Spin-spin correlators on the $\beta$/${\beta }^{\star }$ boundaries in 2D Ising-like models: exact analysis through theory of block Toeplitz determinants

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 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 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.

30 pages. Introduction added; title changed to better reflect the content; the form factor expansion is performed up to three-particle form factors

References in corpus (3)

• Entanglement entropy in quantum spin chains with finite range interaction
• On the boundary form factor program
• Asymptotics of block Toeplitz determinants and the classical dimer model

Cited by in corpus (1)

• Exploring superconformal Yang-Mills theories through matrix Bessel kernels