Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

Abstract

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.

Critical two-dimensional Ising model with free, fixed ferromagnetic, fixed antiferromagnetic, and double antiferromagnetic boundaries

“…The BCs (0000) have been studied in previous work [20,23]. The surface term is given by f surf (0) = …”

“…It was developed to compute the partition function of the Ising model in two dimensions [21,22]. Subsequently, the BP algorithm for the internal energy and specific heat was developed [23]. It is so powerful that very large system sizes and very accurate numerical results can be reached.…”

“…It is so powerful that very large system sizes and very accurate numerical results can be reached. With this algorithm, the calculations have been carried out on square and triangular lattices with free boundaries [19,20,23]. The results of free energy, internal energy, and specific heat are surprisingly accurate to 10 −26 .…”

Efficient Algorithms for the Two-Dimensional Ising Model with a Surface Field

“…The corner's logarithmic corrections agree with the conformal field theory [17] in an accuracy of 10 −14 . Applying these algorithms, the edge and corner terms of the internal energy and specific heat have been obtained numerically for rectangular, triangular, rhomboid, trapezoid and hexagonal shape [18]. The numerical result is so accurate (to the accuracy 10 −28 ), that the exact results on these terms are conjectured.…”

“…We complete the algorithm and proposed the algorithm, in which the BP series and Y − ∆ transformations are included, for the specific heat [18].…”