Complete finite-size eigenvalue spectrum of the homogeneous N-state chiral Potts transfer matrix beyond the superintegrable point

In his 2008 solution of the finite-size spectrum of the chiral Potts model (CPM) with alternating rapidities, Shi-shyr Roan highlighted that the eigenvalue problem for the homogeneous $N$-state chiral Potts model — the one with all vertical rapidities equal — is unsolved except at the superintegrable point, and posed the challenge of obtaining all finite-size eigenvalues as completely as the Ising model, together with the deferred physical interpretation of such spectra in the hermitian quantum-chain and positive-Boltzmann-weight statistical regimes. Later work solved the $\tau_2$-model and the inhomogeneous CPM spectra, by Sklyanin separation of variables and by the off-diagonal Bethe ansatz, but only for generic parameters: the homogeneous non-superintegrable point is a degenerate measure-zero locus not covered by those completeness theorems. The problem is the complete, provably complete characterization of all finite-size eigenvalues of the homogeneous non-superintegrable chiral Potts transfer matrix, together with its physical reading across the two regimes.

Background

The $N$-state chiral Potts model (CPM) is the solvable $\mathbb{Z}_N$-symmetric generalization of the Ising model on the square lattice. Its Boltzmann weights depend on "rapidities" — points on a fixed algebraic curve in variables $x,y$ (a curve of genus $(N-1)(N-2)/2$, which is already positive at $N=3$ and grows with $N$) that satisfy the star-triangle (Yang–Baxter) relation, so that the row-to-row transfer matrices commute. In the six- and eight-vertex models the spectral parameters of commuting transfer matrices are related by simple shifts (the "difference property"), which underlies Baxter's $TQ$-relation method and complete Bethe-ansatz solutions; the chiral Potts model famously lacks this property, which is the main source of its technical difficulty. Bazhanov and Stroganov showed that the CPM is a descendant of the six-vertex model through the $\tau^{(2)}$-model (Baxter–Bazhanov–Stroganov model), a family of Lax operators with two-dimensional auxiliary space and $N$-dimensional cyclic quantum space built from Weyl operators $X,Z$ with $XZ=\omega ZX$, $\omega=e^{2\pi i/N}$. Baxter, Bazhanov and Perk derived the functional relations — a fusion hierarchy in $\tau^{(j)}$-transfer matrices plus a Baxter-type equation — in which the chiral Potts transfer matrix plays the role of Baxter's Q-operator (the operator whose eigenvalues linearize the transfer-matrix eigenvalue problem) for the $\tau^{(2)}$-transfer matrix. These functional relations underlie essentially all exact results for the homogeneous CPM: the free energy (Au-Yang–Jin–Perk 2001, via Baxter's inversion-relation method), the order parameter (Baxter 2005), and the maximum eigenvalue and low-lying excitations in the thermodynamic limit $L\to\infty$ (Baxter 1990; McCoy–Roan 1990). The complete finite-size eigenvalue spectrum is known only at the superintegrable point — the high-symmetry specialization where the model acquires an Onsager-algebra symmetry (the algebra generated by Onsager's original Ising operators, realized here as the closure of the $\tau^{(j)}$-generators) — solved by Albertini, McCoy and Perk and by Baxter. For $N=2$ the model degenerates to the free-fermion Ising case, whose finite-size transfer matrix was completely diagonalized by Kaufman's spinor analysis; this is the standard of "completely solved" invoked below. In 2008 Roan studied the CPM with alternating rapidities (vertical rapidities alternating between two points $p,p'$ on a common rapidity curve, degenerate curves included) and obtained the Bethe equations of the $\tau^{(2)}$-model and explicit formulas for the eigenvalues of the finite-size transfer matrix, recovering the homogeneous superintegrable spectrum at $p=p'$. He stated two limitations of his framework: it requires all rapidities on one rapidity curve, whereas Baxter's generalized ("inhomogeneous") $\tau_2$-model of 2004 allows vertical rapidities off a common curve; and, verbatim, "the solution of the homogenous CPM, except in the superintegrable case, is yet unknown to the best of the author's knowledge" — posing the challenge to obtain all eigenvalues "as they can be solved completely for the finite-size Ising model". His concluding section explicitly restricts the work to the mathematical aspect of the eigenvalue spectrum and defers the physical discussion to future work, distinguishing the hermitian quantum-chain regime from the statistical regime with positive Boltzmann weights, which overlap only at the critical point. Subsequent work solved the spectral problem only at generic (inhomogeneous) parameter values. Grosjean and Niccoli (2012), using Sklyanin's separation of variables (SOV — the representation in which one component $B(\lambda)$ of the monodromy matrix is diagonal, turning the eigenvalue problem into scalar functional equations), proved completeness of Bethe-type equations for the $\tau_2$-model for the most general cyclic representations "for almost all values of the 6N parameters", and characterized the inhomogeneous CPM spectrum on the chiral Potts curves. Xu et al. (2015, 2016) obtained the off-diagonal Bethe ansatz (ODBA — a Bethe-ansatz scheme built from operator product identities and asymptotic behaviour when no reference state exists) solution of the generic $\tau_2$-model with generic site-dependent inhomogeneities, and with arbitrary boundary fields, with numerical evidence for completeness. In all of these theorems the homogeneous chiral Potts point — all rapidities equal — is a degenerate, measure-zero locus where the genericity hypotheses (such as simplicity of the SOV operator $B(\lambda)$) are unavailable, so completeness there is neither covered nor inheritable by a routine limiting argument. The physical interpretation of the finite-size spectra in the two regimes likewise remains undiscussed in the literature.

