Finite-size behavior of the critical Ising model on a rectangle with free boundaries

Abstract

Using the bond-propagation algorithm, we study the Ising model on a rectangle of size M×N with free boundaries. For five aspect ratios, ρ=M/N=1, 2, 4, 8, and 16, the critical free energy, internal energy and specific heat are calculated. The largest size reached is M×N=64×10(6). The accuracy of the free energy reaches 10(-26). Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The fitted bulk free energy density is given by f(∞)=0.92969539834161021499(1), compared with Onsager's exact result f(∞)=0.929695398341610214985.... We confirm the conformal field theory (CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be c=0.5±1×10(-10), compared with the CFT result c=0.5. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding for finite-size scaling is discussed. In the second-order correction of the free energy, we confirm the geometry dependence predicted by CFT and determine a geometry-independent constant beyond CFT. High-order corrections are also obtained.