The square lattice Ising model on the rectangle I: Finite systems

TL;DR

This paper derives an exact formula for the partition function of the square lattice Ising model on a rectangle with open boundaries, highlighting finite-size effects and their relation to the critical Casimir force.

Contribution

It introduces a new exact decomposition of the free energy into a strip part and a residual part, revealing finite-size effects and mapping residual contributions to an effective spin model.

Findings

Exact partition function for arbitrary size and temperature.

Finite-size residual free energy related to Casimir force.

Residual free energy diminishes exponentially for large aspect ratios.

Abstract

The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size $L\times M$ and temperature. We start with the dimer method of Kasteleyn, McCoy & Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, $F(L,M)=F_\mathrm{strip}(L,M)+F_\mathrm{strip}^\mathrm{res}(L,M)$, where the residual part $F_\mathrm{strip}^\mathrm{res}(L,M)$ contains the nontrivial finite-$L$ contributions for fixed $M$. It is given by the determinant of a $\frac{M}{2}\times \frac{M}{2}$ matrix and can be mapped onto an effective spin model with $M$ Ising spins and long-range interactions. While $F_\mathrm{strip}^\mathrm{res}(L,M)$ becomes exponentially small for large $L/M$ or off-critical temperatures, it leads to important finite-size effects such as the critical Casimir force near criticality. The relations to the Casimir potential and the Casimir force are discussed.