Problem Statement

Consider the homogeneous $N$-state chiral Potts model on a finite square lattice with periodic boundary conditions: all vertical rapidities equal to a single point $p$ on a chiral Potts rapidity curve (of the Baxter–Perk–Au-Yang family, degenerate curves included), at generic — non-superintegrable — values of the rapidities and horizontal parameters, with $N\ge 3$ (the $N=2$ member degenerates to the free-fermion Ising case completely solved by Kaufman). The problem, posed as an explicit challenge by Roan (2008), has a primary and a linked secondary objective that together constitute an Ising-style complete solution of the model.

(1) Spectral characterization with completeness (primary). Obtain all eigenvalues of the finite-$L$ transfer matrix in an explicit, directly evaluable form — for example as the full solution set of Bethe-type equations or of the Baxter–Bazhanov–Perk functional relations — valid for arbitrary finite $L$ at generic non-superintegrable rapidities, together with a proof of completeness: a proof that the characterized solution set is exactly the spectrum of the transfer matrix, with every eigenvalue accounted for with its correct multiplicity. The characterization must reproduce the known spectra in the limiting cases where these are known: the superintegrable point (Albertini–McCoy–Perk/Baxter spectrum) and the $N=2$ Ising reduction.

(2) Physical interpretation of the derived spectra (linked deliverable, the discussion Roan deferred). Use the complete spectrum to give the physical reading of the model in the two regimes Roan distinguished, which overlap only at the critical point: (a) the hermitian quantum-chain regime, where the spectra determine the excitation and gap structure of the associated $N$-state quantum spin chain at finite $L$; and (b) the statistical-mechanics regime with positive Boltzmann weights, where the leading eigenvalues control the finite-size corrections to the known thermodynamic free energy and the subdominant, interfacial-tension-carrying eigenvalues. The interpretation should state which parts of the mathematical spectrum are physically realized or relevant in each regime.

The single overarching objective is the complete solution of the homogeneous chiral Potts model in the sense in which the finite-size Ising model is solved: deliverable (2) is the physical content of the spectra obtained in deliverable (1), and acceptance of (1) is the precondition for (2).

References

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