Izmailian N Sh, Hu Chin-Kun
Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Mar;65(3 Pt 2A):036103. doi: 10.1103/PhysRevE.65.036103. Epub 2002 Feb 7.
Let f, U, and C represent, respectively, the free energy, the internal energy, and the specific heat of the critical Ising model on the MxN square lattice with periodic boundary conditions, and f(infinity) represents f for fixed M/N and N-->infinity. We find that f, U, and C can be written as N(f-f(infinity))= summation operator(infinity)(i=1)f(2i-1)/N(2i-1), U=-square root of [2]+ summation operator(infinity)(i=1)u(2i-1)/N(2i-1), and C=8 ln N/pi+ summation operator(infinity)(i=0)c(i)/N(i), i.e., Nf and U are odd functions of N(-1). We also find that u(2i-1)/c(2i-1)=1/square root of [2] and u(2i)/c(2i)=0 for 1 < or = i <infinity and obtain closed form expressions for f, U, and C up to orders 1/N(5), 1/N(5), and 1/N(3), respectively, which implies an analytic equation for c(5).
设(f)、(U)和(C)分别表示具有周期性边界条件的(M\times N)方形晶格上临界伊辛模型的自由能、内能和比热，(f(\infty))表示固定(M/N)且(N\rightarrow\infty)时的(f)。我们发现(f)、(U)和(C)可表示为(N(f - f(\infty)) = \sum_{i = 1}^{\infty} \frac{f_{2i - 1}}{N^{2i - 1}})，(U = -\sqrt{2} + \sum_{i = 1}^{\infty} \frac{u_{2i - 1}}{N^{2i - 1}})，以及(C = \frac{8\ln N}{\pi} + \sum_{i = 0}^{\infty} \frac{c_i}{N^i})，即(Nf)和(U)是(N^{-1})的奇函数。我们还发现对于(1\leq i\lt\infty)，(\frac{u_{2i - 1}}{c_{2i - 1}} = \frac{1}{\sqrt{2}})且(\frac{u_{2i}}{c_{2i}} = 0)，并分别得到了(f)、(U)和(C)直至(1/N^5)、(1/N^5)和(1/N^3)阶的封闭形式表达式，这意味着得到了(c_5)的一个解析方程。
