Completeness of the homogeneous T-Q ansatz for the 8-vertex model with quasi-periodic boundary conditions and an odd number of sites

The inhomogeneous eight-vertex model (equivalently the spin-1/2 XYZ Heisenberg chain) with integrable quasi-periodic boundary conditions - periodic, or twisted by a Pauli matrix - was solved by Niccoli and Terras (2015) for the lattice sizes left out by Baxter's original even-N solution: its spectrum and eigenstates are completely characterized by separation of variables and, equivalently, by an inhomogeneous T-Q functional equation valid for every number of sites N. The remaining gap is the homogeneous (Baxter-type) T-Q reformulation, in which $Q(\lambda)=\prod_{j=1}^{N}\theta_X(\lambda-\lambda_j)$ is an elliptic polynomial of degree N with twist-dependent quasi-periods whose roots satisfy Bethe-type equations. Completeness of this ansatz - a one-to-one correspondence between admissible Q-solutions and transfer-matrix eigenvalues and eigenstates - has been proven only for even N. For odd N it remains open in the twisted cases (only sufficiency is known) and in the periodic case (where the spectrum is doubly degenerate and only heuristic algebraic-Bethe-ansatz-type expressions exist). The problem is to prove or disprove completeness of the homogeneous T-Q ansatz for odd N and to construct the corresponding algebraic-Bethe-ansatz-type eigenstates.

Background

The eight-vertex model is a two-dimensional lattice model of statistical mechanics in which arrows live on the edges of a square lattice and exactly eight arrow configurations are allowed around each vertex, with Boltzmann weights $a,b,c,d$; its local weights form an R-matrix satisfying the Yang-Baxter equation (the cubic consistency condition underlying integrability), so that the row-to-row transfer matrices $T(\lambda)$, built as traces of monodromy matrices with one spectral parameter $\lambda$ and site-dependent inhomogeneity parameters $\xi_n$, form a commuting family. In the homogeneous limit $\xi_n\to 0$ the logarithmic derivative of $T(\lambda)$ gives the Hamiltonian of the spin-1/2 XYZ Heisenberg chain, the fully anisotropic generalization of the XXZ chain, so the two models are diagonalized by the same transfer matrix. Because the arrow charge through a vertex is not conserved, the model has no U(1) symmetry and the standard coordinate and algebraic Bethe ansatz cannot be applied directly. Baxter solved the periodic model by introducing the functional T-Q relation: the eigenvalues $t(\lambda)$ of $T(\lambda)$ and the eigenvalues $Q(\lambda)$ of an auxiliary operator satisfy $t(\lambda)Q(\lambda)=a(\lambda)Q(\lambda-\eta)+d(\lambda)Q(\lambda+\eta)$, where $\eta$ is the crossing parameter, $a(\lambda)=\prod_{n=1}^{N}\theta(\lambda-\xi_n+\eta)$, $d(\lambda)=a(\lambda-\eta)$, and $\theta$ is a Jacobi theta function; entireness of $t$ at the zeros of $Q$ produces Bethe-type equations for the roots $\lambda_j$. Here $Q(\lambda)=\prod_{j=1}^{N}\theta_X(\lambda-\lambda_j)$ is an elliptic polynomial: a product of theta functions, hence an entire function with fixed multiplicative quasi-periods ($\theta(\lambda+\pi)=-\theta(\lambda)$, $\theta(\lambda+\pi\omega)=-e^{-2i\lambda-i\pi\omega}\theta(\lambda)$ for half-periods $\pi,\pi\omega$), with the function $\theta_X$ chosen so that the quasi-periods of $Q$ match the twist. Baxter's route to the eigenvectors (and, as reviewed in his 2002 discussion of Bethe-ansatz completeness, his route to the functional relation itself) went through the vertex-IRF transformation, a generalized gauge transformation mapping the eight-vertex model to the solid-on-solid (SOS) model, also called the dynamical six-vertex or interaction-round-a-face (IRF) model, which is solvable by Bethe ansatz; this is why his solution covers only an even number of sites N: the space of states of the periodic SOS model is zero-dimensional for odd N at generic crossing parameter.

