Spin-operator form factors of the critical Ising chain and their finite volume scaling limits

TL;DR

This work computes spin-operator form factors for the critical Ising chain in the fermionic basis at finite volume and analyzes their finite-volume and scaling-limit behavior. It leverages a near-Cauchy structure of Toeplitz-symbol matrices to derive square-root dressing functions, yielding desingularized integral representations; in the scaling limit these reduce to gamma-function products and Barnes $G$-function normalizations, reproducing the Ising CFT spin-operator matrix elements on the cylinder. A complete finite-$N$ formula is obtained for the form factors and ground-state overlap via Wick contractions and Pfaffians, together with a determinant/Fredholm representation for the two-point function that matches the expected cylinder-CFT correlator. The results establish a precise bridge between lattice critical Ising data and Ising CFT, including exact normalization and a tractable Fredholm-determinant framework for correlation functions.

Abstract

In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical Ising chain at a generic system length $N\in 2Z_{\ge 2}$. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbol in the critical model, and leads to simpler product formulas for the dressing functions in terms of square root functions. These square root products allow fully dis-singularized integral representations. In the finite volume scaling limit, they further reduce to the Binet's second integral for the gamma function logarithm and its Hermite's generalization. As such, all the matrix elements in the scaling limit allow simple product formulas in terms of the gamma function at integer and half-integer arguments, and are rational numbers up to $\sqrt{2}$. They are exactly the spin-operator form factors of the Ising CFT in the fermionic basis, whose explicit forms are much less well known in comparison to the finite-volume form factors in the massive theory. We also fully determine the normalization factor of the spin-operator and show explicitly how the coefficient $G(\frac{1}{2})G(\frac{3}{2})$ appear through a ground state overlap.