  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
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 $E(a)$ and $E(\tilde a)$
for the large $n$ asymptotics still allow explicit representations that
generalize the strong Szeg\"o'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.
