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arXiv:2507.13093 · Journal of High Energy Physics

Asymptotics of spin-spin correlators weighted by fermion number measurements with low rapidity threshold in the 2D Ising free-fermion QFT

Yizhuang Liu

Published Jan 9, 2026• 0 citations
arXivHTMLPDF

Abstract

A bstract In the work, we study the averaged number of massive fermions above a low rapidity threshold Y , underlying the form-factor expansions of the spin-spin two-point correlators at an Euclidean distance r , in the 2D Ising QFT at the free massive fermion point. Despite the on-shell freeness, the spin operators are still far away from being Gaussian, and create particles in the asymptotic states with complicated correlations. We show how the number observables can still be incorporated into the integrable Sinh-Gordon/Painleve-III framework and controlled by linear differential equations with two variables ( r, Y ). We show how the differential equations and the information of two crucial scaling functions arising in the r → 0, $${e}^{Y}r=\mathcal{O}(1)$$ scaling limit , can be combined to fully determine the small- r asymptotics of the observables, in the λ-extended form. The scaling functions, on the other hand, are obtained by summing the exponential form-factor expansions directly, generalizing the traditional Ising connecting computations. We show carefully, how the singularities cancel in the physical value limit λπ → 1 and how the power-corrections that collapse at this value can be resummed. In particular, we show for the physical λ -value, the scaling functions are related to integrated four-point functions in the Ising CFT and continue to control the asymptotics of the number-observables in the scaling limit up to $$\mathcal{O}({r}^{3})$$ .

DOI
10.1007/jhep01(2026)064

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A bstract In the work, we study the averaged number of massive fermions above a low rapidity threshold Y , underlying the form-factor expansions of the spin-spin two-point correlators at an Euclidean distance r , in the 2D Ising QFT at the free massive fermion point. Despite the on-shell freeness, the spin operators are still far away from being Gaussian, and create particles in the asymptotic states with complicated correlations. We show how the number observables can still be incorporated into the integrable Sinh-Gordon/Painleve-III framework and controlled by linear differential equations with two variables ( r, Y ). We show how the differential equations and the information of two crucial scaling functions arising in the r → 0, eYr=O(1){e}^{Y}r=\mathcal{O}(1)eYr=O(1) scaling limit , can be combined to fully determine the small- r asymptotics of the observables, in the λ-extended form. The scaling functions, on the other hand, are obtained by summing the exponential form-factor expansions directly, generalizing the traditional Ising connecting computations. We show carefully, how the singularities cancel in the physical value limit λπ → 1 and how the power-corrections that collapse at this value can be resummed. In particular, we show for the physical λ -value, the scaling functions are related to integrated four-point functions in the Ising CFT and continue to control the asymptotics of the number-observables in the scaling limit up to O(r3)\mathcal{O}({r}^{3})O(r3) .
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⋆. Spin-spin correlators in such constructions without reflection symmetry with respect to translation-invariant directions are usually represented as 2×2 block Toeplitz determinants which are usually significantly harder than the scalar (1×1 block) versions. Nevertheless, we show that for the specific β/β⋆ boundaries considered in this work, the symbol matrices allow explicit commutative Wiener-Hopf factorizations. As a result, the constants E(a) and E(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.
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A program is proposed to study numerically the correlation functions in massive integrable 2D relativistic field theories. It relies crucially on the exact form factors of fields which can be reconstructed from the factorized scattering data. The correlation functions are expressible as infinite sums over intermediate asymptotic states. We suggest using computer power to perform the summation numerically. The convergence of the sum is tested for the simplest example of the scaling Ising spin-spin correlations (without magnetic field).
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In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length N∈2Z≥2N\in 2Z_{\ge 2}N∈2Z≥2​. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to 2\sqrt{2}2​. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient G(12)G(32)G(\frac{1}{2})G(\frac{3}{2})G(21​)G(23​) appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder.
Abstract In the neighbourhood of the critical point, the correlation length of the spin–spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation N between spins goes to infinity, but the scaling variable remains fixed, where t is the coupling, and t = 1 the critical point. Previous work has specified these scaling functions (there is one for the critical point being approached from above, and another if approached from below) in terms of transcendents defined by a particular -form of the degenerate Painlevé V equation. For the diagonal–diagonal correlation, we characterise the first two leading large N correction terms to the scaling functions—these occur at orders N −1 and N −2 —in terms of solutions of a second order linear differential equation with coefficients given in terms of these transcendents, and show how they can be computed. We show that the order N −1 is trivial and can be eliminated through appropriate variables so that the leading non-trivial correction is of order N −2 . In this respect our solution is the first non-trivial correction to the scaling limit pair-correlation results given in the earlier literature, but restricted to the case with both spins on a diagonal.