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Communications in Theoretical Physics

A Modified Recursion Method for One-Dimensional Potts Model

Yang Zhan-Ru

Published Mar 1, 1985• 1 citation
Publisher

Abstract

A modified recursion method is proposed and used to produce the exact result of the partition function for a one-dimensional Potts model with either free boundary or periodic boundary. The method is compared with both transfer matrix method and Marchi and Vila's method. It may perhaps provide a possible way to solve the Potts model on two-dimensional square lattice.

DOI
10.1088/0253-6102/4/2/157

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A modified recursion method is proposed and used to produce the exact result of the partition function for a one-dimensional Potts model with either free boundary or periodic boundary. The method is compared with both transfer matrix method and Marchi and Vila's method. It may perhaps provide a possible way to solve the Potts model on two-dimensional square lattice.
We present exact calculations of the partition function for the q-state Potts model for general q, temperature and magnetic field on strips of the square lattices of width Ly=2L_{y}=2Ly​=2 and arbitrary length $L_x = m $ with periodic longitudinal boundary conditions. A new representation of the transfer matrix for the q-state Potts model is introduced which can be used to calculate the determinant of the transfer matrix for an arbirary m×mm \times mm×m lattice with periodic boundary conditions.
Abstract The Ising model is a well known statistical model which can be solved exactly by various methods. The most familiar one is the transfer matrix method. Sometimes it can be difficult to approach the open boundary case rather than periodic boundary ones in higher dimensions. But physically it is more intuitive to study the open boundary case, as it gives a closer view of the real system. We have introduced a new method called the pairing method to determine the exact partition function for the simplest case, a 1D Ising lattice. This method simplifies the problem's complexities and reduces it to a pure combinatorial problem. The study also reveals that it is possible to apply this pairing method in the case of a 2D square lattice. The obtained results agree perfectly with the values in the literature and this new approach provides an algorithmic insight to deal with such problems.
The partition function of the two-dimensional Ising model on a square lattice with nearest-neighbour interactions and periodic boundary conditions is investigated. Kaufman [Phys. Rev. 76, 1232--1243 (1949)] gave a solution for this function consisting of four summands. The summands are rewritten as functions of a low-temperature expansion variable, resulting in polynomials with integer coefficients. Considering these polynomials for system sizes 2m×2n2^m\times 2^n2m×2n (m,n∈Nm,n\in\Nm,n∈N), a variety of recursion relations in m,nm,nm,n are found. The recursions reveal a rich structure of the partition function and can be employed to render the computer algebra calculation of the microcanonical partition function more efficient.
We have evaluated numerically the zeros of the partition function of the qqq-state Potts model on the square lattice with reduced interactions KKK. On the basis of our numerical results, we conjecture that, both for finite planar self-dual lattices and for lattices with free or periodic boundary conditions in the thermodynamic limit, the zeros in the Re(x)>0\mathrm{Re}(x)>0Re(x)>0 region of the complex $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({e}^{K}\ensuremath{-}1)/\sqrt{q}$ plane are located on the unit circle ∣x∣=1|x|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1∣x∣=1.