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

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 and temperature. We start with the dimer method of Kasteleyn, McCoy and 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, , where the residual part contains the nontrivial finite-L contributions for fixed M. It is given by the determinant of a matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. While 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.