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arXiv:2608.19274

Exact expressions of correlation functions between two spins in the boundary row of the two-dimensional rectangular Ising model with periodic-free boundary conditions and finite size

Mei Tao

Published Aug 18, 2026• 0 citations
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Abstract

We present exact expressions of correlation functions between two spins in the boundary row of the two-dimensional rectangular Ising model with periodic-free boundary conditions and finite size. Some properties of exact expressions obtained are discussed. The fundamental property of the Ising model that the long range order emerge as the temperature decrease is shown clearly; expressions of the correlation functions in the thermodynamic limit varies depending on the order of taking limits of two parameters L and N; the impact of different sizes on correlation functions is illustrated with the aid of diagrams. Expressions of the correlation function discussed in this paper in the thermodynamic limit has been presented by previous researchers, and we prove that expressions obtained in this article is identical in form to expressions provided by previous researchers in the thermodynamic limit.

DOI
10.48550/arxiv.2608.19274

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We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation functions of the model with periodic-periodic boundary conditions on a finite lattice. The corresponding forms in the thermodynamic limit are presented, which show the short-range order. Then, we present the exact expression of the correlation function of the two farthest pair of spins in a column of the model with periodic-free boundary conditions on a finite lattice. Again, the corresponding form in the thermodynamic limit is discussed, from which the long-range order clearly emerges as the temperature decreases.
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