Completing the spectral theory of the degenerate chiral Potts models: eigenvectors and completeness, the $k'=0$ commuting Q-operator problem, and thermodynamics

The generalized $\tau^{(2)}$-model (Bazhanov–Stroganov model) is equivalent, outside two pseudovacuum loci treatable by algebraic Bethe ansatz, to the $N$-state chiral Potts model (CPM) with two alternating vertical rapidities, once the degenerate CPMs at $k'=1$ (self-dual curves) and $k'=0$ are included; at the $k'=1$ loci the degenerate CPM transfer matrices $T,\hat T$ then serve as the Baxter $Q_R$-, $Q_L$-operators and the full chiral Potts functional relations hold, while at the $k'=0$ loci the weights fail the star-triangle relation and $[Q_R,Q_L]\ne0$ (Roan, 2007). The spectral program announced there is only partially done: the Bethe equations of the $\tau^{(2)}$-model and the finite-size transfer-matrix eigenvalue formulas are now established for the $k'=1$ degenerate self-dual curves (Roan, 2008), but the $k'=0$ ('zero-temperature') $\tau^{(2)}$-models were explicitly excluded, and separation-of-variables completeness proofs cover only the generic rapidity curves. The surviving open problem has three linked components sharing one objective — completing the functional-relation/Q-operator spectral theory of the degenerate chiral Potts models and of the generalized $\tau^{(2)}$-model at all parameter loci: (i) construct the eigenvectors of the degenerate ($k'=1$ self-dual) models and prove completeness of the Bethe-type spectral characterization; (ii) at the $k'=0$ loci, where the degenerate weights fail the star-triangle relation and $[Q_R,Q_L]\ne0$, decide whether a commuting Baxter-type Q-operator exists and construct one or prove impossibility, and give a spectral resolution of these models; (iii) derive the thermodynamic quantities (free energy, excitation spectrum, phase structure) of the degenerate models. This core remains open in the follow-up literature (Roan's series through 2013, the Grosjean–Niccoli–Maillet separation-of-variables program, and adjacent works through 2026).

Background

The $N$-state chiral Potts model (CPM) is a two-dimensional solvable lattice model whose Boltzmann weights satisfy the star-triangle relation — the Yang–Baxter equation of this setting — which guarantees that the row-transfer matrices commute and can in principle be diagonalized exactly. Its weights depend on 'rapidities' constrained to a high-genus algebraic curve with parameters $k,k'$ obeying $k^2+k'^2=1$, where $k'$ plays the role of temperature. Unlike the six-vertex model, the CPM weights do not depend only on rapidity differences (the 'non-difference property'), which is the main source of technical difficulty in its spectral analysis.

The solvability of the CPM runs through an auxiliary model discovered by Bazhanov and Stroganov: the $\tau^{(2)}$-model is the trace of an L-operator acting on a two-dimensional auxiliary space and on the $N$-cyclic quantum space spanned by $|n\rangle$, $n\in\mathbb{Z}_N$, with Weyl operators $X|n\rangle=|n+1\rangle$, $Z|n\rangle=\omega^n|n\rangle$, $\omega^N=1$; equivalently it is the transfer matrix of a cyclic representation of the six-vertex Yang–Baxter algebra, and it is the second element of a fusion hierarchy (a nested family of transfer matrices obtained by fusing the auxiliary representation, whose functional relations close at level $N$). Baxter, Bazhanov and Perk derived the functional relations (the inversion relation for $T\hat T$ and the fusion relations) connecting the CPM transfer matrices $T,\hat T$ to $\tau^{(2)}$, and Baxter used them to compute the eigenvalues of the CPM transfer matrix and, by the inversion-relation method, its thermodynamics. In Baxter's TQ framework, the CPM transfer matrices $T,\hat T$ themselves act as the right and left Q-operators ($Q_R$, $Q_L$) of the $\tau^{(2)}$-model: a Q-operator is a transfer matrix commuting with the model of interest whose eigenvalues obey a two-term Baxter equation with the transfer-matrix eigenvalues, reducing the spectral problem to Bethe-type equations.

Roan (arXiv:0710.2764) completed the equivalence program: a 'generic' five-parameter generalized $\tau^{(2)}$-model is gauge/scale equivalent to a three-parameter $\tau^{(2)}$-model inside the CPM with two alternating vertical rapidities, and every non-generic locus, except two pseudovacuum loci where the algebraic Bethe ansatz (ABA — the reference-state/creation-operator method) applies, reduces to a 'degenerate chiral Potts model'. At $k'=1$ there are three degenerate rapidity curves, $x^N+y^N=x^Ny^N$, $x^N+y^N=1$ with $\mu^N=1$, and $x^N+y^N=0$ with $\mu^N=\pm1$, whose weights are self-dual solutions of the star-triangle relation; there the full CPM functional relations hold and $Q_R,Q_L$ commute, so the commuting Q-operator construction survives. As a by-product, XXZ spin chains built on cyclic representations of the quantum group $U_q(sl_2)$ at $q^N=1$ ($N$ odd) are identified either with the superintegrable CPM (the special rapidity choice with extra Onsager-algebra symmetry; the Onsager algebra is the infinite-dimensional Lie algebra Onsager used to solve the Ising model) or with the $k'=1$ degenerate self-dual model. But at the $k'=0$-type loci — parameters with $c^N\ne1$ and $(a'^N-b^N)(a^N-b'^N)=0$ in the L-operator parametrization $(a,b,a',b',c)$ — the degenerate weights violate the star-triangle relation, there is no common rapidity variable for $T$ and $\hat T$, and $[Q_R,Q_L]\ne0$, so no commuting family of Q-operators is obtained there.

