Bethe Equation of $tau^{(2)}$-model and Eigenvalues of Finite-size Transfer Matrix of Chiral Potts Model with Alternating Rapidities

We establish the Bethe equation of the $\tau^{(2)}$-model in the $N$-state chiral Potts model (including the degenerate selfdual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral Potts model are computed by use of functional relations. The significance of the “alternating superintegrable” case of the chiral Potts model is discussed, and the degeneracy of $\tau^{(2)}$-model found as in the homogeneous superintegrable chiral Potts model.

💡 Research Summary

The paper addresses the long‑standing problem of obtaining the exact spectrum of the finite‑size transfer matrix of the N‑state chiral Potts model (CPM) when the vertical rapidities alternate between two values, a situation that had not been fully treated in the literature. The authors start by recalling the τ^(2)‑model, a five‑parameter family of L‑operators built from Weyl operators X and Z, whose ω‑twisted trace defines τ^(2)(t)=A_L(ωt)+D_L(ωt). This model is known to be a descendant of the six‑vertex model and to satisfy a set of functional (fusion) relations: τ^(2)(ω^{j‑1}t) τ^{(j)}(t)=z(ω^{j‑1}t) X τ^{(j‑1)}(t)+τ^{(j+1)}(t), together with a boundary condition involving τ^{(N+1)}. The authors specialize to the case where the τ^(2)‑model parameters correspond to a pair of alternating rapidities p and p′ that lie on either the generic rapidity curve W_{k′} or one of its degenerate self‑dual limits (W′_1, W″_1, W‴_1). The rapidities are parametrized by (x,y,μ) and related variables t=xy, λ=μ^N, etc., which allow the authors to write the quantum determinant z(t) in a compact form involving the rapidities of the two alternating rows.

Next, the chiral Potts transfer matrix T_{p,p′}(q) is introduced. Its Boltzmann weights W_{p,q}(n) satisfy the star‑triangle relation, guaranteeing integrability. The transfer matrix commutes with the spin‑shift operator X and obeys simple transformation rules under the automorphisms of the rapidity curves. The functional relations between T_{p,p′}(q) and τ^(2)(t) are the same as those previously derived for the homogeneous case, but now they involve the alternating rapidities explicitly.

The core technical achievement is the derivation of a Bethe‑Ansatz‑type equation for the τ^(2)‑model in the alternating‑rapidity setting. By assuming that the eigenpolynomial Q(t)=∏{ℓ=1}^M (1−v_ℓ t) satisfies the τ‑relation, and applying the Wiener‑Hopf factorization technique (originally used for the homogeneous model), the authors obtain the Bethe equations
 v_ℓ^N = −∏{k=1}^L (t_{p_k}−v_ℓ)/(t_{p′k}−v_ℓ)·(y{p_k} y_{p′k})/(x{p_k} x_{p′_k}),
where L is the lattice width and the set {p_k, p′_k} encodes the alternating rapidities along the vertical direction. This equation reduces to the known homogeneous Bethe equation when p=p′, but in the alternating case the ratio of rapidities introduces a non‑trivial phase that reflects the lack of rapidity difference‑property.

Having solved the Bethe equations (or at least characterized their solutions), the authors express the eigenvalues of the finite‑size CPM transfer matrix as
 Λ(q)=∏{ℓ=1}^M (t_q−v_ℓ)/(t_q−ω v_ℓ)·Φ(q),
where Φ(q) is a known scalar factor built from the parameters α_q and α{q†} that appear in the quantum determinant and are themselves products of the z‑functions evaluated at shifted arguments. The factor Φ(q) simplifies dramatically in the “alternating superintegrable” regime, i.e. when the rapidity curve parameters satisfy k′=0 or k′=±1, because the α’s become constants. In this regime the Bethe equations collapse to v_ℓ^N=−1, so the roots v_ℓ are simply the N‑th roots of (−1). Consequently, the τ^(2)‑model exhibits an N‑fold degeneracy: different Bethe solutions correspond to the same eigenvalue of τ^(2) and of the transfer matrix. This degeneracy mirrors the Onsager‑algebra symmetry found in the homogeneous superintegrable CPM and explains the multiplicities observed in earlier works.

The paper also discusses how the functional relations derived for the alternating case remain valid for the degenerate self‑dual curves, confirming that the whole framework is robust under the various limits of the rapidity parameters. The authors provide explicit expressions for the normalized transfer matrix, its conjugate, and the associated functional equations, extending the known homogeneous formulas to the alternating setting.

In the concluding section, the authors emphasize that their results demonstrate the universality of the τ^(2)‑model Bethe Ansatz: it works not only for homogeneous rapidities but also for the more general alternating and degenerate configurations. The explicit finite‑size eigenvalue formulas open the way to exact finite‑size scaling analyses, numerical checks, and possibly the computation of correlation functions in non‑homogeneous chiral Potts models. Moreover, the identification of the alternating superintegrable point as a highly symmetric, degenerate case suggests new avenues for exploring algebraic structures (such as the Onsager algebra) in broader classes of integrable lattice models. Overall, the work significantly advances the analytical toolbox available for the chiral Potts model and related integrable systems.