Recursion relations for the partition function of the two-dimensional Ising model

2006/05/23 by Michael Kästner, Kastner, Michael
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.cond-mat/0605568

openalex publication_date 2006/05/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

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 2^m× 2ⁿ (m,n∈\N), a variety of recursion relations in m,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.

Citations

• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition

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