What was known for the genuinely solvable CPM sets the standard the degenerate models should meet: eigenvalue formulas (Baxter 1990), excitation spectrum and phase structure (McCoy–Roan 1990), and, for eigenvectors, only partial results — the superintegrable case is understood through its Onsager- and loop-algebra symmetries (Roan 2010; Au-Yang–Perk), with a completeness proof essentially limited to the three-state superintegrable model (Dasmahapatra–Kedem–McCoy 1993). An alternative to the Q-operator route is Sklyanin's separation of variables (SOV — rewriting the spectral problem in a basis where one monodromy-matrix entry is diagonal), available for the Bazhanov–Stroganov model since von Gehlen–Iorgov–Pakuliak–Shadura (2006); Grosjean–Niccoli (2012) used it to characterize the complete spectrum (eigenvalues and eigenstates) of the $\tau^{(2)}$-model for the most general cyclic representations and to prove completeness of Bethe-type equations — but for the chiral Potts transfer matrices their completeness theorems cover only self-adjoint representations on the generic rapidity curves, not the degenerate curves, and they construct no commuting Q-operator. Roan's own follow-up established the $\tau^{(2)}$ Bethe equations and the finite-size CPM eigenvalue formulas for alternating rapidities including the $k'=1$ degenerate curves, while explicitly excluding the $k'=0$ models and deferring all physical (thermodynamic) discussion.

Problem Statement

Complete the functional-relation / Q-operator spectral theory of the degenerate chiral Potts models and of the generalized $\tau^{(2)}$-model at the parameter loci where the equivalence program of Roan (arXiv:0710.2764) remains unfinished. The three components below serve this single objective — bringing the degenerate models to the same standard of solution as the generic solvable chiral Potts model — and are scientifically linked through the shared functional relations and the role of the chiral Potts transfer matrices as Q-operators.

(i) Eigenvectors and completeness for the degenerate ($k'=1$ self-dual) models. On the three degenerate rapidity curves $x^N+y^N=x^Ny^N$; $x^N+y^N=1$ with $\mu^N=1$; and $x^N+y^N=0$ with $\mu^N=\pm1$, the full chiral Potts functional relations hold, the $\tau^{(2)}$ Bethe equations are established, and the finite-size transfer-matrix eigenvalue formulas are known (Roan 2008). Construct the eigenvectors of the transfer matrices $T,\hat T$ of these degenerate models for arbitrary finite size and arbitrary alternating vertical rapidities, and prove completeness of the spectral characterization: that the solutions of the Bethe-type equations, together with the quantum-number constraints, account for the entire spectrum with the correct multiplicities, including at loci where eigenspaces degenerate.

(ii) The $k'=0$ loci. For the generalized $\tau^{(2)}$-models with $c^N\ne1$ and $(a'^N-b^N)(a^N-b'^N)=0$, the degenerate weights violate the star-triangle relation, no common rapidity variable exists for $T$ and $\hat T$, and $[Q_R,Q_L]\ne0$, so the standard construction yields no commuting family of Baxter-type Q-operators. Determine whether a commuting Baxter-type Q-operator compatible with the fusion and inversion relations exists for these $\tau^{(2)}$-models — construct one explicitly, or prove that none exists — and in either case provide a spectral resolution of these models (eigenvalues and eigenvectors, with completeness) that fulfils the role the chiral Potts transfer matrices fulfil at every other locus where they serve as commuting Q-operators.

(iii) Thermodynamics of the degenerate models. From the functional-relation program, derive the thermodynamic quantities of the degenerate models in the large-lattice limit — the free energy, the excitation spectrum, and the phase structure — in the spirit of the corresponding results for the generic solvable chiral Potts model.

An answer is complete when all three components are settled for general $N$ (odd and even) and arbitrary chain length; results restricted to small $N$, or to the superintegrable sublocus, settle only special cases.

References

1. Shi-shyr Roan, On the Equivalent Theory of the Generalized τ^{(2)}-model and the Chiral Potts Model with two Alternating Vertical Rapidities, preprint arXiv:0710.2764 (2007, v3 2008), DOI 10.48550/arXiv.0710.2764, https://arxiv.org/abs/0710.2764 (source of the open problem)
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