Title: Spin-spin correlators on the $β$/$β^{\star}$ boundaries in 2D Ising-like models: asymptotics of block Toeplitz determinants and "anomalous contributions'' in the scaling region

Abstract: In this work, we investigate quantitative properties of correlation functions on the boundaries between two 2D Ising-like models with dual parameters $\beta$ and $\beta^{\star}$. 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 normally significantly harder than the scalar ($1\times 1$ block) versions. Nevertheless, we show that for the specific $\beta/\beta^{\star}$ 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 allow simple explicit representations. 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 ``anomalous contributions'' in the re-scaled correlators.