Abstract
Oitmaa(1981) 曾計算易辛模型 (Ising model) 在具有最近鄰及次近鄰交互作用 的方形晶格上磁化率高溫級數展開至 11 項。我們將此級數擴展至 18 項在各向同性 的情形，級數的前 11 項和 Oitmaa所得的完全一致。曾錦桐(1996) 求得同樣的級數 至 12 項，但級數的後 2 項和我們所得的級數有些微差距。我們利用一種由 Sykes 提出的圖形計算理論，來幫助我們使用電腦來計算級數所需的晶格常數。李賜福 (1996) 計算易辛模型在同樣的晶格上自發磁化強度及磁化率低溫級數 展開分別至 24 及 20 項。 在這篇論文中，我們想知道易辛模型在含次近鄰交互作用的方形晶格與只含最近 鄰交互作用的方形晶格否俱屬同一普適種類。按照區分普適種類的假定，系統的臨界 性質與它的空間維度及序參量(order parameter) 的個數有關，而對短程力而言，交 互作用的方式與臨界特性無關。至目前為止，所有已解出的平面晶格易辛模型自發磁 化強度的臨界指數均具有 Beta = 0.125 的結果。關於零磁場磁化率則尚未有精確解 ，但由已知的級數解及一些嚴格的證明顯示，臨界指數具有 Gamma = 1.75 的結果 。 我們利用 Padeapproximants 來分析級數，估計得到 Gamma = 1.751 (+ -) 0.002 ，而李賜福得到 Beta = 0.125 (+ -) 0.003 的結果。我們利用在本研究中所 估計到的臨界點值 Wc = 0.187937 ，從李賜福所得的磁化率低溫級數中取定見分析 得到 Gamma = 1.785。上述的結果顯示，所得到的臨界指數 Beta 和 Gamma 與預期的值極為接近。因此本研究證實了普適種類的假定，易辛模型在含次近鄰交互 作用的方形晶格與只含最近鄰交互作用的方形晶格應同屬於二維易辛系統普適種類。The high-temperature series expansion of zero-fieldsusceptibility of Ising system on a square lattice with nearestand next nearest neighbor interactions had been derived up tothe 11th order by J.Oitmaa(in 1981). We extended this series tothe 18th order in a special case where the interactions areisotropic, and Oitmaa's results are confirmed. Tseng(in 1996)extended the same series to the 12th order, but the last twoterms are slightly different from ours. Lee(in 1996)calculated the low-temperature series expansions of thespontaneous magnetization and zero-field susceptibility ofIsing model on the same lattice to the 24th and 20th orderrespectively. We use a special counting theorem (Sykes, Nagleand Temperley) which can reduce the numeration of latticeconstants and is suited for computer calculation. In thisdissertation, we want to know whether the critical exponents ofthe Ising model on the square lattice with interactions beyondfirst neighbors belongs to the same universality class withtwo-dimemsional Ising model. So far the zero-fieldsusceptibility of two-dimensional Ising model is stillunsolved, but the critical exponent $\gamma \;(T>T_c)$ and$\gamma ^{\prime} \;(T<T_c)$ are considered to be 1.75, whilethe exact value of the critical exponent $\beta = 0.125$. Weuse the Pad\'{e} approximants to estimate the criticalexponent, and find that $\gamma = 1.751 \pm 0.002$, while Leeconcluded that $\beta =0.125 \pm 0.003$. A biased estimatevalue of $\gamma ^{\prime}\approx 0.785$ is obtained from Lee'sseries by using the critical point $w_c=\tanh (J/kT) =0.187937$which is estimated from this research. These results providestrong evidence that $\gamma$ and $\beta$ both take the valuewhich are very near the universal value. Therefore our resultsare consistent with the universality hypothesis which predictsthat the Ising model on the square lattice with multi-pairsinteractions belong to the same universality class of two-dimensional Ising model with nearest pair interaction only.