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 $L\times M$ 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.

30 pages, 4 figures, see also arXiv:1701.08722

References in corpus (5)

• Aspect-ratio dependence of thermodynamic Casimir forces
• The bulk, surface and corner free energies of the square lattice Ising model
• The square lattice Ising model on the rectangle II: Finite-size scaling limit
• Direct simulation of critical Casimir forces
• Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios

Cited by in corpus (15)

• Critical Casimir Effect: Exact Results
• The bulk, surface and corner free energies of the square lattice Ising model
• The square lattice Ising model on the rectangle II: Finite-size scaling limit
• Energy correlations of non-integrable Ising models: The scaling limit in the cylinder
• Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios
• Anisotropic scaling of the two-dimensional Ising model I: the torus
• Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit
• Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields
• The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants
• Transfer-matrix summation of path integrals for transport through nanostructures
• The Fate of Ernst Ising and the Fate of his Model
• Exact solution of the critical Ising model with special toroidal boundary conditions
• Approximate thermodynamics of the two-dimensional Ising model in an external magnetic field
• The boundary disorder correlation for the Ising model on a cylinder
• Random Matrices and the Free Energy of Ising-Like Models with Disorder