Spin-spin correlators on the $β$/$β^{\star}$ boundaries in 2D Ising-like models: exact analysis through theory of block Toeplitz determinants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908477254795264 |
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| author | Liu, Yizhuang |
| author_facet | Liu, Yizhuang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00550 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spin-spin correlators on the $β$/$β^{\star}$ boundaries in 2D Ising-like models: exact analysis through theory of block Toeplitz determinants Liu, Yizhuang Statistical Mechanics High Energy Physics - Theory Mathematical Physics 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. |
| title | Spin-spin correlators on the $β$/$β^{\star}$ boundaries in 2D Ising-like models: exact analysis through theory of block Toeplitz determinants |
| topic | Statistical Mechanics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2405.00550 |