Exact partition functions of the ising model on L × L square lattices with free boundary conditions up to L = 22

Abstract

The exact partition functions of the Ising model on L × L square lattices with free boundary conditions are evaluated up to L = 22 for the first time by using a microcanonical transfer matrix and counting all possible spin states. The total number of states is 222×22 = 2484 ≈ 5.0 × 10145 for L = 22. Using the partition function zeros in the complex temperature plane of the Ising model on L× L square lattices with free boundary conditions, we analyze the critical behavior of the square-lattice Ising antiferromagnet and compare them with the results for periodic boundary conditions (the most popular boundary conditions). Our results show clearly that free boundary conditions are much more efficient than periodic boundary conditions in the study of an antiferromagnet.