Secular equations and eigenvalue structure of the Onsager algebras generated by the n-state generalized Jordan-Wigner constructions (Minami's Theorems 2 and 3)

Minami (2021) proved that operators built from the algebraic generalization of the Jordan-Wigner transformation, and from their n-state clock generalizations satisfying the generalized Clifford relations, can be combined into pairs A_0, A_1 that obey the Dolan-Grady relations and hence generate the Onsager algebra. For the simplest construction (Theorem 1) the generated algebra closes uniformly, $A_{l+2N}=A_l$, yielding a quotient of the Onsager algebra with known spectral consequences for the superintegrable Hamiltonian $H=A_0+\lambda A_1$. For the two novel constructions of that paper — the power-combination construction with $n=3k$ (Theorem 2, Dolan-Grady constant $C=-27$) and the weighted-sum construction (Theorem 3, $C=n^2$) — the finite closure relations (secular equations) and the possible form of the eigenvalues of $H=A_0+\lambda A_1$ were explicitly left open.

Background

The Onsager algebra, introduced in Onsager's 1944 exact solution of the two-dimensional Ising model, is the infinite-dimensional Lie algebra generated by elements $A_l$ and $G_m$ ($l,m\in\mathbb{Z}$) with defining relations

$[A_l,A_k]=4G_{l-k},\qquad [G_m,A_l]=2A_{l+m}-2A_{l-m},\qquad [G_m,G_k]=0.$

Dolan and Grady (1982) showed that a pair of operators $A_0,A_1$ satisfying the cubic relations $[A_0,[A_0,[A_0,A_1]]]=C[A_0,A_1]$ and $[A_1,[A_1,[A_1,A_0]]]=C[A_1,A_0]$ (the Dolan-Grady condition, with $C$ a constant) generates the whole Onsager algebra, and that the Hamiltonian $H=A_0+\lambda A_1$ then belongs to an infinite family of mutually commuting operators, i.e. $H$ is superintegrable. Davies (1990) determined the general structure of finite-dimensional representations: in each Onsager sector (an invariant block on which the algebra acts irreducibly) the representation is characterized by a polynomial $h(z)=\sum_{m=-n}^{n}\alpha_m z^{m+n}$ whose roots lie on the unit circle, the algebra closes through the relation $\sum_m \alpha_m A_m=0$, and on irreducible representations the eigenvalues of $H=A_0+\lambda A_1$ take the form

$(\alpha+\beta\lambda)+\sum_{j=1}^{n}4m_j\sqrt{1+\lambda^2+2\lambda\cos\theta_j},\qquad m_j=0,\pm1,\ldots,\pm s_j,$

where $e^{i\theta_j}$ are the roots of $h(z)$. Determining the closure relation — called the secular equation in this context — for a concrete family of operators is what fixes the angles $\theta_j$, the sector data, and hence the actual spectrum.

Minami (2021) connected this framework to the algebraic generalization of the Jordan-Wigner transformation. Recall that the Jordan-Wigner transformation maps a spin-1/2 chain to free fermions using only the relations $\eta_j\eta_{j+1}=-\eta_{j+1}\eta_j$, $\eta_j^2=1$, and $\eta_j\eta_k=\eta_k\eta_j$ for $|j-k|>1$ among the interaction operators $\eta_j$. Minami's generalization replaces these by the n-state (generalized Clifford) relations

$\eta_j\eta_{j+1}=\omega\eta_{j+1}\eta_j,\qquad \eta_j^n=1,\qquad \omega=e^{2\pi i/n},$

