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arXiv 0907.0925cond-mat.stat-mech
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格条上磁场中Potts模型的配分函数和转移矩阵的结构

Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

Shu-Chiuan Chang, Robert Shrock

备注：
Reference added to a relevant paper by F. Y. Wu
发表信息：
J. Stat. Phys. 137, 667-699 (2009)
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摘要：

我们确定的一般结构的配分函数的$q$状态Potts模型在外部磁场，$Z(G,q,v,w)$为任意$q$，温度变量$v$，和磁场变量$w$，循环，莫比乌斯，和自由带状图$G$的正方形（sq），三角形（三），和蜂窝状（hc）格子宽度$L_y$和任意大长度$L_x$。对于循环的情况下，我们证明了配分函数具有$Z(Λ,L_y \times L_x,q,v,w)=\sum_{d=0}^{L_y} \tilde c^{(d)} Tr[(T_{Z,Λ,L_y,d})^m]$的形式，其中$Λ$表示格型，$\tilde c^{(d)}$是指定的多项式次数$d$在$q$，$T_{Z,Λ,L_y,d}$是相应的转移矩阵，和$m=L_x$（$L_x/2$）的$Λ=sq, tri (hc)$，分别。对于莫比乌斯带，给出了类似的公式，而对于自由带，仅出现$T_{Z,Λ,L_y,d=0}$。我们展示了一种计算任意$L_y$的$T_{Z,Λ,L_y,d}$的方法，并给出了说明性的例子。对于任意$L_y$，给出了具有$d=L_y$和$d=L_y-1$的$T_{Z,Λ,L_y,d}$的显式结果。我们找到了行列式$det(T_{Z,Λ,L_y,d})$的非常简单的公式。我们还给出了正方形格的自对偶循环条的结果。

英文摘要：

We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, Möbius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(Λ,L_y \times L_x,q,v,w)=\sum_{d=0}^{L_y} \tilde c^{(d)} Tr[(T_{Z,Λ,L_y,d})^m]$, where $Λ$ denotes the lattice type, $\tilde c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,Λ,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $Λ=sq, tri (hc)$, respectively. An analogous formula is given for Möbius strips, while only $T_{Z,Λ,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,Λ,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,Λ,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,Λ,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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