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

TL;DR

This paper derives explicit formulas for spin-operator form factors in the critical Ising chain, connecting finite volume results to conformal field theory limits using integral representations and matrix properties.

Contribution

It provides a self-contained derivation of spin-operator matrix elements and their scaling limits, offering explicit formulas and connections to CFT that were previously computed iteratively.

Findings

Explicit product formulas for spin form factors in the Ising CFT

Demonstration of convergence of two-point correlators to CFT predictions

Identification of rational number structure up to rac{2}

Abstract

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 $N\in 2Z_{\ge 2}$. 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 $\sqrt{2}$. We also 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. 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.