and σj interact with energy –Jpδ(σi, σj), where
⎧1, δ(σ i , σ j ) = ⎨ ⎩0,
σi = σ j, σi ≠ σ j.
Let us suppose that an external field, H, is acting on state 1. The total energy is then given by
E = −J pΣ (i, j )δ(σ i , σ j ) − H Σ i δ(σ i ,1). We assume that nonmagnetic atoms (“impurities”) are located at some lattice sites. Let b be the fraction of spins; accordingly, 1 – b is the fraction of impurities in the lattice. We can consider two types of impurities: “frozenin” stationary impurities that are distributed at random among lattice sites without correlation and “mobile” impurities that can move over sites and are in thermodynamic equilibrium with the matrix. The model with frozen-in impurities is most interesting because most magnets with impurities belong precisely to this type. Unfortunately, the exact solution of the problem with frozen-in impurities cannot be obtained even for simple lattices. It will be shown below, however, that the exact solution to the problem with mobile impurities can be obtained on the Bethe lattice. This solution, which is probably interesting as such, makes it possible to analyze the behavior of a system with frozen-in impurities. For mobile impurities, the correlation (covariation) in the arrangement of impurities at neighboring lattice sites can be calculated. Imposing the condition of vanishing on this cor-
relation, we obtain a impurity distribution that we called the “pseudochaotic” distribution. Although such a distribution of impurities over lattice sites is not completely random, the percolation threshold (e.g., for a pseudochaotic distribution on the Bethe lattice) coincides with the threshold for frozen-in impurities. We believe that the behavior of a system with pseudochaotic mobile impurities is a good approximation for a magnet with frozen-in impurities. Thus, we consider the Potts model with mobile impurities. Let us suppose that variables σi can assume, apart from values of 1, 2, …, n, a zero value when a nonmagnetic impurity is located at a site. We assume that the fo
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