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

## TL;DR

This paper provides an exact calculation of finite-size effects on the free energy of the square lattice Ising model on a rectangle, confirming conformal field theory predictions and revealing corner effects at criticality.

## Contribution

It derives exact, rapidly converging series for Casimir scaling functions in the finite-size scaling limit, including corner effects at the critical point.

## Key findings

- Confirmed conformal field theory predictions for Casimir effects
- Derived exact series for Casimir potential and force
- Identified logarithmic divergence due to corners at criticality

## 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.