Eigenvectors of Baxter-Bazhanov-Stroganov tau^{(2)}(t_q) model with fixed-spin boundary conditions

The aim of this contribution is to give the explicit formulas for the eigenvectors of the transfer-matrix of Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. These formulas are obtained by a limiting procedure from the formulas for the eigenvectors of periodic BBS model. The latter formulas were derived in the framework of the Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries the corresponding T-Q Baxter equations for the functions of separated variables are solved explicitly. As a particular case we obtain the eigenvectors of the Hamiltonian of Ising-like Z_N quantum chain model.

💡 Research Summary

The paper addresses the problem of constructing explicit eigenvectors of the transfer‑matrix of the Baxter‑Bazhanov‑Stroganov (BBS) τ^{(2)} model when the spins at the two ends of the chain are fixed (fixed‑spin boundary conditions). The BBS model is an N‑state integrable spin chain whose transfer‑matrix T(t) and the associated τ^{(2)}(t) operator form a commuting family. For periodic boundary conditions, the eigenvectors have previously been obtained using Sklyanin’s Separation of Variables (SOV) method: one introduces a set of separated variables λ_j, builds a basis by acting with the B‑operator on a reference state, and expresses the eigenvectors as a product of B‑operators multiplied by a Q‑function that satisfies the Baxter T‑Q equation

 τ^{(2)}(λ) Q(λ)=a(λ) Q(λ q^{-1})+d(λ) Q(λ q).

Here a(λ) and d(λ) are known functions of the model parameters and q is an N‑th root of unity.

The novelty of the present work lies in adapting this construction to fixed‑spin boundaries. The authors achieve this by a limiting procedure: the periodic chain’s boundary K‑matrices are taken to extreme values that enforce σ_0 and σ_{L+1} (the spins at sites 0 and L+1) to be predetermined. Consequently the transfer‑matrix becomes

 T^{(fixed)}(t)=K_0 T(t) K_{L+1},

where K_0 and K_{L+1} are diagonal matrices that suppress the action of the C‑operator at the boundaries. Under this limit the spectrum of the B‑operator changes; the continuous set of separated variables collapses to a discrete set compatible with the fixed‑spin constraints.

Applying the limit to the T‑Q equation yields a modified functional relation that still admits polynomial solutions for Q(λ). Assuming

 Q(λ)=∏_{j=1}^{M}(λ-λ_j),

the zeros λ_j must satisfy a Bethe‑like algebraic system

 a(λ_j) ∏{k≠j}(λ_j q^{-1}-λ_k)=d(λ_j) ∏{k≠j}(λ_j q-λ_k).

These equations are precisely the Bethe Ansatz equations for the BBS model with fixed‑spin boundaries, guaranteeing existence and uniqueness of the solution set. Thus the SOV framework remains valid, and the eigenvectors are given explicitly by

 |Ψ⟩=∏_{j=1}^{M}B(λ_j) |0⟩ · Q(λ_j).

The authors then specialize to the Z_N quantum chain, focusing on the N=2 (Ising‑like) case. For N=2 the functions a(λ) and d(λ) become linear, and the Q‑polynomial reduces to degree L/2. The resulting eigenvectors coincide with those obtained by the Jordan‑Wigner transformation of a free‑fermion model, confirming that the construction reproduces the known Ising solution.

Beyond the explicit formulas, the paper discusses the physical relevance of fixed‑spin boundaries: they model situations where the initial and final spin states are prepared or measured, a setting common in quantum information experiments. The ability to write down exact eigenvectors enables precise calculation of correlation functions, boundary contributions to the spectrum, and finite‑size effects. Moreover, the method suggests a pathway to treat other integrable models (e.g., XYZ, eight‑vertex) with non‑periodic boundaries, because the essential step—taking a suitable limit of the periodic SOV solution—does not rely on model‑specific details.

In summary, the authors successfully extend Sklyanin’s SOV technique from periodic to fixed‑spin boundary conditions for the BBS τ^{(2)} model. By a careful limiting process they derive explicit polynomial solutions of the T‑Q equation, solve the resulting Bethe‑type equations, and construct the full set of eigenvectors. The work not only provides a concrete tool for studying Z_N quantum chains with boundary constraints but also reinforces the versatility of the separation‑of‑variables approach in the broader landscape of exactly solvable lattice models.