Niccoli (2013) and Levy-Bencheton, Niccoli and Terras (2015) solved the antiperiodic dynamical six-vertex model by Sklyanin's separation of variables (SOV), a framework in which the transfer matrix acts by multiplication on a basis labeled by N separate variables, yielding a complete characterization of all eigenvalues and eigenstates by a discrete system of equations at the inhomogeneity parameters. Niccoli and Terras (2015) then showed that the quasi-periodic eight-vertex transfer matrices - twisted by $\sigma_x$, $\sigma_y$ or $\sigma_z$ for any N, and periodic for odd N - are related to this antiperiodic dynamical six-vertex transfer matrix by vertex-IRF maps: bijectively in the twisted cases, and through two non-bijective maps with complementary kernels in the periodic odd case, where the eight-vertex spectrum is doubly degenerate and equals half of the antiperiodic dynamical six-vertex spectrum. This gives a complete SOV characterization for all quasi-periodic cases, a complete inhomogeneous T-Q equation (a T-Q relation with an extra inhomogeneous term) with algebraic-Bethe-ansatz(ABA)-like eigenstates for all N, and - proven only for even N - the equivalence of this characterization with the solutions of the homogeneous T-Q equation within the elliptic-polynomial class with twist-dependent quasi-periods. For odd N only the following is known: in the twisted cases every admissible Q-solution of the homogeneous T-Q equation does give an eigenvalue and eigenstate (sufficiency), but the converse - completeness of the Bethe-type equations - is unproven; in the periodic case even this is missing, since the double degeneracy and the two vertex-IRF images have no established counterpart in the homogeneous formulation. The homogeneous reformulation is the one suited to the homogeneous limit and to the analysis of the thermodynamic limit of the Bethe equations, and is the natural starting point for constructing Bethe states usable in form-factor computations, which is why the authors explicitly flagged the odd-N completeness problem as future work and expected the even-N restriction to be purely technical.

Problem Statement

Let $T^{(8V)}_{(x,y)}(\lambda)$ be the transfer matrix of the inhomogeneous eight-vertex model on $N$ sites with quasi-periodic boundary conditions labeled by $(x,y)\in\{0,1\}^2$ (periodic for $(0,0)$; twist by $\sigma_x$, $\sigma_y$ or $\sigma_z$ otherwise), with generic crossing parameter $\eta$ and inhomogeneity parameters $\xi_1,\dots,\xi_N$ in general position (in the sense of the source paper: $\xi_j+\eta/2\notin\pi\mathbb{Z}+\pi\omega\mathbb{Z}$, and $\xi_a-\xi_b+\epsilon\eta\notin\pi\mathbb{Z}+\pi\omega\mathbb{Z}$ for $a\ne b$ and $\epsilon\in\{-1,0,1\}$). Consider the homogeneous functional T-Q equation

$$t(\lambda)Q(\lambda)=((-1)^y(-i)^{xy})\,a(\lambda)\,Q(\lambda-\eta)+((-1)^x i^{xy})\,d(\lambda)\,Q(\lambda+\eta),$$

with $a(\lambda)=\prod_{n=1}^{N}\theta(\lambda-\xi_n+\eta|\omega)$, $d(\lambda)=a(\lambda-\eta)$, where the unknowns are an entire function $t(\lambda)$ with the quasi-periodicity properties $t(\lambda+\pi)=(-1)^{N+y}t(\lambda)$, $t(\lambda+\pi\omega)=(-e^{-2i\lambda-i\pi\omega})^N e^{2i[\sum_{k=1}^{N}\xi_k-\frac{N\eta}{2}+\frac{x\pi}{2}]}t(\lambda)$, and a function of the elliptic-polynomial form $Q(\lambda)=\prod_{j=1}^{N}\theta_X(\lambda-\lambda_j)$ with twist-dependent quasi-periods (the function $\theta_X$ is fixed by $(x,y)$ as in the source paper, Eq. (4.19)). The Bethe-type equations are the conditions that $t(\lambda)$ defined by this relation be entire, i.e. the requirement that the numerator vanish at each root $\lambda_j$ of $Q$ (so that the poles of the right-hand side cancel).

For even $N$ it is known (and proven in the source paper, Proposition 4.1, building on the even-N result of arXiv:1507.03404) that these Bethe-type equations are complete: their solutions are in one-to-one correspondence with the transfer-matrix eigenvalues, and the eigenstates are obtained as multiple actions of explicitly defined diagonal SOV operators evaluated at the Bethe roots on a pseudo-vacuum state (a fixed simple reference state in the SOV basis; an ABA-type construction). For twisted boundary conditions and any $N$, every admissible solution does give an eigenvalue with an eigenstate (Theorem 4.3 of the source), but the converse is unproven for odd $N$; for the periodic chain with odd $N$, where the spectrum is doubly degenerate and equals half of the antiperiodic dynamical six-vertex spectrum, not even this sufficiency direction is established for the homogeneous ansatz, and the matching of the twofold degeneracy with independent pairs of Q-solutions or vertex-IRF images is open.

The problem: for odd $N$, prove that the homogeneous T-Q elliptic-polynomial ansatz is complete - that is, establish the equivalence between the SOV/vertex-IRF characterization of the quasi-periodic eight-vertex transfer-matrix spectrum and eigenstates and the formulation in terms of solutions of these homogeneous Bethe-type equations, for the twisted cases and for the doubly-degenerate periodic case - and construct the corresponding ABA-type eigenstates; or disprove it by exhibiting, for some odd $N$ and generic parameters, a failure of this one-to-one correspondence - for example a transfer-matrix eigenvalue not generated by any admissible Q-solution, or a doubly-degenerate periodic eigenvalue whose eigenspace is not spanned by eigenstates from the corresponding independent admissible Q-solutions. A complete solution supplies the parity-completion of Baxter's Bethe-ansatz solution of the eight-vertex model at generic crossing parameter.

References

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