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  • Open Access

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

De-Zhang Li1 and Xin Wang2,3,*

  • *Contact author: x.wang@cityu.edu.hk
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APS Open Sci. 1, 000125 – Published 4 September, 2026

DOI: https://doi.org/10.1103/1dby-z8np

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.

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APS Open Sci. 1, 000125– Published 4 September, 2026
Vol. 1
  • Received 18 April 2026
  • Accepted 25 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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