Factorized finite-size Ising model spin matrix elements from Separation of Variables

Using the Sklyanin-Kharchev-Lebedev method of Separation of Variables adapted to the cyclic Baxter–Bazhanov–Stroganov or $\tau^{(2)}$-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.

💡 Research Summary

The paper presents a rigorous derivation of factorized expressions for spin‑matrix elements of the finite‑size two‑dimensional Ising model by exploiting the Separation of Variables (SOV) method originally developed by Sklyanin, Kharchev, and Lebedev. The authors adapt this framework to the cyclic Baxter‑Bazhanov‑Stroganov (τ²) model, which is known to be equivalent to the Ising model on a square lattice with periodic boundary conditions.

The work proceeds in several logical stages. First, the τ²‑model is introduced as a quantum integrable system whose monodromy matrix is built from L‑operators satisfying the Yang‑Baxter equation. The periodic boundary condition imposes a cyclic symmetry that makes the model amenable to a “separated‑variables” description. The authors recall the Sklyanin construction of a set of commuting operators {x₁,…,x_N} (the separated variables) whose eigenvalues parametrize the spectrum of the transfer matrix T(λ).

Second, they apply the Sklyanin‑Kharchev‑Lebedev (SKL) version of SOV to the τ²‑model. By diagonalising the operator B(λ) (one of the entries of the monodromy matrix) they obtain a functional equation for the Baxter Q‑function, namely the T‑Q relation
 T(λ) Q(λ) = a(λ) Q(λ−η) + d(λ) Q(λ+η),
where a(λ) and d(λ) are known scalar functions determined by the L‑operators and η is the crossing parameter. The zeros {θ_k} of Q(λ) are identified with the separated variables and satisfy a set of Bethe‑type equations that are automatically compatible with the periodicity of the model.

Third, the central technical achievement is the expression of a local spin operator σᶻ_j in terms of the separated variables. Using the explicit representation of the eigenvectors in the SOV basis, the matrix element ⟨α|σᶻ_j|β⟩ is reduced to a ratio of two determinants. The determinant in the numerator factorises into a Vandermonde‑type product ∏_{k<l} sin