Simplified Transfer Matrix Approach in the Two-Dimensional Ising Model with Various Boundary Conditions

TL;DR

This paper extends a simplified transfer matrix method for the 2D Ising model to various boundary conditions, enabling better analysis of finite-size effects and relating to Brascamp-Kunz boundary conditions.

Contribution

It generalizes a recent transfer matrix solution to multiple boundary conditions and introduces a method to study finite-size scaling through linear combinations of partition functions.

Findings

Derived explicit transfer matrix solutions for different boundary conditions.

Established a relation between combined partition functions and Brascamp-Kunz boundary conditions.

Proposed a new approach for analyzing finite-size effects in the 2D Ising model.

Abstract

A recent simplified transfer matrix solution of the two-dimensional Ising model on a square lattice with periodic boundary conditions is generalized to periodic-antiperiodic, antiperiodic-periodic and antiperiodic-antiperiodic boundary conditions. It is suggested to employ linear combinations of the resulting partition functions to investigate finite-size scaling. An exact relation of such a combination to the partition function corresponding to Brascamp-Kunz boundary conditions is found.