again with $\eta_j\eta_k=\eta_k\eta_j$ for $|j-k|>1$ and a cyclic boundary condition $\eta_{M+j}=\eta_j$ with $M=2N$ operators. He proved three constructions of pairs $A_0,A_1$ satisfying the Dolan-Grady condition and therefore generating quotients of the Onsager algebra. Theorem 1: $A_0=\sum_j\eta_{2j-1}$ and $A_1=\sum_j\eta_{2j}$, with $C=16$; here the generated algebra closes uniformly, $A_{l+2N}=A_l$, giving a quotient Onsager algebra with known spectral consequences. Theorem 2: $A_0=\sum_j(\eta_{2j-1}^{\,n-k}-\eta_{2j-1}^{\,k})$ and $A_1=\sum_j(\eta_{2j}^{\,n-k}-\eta_{2j}^{\,k})$ with $n=3k$; the Dolan-Grady condition holds with $C=-27$ when $k$ is not divisible by 3 (when $k$ is divisible by 3 the two generators commute and no nontrivial algebra is generated). Theorem 3: $A_0=\sum_j\sum_{k=1}^{n-1}\eta_{2j-1}^{\,k}/(1-\omega^{-k})$ and $A_1=\sum_j\sum_{k=1}^{n-1}\eta_{2j}^{\,k}/(1-\omega^{-k})$, with $C=n^2$; concrete representations include Baxter's clock chain and, for $\eta_{2j-1}=Z_j$ and $\eta_{2j}=X_jX_{j+1}^{\dagger}$ with clock matrices $X,Z$ obeying $XZ=\omega ZX$, the superintegrable chiral Potts chain of von Gehlen and Rittenberg (1985).

For the Theorem-3 algebra in the clock representation, the closure data are equivalent to the eigenvalue data of the superintegrable chiral Potts chain, which since Albertini, McCoy and Perk (1989) have been obtained by solving functional equations sector by sector (the Baxter-Albertini-McCoy-Perk, or BAMP, polynomials, whose zeros determine the sector angles); closed-form secular equations are not known, and completeness of the eigenvalue spectrum has been proven only for $n=3$ (Dasmahapatra, Kedem and McCoy, 1993). For Theorem 2 no representation-level treatment of the closure exists at all.

Problem Statement

Derive the secular equations — that is, the finite closure relations among the Onsager generators $A_l$ and $G_m$ (the analogue of the uniform closure $A_{l+2N}=A_l$ of Theorem 1, equivalently the coefficients $\alpha_m$ of the closure relation $\sum_m \alpha_m A_m=0$ or of Davies' closure polynomial $h(z)$) — for the Onsager algebras generated by the two constructions of Minami (2021):

1. Theorem 2: $A_0=\sum_j(\eta_{2j-1}^{\,n-k}-\eta_{2j-1}^{\,k})$ and $A_1=\sum_j(\eta_{2j}^{\,n-k}-\eta_{2j}^{\,k})$ with $n=3k$ and $k$ not divisible by 3 (Dolan-Grady constant $C=-27$);

2. Theorem 3: $A_0=\sum_j\sum_{k=1}^{n-1}\eta_{2j-1}^{\,k}/(1-\omega^{-k})$ and $A_1=\sum_j\sum_{k=1}^{n-1}\eta_{2j}^{\,k}/(1-\omega^{-k})$ (Dolan-Grady constant $C=n^2$);

in both cases starting only from the generalized Clifford relations $\eta_j\eta_{j+1}=\omega\eta_{j+1}\eta_j$ (with $\omega=e^{2\pi i/n}$) and $\eta_j^n=1$, together with $\eta_j\eta_k=\eta_k\eta_j$ for $|j-k|>1$ and the cyclic boundary condition $\eta_{M+j}=\eta_j$ for $M=2N$ operators. Then specify, for all cases, the possible form of the eigenvalues of the superintegrable Hamiltonian $H=A_0+\lambda A_1$: which closure polynomials $h(z)$, angle sets $\{\theta_j\}$, and sector multiplicities can occur, consistently with the general eigenvalue formula for finite-dimensional Onsager-algebra representations. These deliverables form a single objective — identifying which finite-dimensional quotients of the Onsager algebra the two constructions generate and the spectral structure that follows, the closure relations being what determine the eigenvalue data. A rigorous proof that a given family admits no uniform finite closure, together with a characterization of how its closure data depend on the representation, resolves the problem for that family in the negative direction.

References

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