Solution of the Ising model with Brascamp-Kunz boundary conditions by the transfer matrix method

TL;DR

This paper derives the exact solution of the square lattice Ising model with Brascamp-Kunz boundary conditions using the transfer matrix method, providing new insights into boundary effects and critical phenomena.

Contribution

It introduces a transfer matrix approach to solve the Ising model with Brascamp-Kunz boundary conditions via a novel boundary transformation and fermionic representation.

Findings

Exact partition function derived for Brascamp-Kunz boundary conditions

Fisher zeros analytically calculated and critical point identified

Comparison of transfer matrix approaches for different boundary conditions

Abstract

The square lattice Ising model under the Brascamp-Kunz boundary conditions is a well-known exactly solvable lattice model. The exact solution of this system has been derived within the framework of Pfaffian-type method. In this paper we provide a derivation for the solution by the Schultz-Mattis-Lieb method in the transfer matrix formalism. We set special interactions on the boundaries and take certain limit of these interactions, so that the system under the Brascamp-Kunz boundary conditions is transformed into another system under the toroidal boundary conditions. The Schultz-Mattis-Lieb method is applied to the mapping system and the partition function is exactly solved in the fermionic representation. The Fisher zeros are analytically calculated and the physical critical point is identified. We also discuss the difference between the transfer matrix approaches to the Brascamp-Kunz and to the toroidal boundary conditions. Our work introduces a member to the family of transfer-matrix-based studies for Ising model under various boundary conditions.