The square lattice Ising model on the rectangle II: Finite-size scaling limit

Abstract

Based on the results published recently [J. Phys. A: Math. Theor. 50, 065201 (2017)], the universal finite-size contributions to the free energy of the square lattice Ising model on the $L\times M$ rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit $L,M\to \infty$, $T\to T_{\mathrm{c}}$, with fixed temperature scaling variable $x\propto (T/T_{\mathrm{c}}-1)M$ and fixed aspect ratio $\rho \propto L/M$. We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point $T=T_{\mathrm{c}}$ we confirm predictions from conformal field theory by Cardy & Peschel [Nucl. Phys. B 300, 377 (1988)] and by Kleban & Vassileva [J. Phys. A: Math. Gen. 24, 3407 (1991)]. The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.

31 pages, 6 figures, second part of arXiv:1609.01963, some text and references added, several small errors fixed, figure 5 changed, accepted

References in corpus (9)

• The Casimir effect: from quantum to critical fluctuations
• Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations
• Monte Carlo simulation results for critical Casimir forces
• Aspect-ratio dependence of thermodynamic Casimir forces
• Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces
• The square lattice Ising model on the rectangle I: Finite systems
• The bulk, surface and corner free energies of the square lattice Ising model
• 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 (10)

• Critical Casimir Effect: Exact Results
• The square lattice Ising model on the rectangle I: Finite systems
• 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
• 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
• Exact solution of the critical Ising model with special toroidal boundary conditions
• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• The boundary disorder correlation for the Ising model on a cylinder