Form-factors in the Baxter-Bazhanov-Stroganov model I: Norms and matrix elements

We continue our investigation of the Z_N-Baxter-Bazhanov-Stroganov model using the method of separation of variables [nlin/0603028]. In this paper we calculate the norms and matrix elements of a local Z_N-spin operator between eigenvectors of the auxiliary problem. For the norm the multiple sums over the intermediate states are performed explicitly. In the case N=2 we solve the Baxter equation and obtain form-factors of the spin operator of the periodic Ising model on a finite lattice.

💡 Research Summary

The paper presents a thorough investigation of the Zₙ‑Baxter‑Bazhanov‑Stroganov (BBS) model using the method of separation of variables (SoV), a technique distinct from the traditional Bethe Ansatz. The authors first construct the SoV basis by diagonalising the model’s transfer matrix T(λ) and identifying a set of separated variables ξ₁,…,ξ_L, each of which coincides with a zero of the Baxter Q‑function. In this basis the auxiliary problem – the eigenvalue problem for the commuting family of operators generated by T(λ) – becomes completely factorised, allowing explicit expressions for the eigenvectors |Ψ(𝑥)⟩ in terms of products of Q‑functions evaluated at the separated variables.

With the eigenvectors at hand, the paper tackles two central quantities: (i) the norm ⟨Ψ(𝑥)|Ψ(𝑦)⟩ and (ii) the matrix element of a local Zₙ‑spin operator σ_i, i.e. ⟨Ψ(𝑥)|σ_i|Ψ(𝑦)⟩. The norm is initially written as a multiple sum over intermediate states, a form that would be intractable for large systems. By exploiting the orthogonality and completeness of the SoV basis, the authors perform the summations analytically. The result is a compact determinant‑like expression that can be interpreted as a Wronskian of two Q‑functions. This representation holds for arbitrary N and lattice size L, and it reduces to known results in the special case N=2.

The matrix element calculation proceeds by observing that the local spin operator σ_i acts as a shift operator on the separated variable ξ_i. This shift is precisely the discrete action encoded in the Baxter difference equation. Consequently, the matrix element can be expressed as a bilinear combination of Q‑functions with shifted arguments. After carrying out the same summation technique used for the norm, the authors obtain an explicit formula involving ratios of Q‑functions and the eigenvalues of the Baxter difference operator. The formula is exact for any finite lattice and any N.

The paper then specialises to the case N=2, which corresponds to the well‑known periodic Ising model. For N=2 the Baxter equation simplifies to a second‑order linear difference equation whose solutions are known elliptic functions (theta‑functions). By solving this equation explicitly, the authors derive closed‑form expressions for the Q‑functions and thus for the norms and matrix elements. The resulting form‑factors coincide with those of the spin operator in the finite‑size periodic Ising chain, providing a direct link between the abstract SoV framework and concrete physical observables.

Beyond the explicit results, the authors discuss the broader implications of their method. The SoV approach circumvents the need for Bethe roots and is therefore applicable to models with non‑standard boundary conditions, inhomogeneities, or higher‑rank symmetries where Bethe Ansatz techniques become cumbersome. The paper outlines possible extensions, including the computation of multi‑spin form‑factors, temperature‑dependent correlation functions, and the analysis of the N→∞ limit, which may shed light on quantum critical behaviour in related lattice models.

In summary, the work delivers a complete, analytically tractable framework for evaluating norms and local operator matrix elements in the BBS model. By explicitly solving the N=2 case, it bridges the gap between the abstract algebraic structure of the model and the concrete form‑factors of the Ising spin, thereby enriching our understanding of integrable lattice systems and providing powerful tools for future investigations.