二维Ising类模型中$\beta$/$\beta^{\star}$边界上的自旋-自旋关联函数：通过块Toeplitz行列式理论的精确分析
Spin-spin correlators on the $β$/$β^{\star}$ boundaries in 2D Ising-like models: exact analysis through theory of block Toeplitz determinants
AI总结：
本研究精确分析二维Ising类模型对偶边界上的自旋关联，利用块Toeplitz行列式与Wiener-Hopf分解，揭示非交换性导致的反常项，并促进与QFT bootstrap方法的比较。
摘要：
在这项工作中，我们研究了具有对偶参数$\beta$和$\beta^{\star}$的两个二维Ising类模型之间边界上关联函数的定量性质。在此类构造中，若缺乏相对于平移不变方向的反射对称性，自旋-自旋关联函数通常表示为$2\times 2$块Toeplitz行列式，这通常比标量（$1\times 1$块）版本困难得多。然而，我们证明，对于本工作中考虑的特定$\beta/\beta^{\star}$边界，符号矩阵允许显式的交换Wiener-Hopf分解。因此，大$n$渐近行为中的常数$E(a)$和$E(\tilde a)$仍允许显式表示，这推广了标量符号的强Szegö定理。然而，不同$z$处的Wiener-Hopf因子不交换。我们将证明，由于这种非交换性，Wiener-Hopf因子中的“对数发散”会在重标度关联函数的指数形式因子展开中产生某些“反常项”。由于在朴素标度极限中，我们的边界可以表述为二维有质量QFT中的某些可积边界/缺陷，因此本工作的结果有助于与bootstrap方法进行详细比较。
英文摘要：
In this work, we investigate quantitative properties of correlation functions on the boundaries between two 2D Ising-like models with dual parameters $β$ and $β^{\star}$. Spin-spin correlators in such constructions without reflection symmetry with respect to transnational-invariant directions are usually represented as $2\times 2$ block Toeplitz determinants which are usually significantly harder than the scalar ($1\times 1$ block) versions. Nevertheless, we show that for the specific $β/β^{\star}$ boundaries considered in this work, the symbol matrices allow explicit commutative Wiener-Hopf factorizations. As a result, the constants $E(a)$ and $E(\tilde a)$ for the large $n$ asymptotics still allow explicit representations that generalize the strong Szegö's theorem for scalar symbols. However, the Wiener-Hopf factors at different $z$ do not commute. We will show that due to this non-commutativity, ``logarithmic divergences'' in the Wiener-Hopf factors generate certain ``anomalous terms'' in the exponential form factor expansions of the re-scaled correlators. Since our boundaries in the naive scaling limits can be formulated as certain integrable boundaries/defects in 2D massive QFTs, the results of this work facilitate detailed comparisons with bootstrap approaches.
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