The critical two dimensional Ising model on right-triangle-shaped square lattices

Abstract

Using the bond-propagation algorithm, we study the critical Ising model on the square lattice with the shape of right triangle and free boundaries. The length of short leg is N and that of long leg is M. The aspect ratio between the two legs ρ=M∕N is fixed. For six ratios: ρ=1,2,3,4,8,16, the critical free energy density, internal energy density and specific heat on the lattices with 30≤N≤600 are calculated. 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 corner term in free energy density proportional to lnN is consistent to the conformal field theory in the accuracy of 10−10 at the least. It is found that the edge terms in the internal energy proportional to lnN related to the hypotenuse are the same for ρ=1,2,3,4,8,16, i.e. this term is geometry independent. However the edge terms in the specific heat proportional to lnN are geometry dependent, i.e. they are different for different ρ. The corner terms in the internal energy and specific heat are also obtained.