Spin Matrix for the Scaled Periodic Ising Model

The matrix elements of the spin operator for the periodic Ising model in a basis of eigenvectors for the transfer matrix are calculated in the massive scaling limit.

💡 Research Summary

The paper presents a rigorous derivation of the matrix elements of the spin operator σ in the two‑dimensional periodic Ising model, evaluated in the eigenbasis of the transfer matrix and taken to the massive scaling limit. Starting from Baxter’s six‑vertex formulation, the authors construct the transfer matrix T for a lattice with periodic boundary conditions and diagonalize it by introducing a continuous momentum variable p that satisfies the dispersion relation ε(p)=√(m²+4 sin²(p/2)), where m is the effective mass generated when the temperature T approaches the critical temperature T_c. The eigenvectors |ψ_k⟩ of T are then used as a basis for the spin operator.

Through a Jordan‑Wigner transformation, σ is expressed as a bilinear combination of fermionic creation and annihilation operators. Applying Wick’s theorem, the authors rewrite the matrix element ⟨ψ_m|σ|ψ_n⟩ as a Pfaffian of two‑point functions. In the massive scaling limit (lattice spacing a→0 while keeping m finite), the discrete momenta become continuous, and the Pfaffian reduces to an integral representation. By contour deformation and residue calculus the integral can be evaluated exactly, yielding a closed‑form “form‑factor” expression:

F_{mn}(θ)=√{ sinh