Spin-spin correlators on the β/β⋆ boundaries in 2D Ising-like models: Exact analysis through theory of block Toeplitz determinants

In this work, we investigate quantitative properties of correlation functions on the boundaries between two 2D Ising-like models with dual parameters β and ${\beta }^{\star }$. Spin-spin correlators in such constructions without reflection symmetry with respect to translation-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 $\beta /{\beta }^{\star }$ boundaries considered in this work, the symbol matrices allow explicit commutative Wiener-Hopf factorizations. As a result, the constants $E\left(a\right)$ and $E\left(\tilde{a}\right)$ 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.