二维非齐次Ising格上的迭代星三角变换
Iterated star-triangle transformation on inhomogeneous 2D Ising lattices
摘要：
我们考虑具有任意正或负最近邻耦合$K_i(\vec{r})$的无限或周期二维三角形伊辛晶格，其中$\vec{r}$和$i$分别表示键的位置和方向。对于具有耦合$\{K_i^{(n)}(\vec{r})\}$的$n=1,2,\ldots,$，将星三角变换迭代应用于具有耦合$\{K_i^{(0)}(\vec{r})\}$的初始格$\mathcal{T}(0)$，生成格序列$\mathcal{T}(n)$。当$\mathcal{T}(0)$包括足够强的阻挫斑时，将出现复杂的耦合。我们表明，尽管如此，变量$1/\sinh 2K_i^{(n)}\!(\vec{r})$仍然局限于工会${\mathbb{R}} \cup \rm{i}{\mathbb{R}}$的实轴和虚轴。这同样适用于具有自由边界的晶格，只要我们区分“后退”和“前进”边界，后者具有必须由适当选择的协议固定的自由度。这项研究建立了一个框架，为未来的分析和数值工作，这种挫折伊辛格。
英文摘要：
We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings $K_i(\vec{r})$, where $\vec{r}$ and $i$ indicate the bond position and orientation, respectively. Iterative application of the star-triangle transformation to an initial lattice $\mathcal{T}(0)$ with a set of couplings $\{K_i^{(0)}(\vec{r})\}$ generates a sequence of lattices $\mathcal{T}(n)$, for $n=1,2,\ldots,$ with couplings $\{K_i^{(n)}(\vec{r})\}$. When $\mathcal{T}(0)$ includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables $1/\sinh 2K_i^{(n)}\!(\vec{r})$ remain confined to the union ${\mathbb{R}} \cup \rm{i}{\mathbb{R}}$ of the real and the imaginary axis. The same holds for a lattice with free boundaries, provided we distinguish between "receding" and "advancing" boundaries, the latter having degrees of freedom that must be fixed by an appropriately chosen protocol. This study establishes a framework for future analytic and numerical work on such frustrated Ising lattices.
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