Erratum: Finite-size behavior of the critical Ising model on a rectangle with free boundaries [Phys. Rev. E 86, 041149 (2012)]

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.

The bond-propagation (BP) algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works [X.-T. Wu et al Phys. Rev. E 86, 041149 (2012) and Phys. Rev. E 87, 022124 (2013)], we reach much higher accuracy (10 −26 ) of the internal energy and specific heat,compared to the accuracy 10 −11 of the internal energy and 10 −9 of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of some edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than 1/S, with S the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than 1/S for the internal energy, and logarithmic correction terms of all orders for the specific heat.

No abstract

Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches 10(-26). Based on accurate data on several finite systems with linear size up to N=2000, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles π/3, π/2, and 2π/3 are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal.