Title: 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.