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

Abstract

The bond-propagation 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 (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy ($10^{-28}$) 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 all 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.

Appendices

Appendix 1: Raw Data

In this appendix, we present some selected raw data of internal energy density on square lattice in Table 25. With these data, the future readers can check out fittings. Similarly, we present some selected raw data for specific heat calculated by BP algorithm in Table 26. In Table 27, we present some selected raw data of the internal energy density on triangle lattice for the five shapes. The Table 28 are some selected raw data of the specific heat on triangle lattice for the five shapes.

Table 25 Some selected raw data of the free energy density for $\rho =1,2,4,8,16$

Table 26 Some selected raw data of the specific heat for $\rho =1,2,4,8,16$

Table 27 Some selected raw data of the internal energy density for the five shapes

Table 28 Some selected raw data of the specific heat for the five shapes

Appendix 2: Some Data Beyond the Fitting Size Range

In Sect. 3.1, we fit the critical internal energy density on square lattice with the size range from $N\ge 30$. We can predict the data outside this range. This would provide one additional diagnostic on our choice of functional form.

In Table 29, we present the internal energy density obtained by the fitting formula Eq. (18) with size $N=5,10,15,20$. For comparison, we also calculate these data with BP algorithm. Since the sizes of lattice are very small, the errors due to the accumulation of round-off error are less than $10^{-31}$. Because the fitting formula Eq. (18) is only expanded to $15$th order, the truncation error should be proportional to $N^{-16}$. They are about $6.0\times 10^{-12}$, $10^{-16}$, $10^{-19}$ and $2.0\times 10^{-21}$ for $N=5,10,15,20$ respectively. These truncation errors are much larger than the errors in the BP algorithm. Therefore we can take the data obtained with BP algorithm as more accurate values to check the truncation errors of the data obtained by the fitting formula Eq. (18). Take the internal energy density as an example, see Table 29. The absolute values of difference between the data obtained with Eq. (18) and data obtained with BP algorithm are about $4\times 10^{-10}$, $3 \times 10^{-13}$, $3\times 10^{-17}$, and $5\times 10^{-20}$ for $N=5,10,15$, and $20$, for $\rho =1$. They agree with the truncation errors estimated above approximately. It is so for other cases.

Table 29 The internal energy density of the systems with size beyond our fitting data. The left column are obtained with the fitting formula (18) and the right column are obtained with BP algorithm

In Table 30, we present the specific heat obtained by the fitting formula Eq. (24) and the data by BP algorithm with size $N=5,10,15,20$ (Tables 31, 32).

Table 30 The specific heat with size beyond our fitting data. The left column are obtained with the fitting formula (24) and the right column are obtained with BP algorithm

Table 31 Some internal energy density for five shapes on the triangle lattice with size beyond our fitting range

Table 32 Some specific heat for five shapes on the triangle lattice with size beyond our fitting range