Proof of the inhomogeneous eigenvalue conjecture for the supersymmetric eight-vertex model on a strip with open boundary conditions (Hagendorf–Liénardy Conjecture 5.1)

At the supersymmetric point of the eight-vertex model, where the vertex weights obey $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ (equivalently, crossing parameter $\eta=\pi/3$ ), Hagendorf and Liénardy proved in 2020 that the homogeneous transfer matrix of the model on a strip with open boundary conditions and specific non-diagonal $K$ -matrices possesses the remarkably simple, non-degenerate, largest eigenvalue $\Lambda_L=(a+b)^{2L}\operatorname{tr}(K^+K^-)$ . Their Conjecture 5.1 extends this to the inhomogeneous model, in which each of the $L$ vertical lines carries its own spectral parameter $u_j$ : the double-row transfer matrix is conjectured to have eigenvalue $\Lambda_L=\operatorname{tr}(K^+(u)K^-(u))\prod_{j=1}^{L}(a(u+u_j)+b(u+u_j))(a(u-u_j)+b(u-u_j))$ . The conjecture was verified numerically for small $L$ in the trigonometric limit $p\to0$ , and that limit was subsequently proven using boundary quantum Knizhnik–Zamolodchikov equations; however, the conjecture itself — the elliptic regime at generic nome $p$ — remains unproven, as does its periodic-boundary counterpart dating from 2010–2013. This problem asks for a rigorous proof of Conjecture 5.1, which is the stated route to rigorous eigenvector analysis, sum rules, exact finite-size correlation functions, and connections to alternating-sign-matrix combinatorics and the Painlevé VI equation.

Background

The eight-vertex model is a statistical-mechanical model on the square lattice in which arrows on edges are conserved at each vertex, leaving eight local configurations with weights $a$ , $b$ , $c$ , $d$ . Its row-to-row transfer matrix $T$ builds the partition function row by row, and its largest eigenvalue determines the free energy per site. Baxter introduced a parametrisation of the weights by Jacobi theta functions depending on a spectral parameter $u$ , a crossing parameter $\eta$ , and an elliptic nome $p$ with $0<p<1$ ; the limit $p\to0$ is the trigonometric (six-vertex) limit. On a strip with $L$ vertical lines and open boundary conditions, the transfer matrix is of Sklyanin double-row type, $T(u)=\operatorname{tr}_0 K_0^+(u)\,U_{0,[1,L]}(u)\,K_0^-(u)\,\bar U_{0,[1,L]}(u)$ , where $U$ and $\bar U$ are ordered products of the eight-vertex $R$ -matrix (the local vertex weights in operator form) between an auxiliary space and the $L$ physical spaces, and the $K$ -matrices are $2\times2$ solutions of the reflection equation that encode the boundary weights. In the inhomogeneous model the $j$ -th vertical line carries its own spectral parameter $u_j$ , so the $R$ -matrices in $U$ and $\bar U$ are evaluated at $u\pm u_j$ .

A distinguished locus is the supersymmetric eight-vertex model, where the weights obey $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ ; in the theta-function parametrisation this corresponds to $\eta=\pi/3$ . The name comes from a dynamic lattice supersymmetry of the associated open XYZ spin chain — the anisotropic Heisenberg spin- $1/2$ chain with boundary magnetic fields on the matching anisotropy line. Its supercharge $Q$ (a nilpotent local operator) and adjoint $Q^\dagger$ define supersymmetry singlets, the common vectors annihilated by $Q$ and $Q^\dagger$ ; for the relevant parameters these singlets are the ground states of the chain.

In 2001 Stroganov conjectured that, for periodic boundary conditions and an odd number $L=2n+1$ of vertical lines, the homogeneous transfer matrix of this model possesses the remarkably simple, doubly degenerate eigenvalue $\Theta_n=(a+b)^{2n+1}$ . This conjecture stimulated work connecting the model to enumerative combinatorics (alternating sign matrices, the objects enumerated by the ground-state components), to functional equations, and to the Painlevé VI equation (certain polynomials appearing at special spectral values are tau functions of Painlevé VI). Hagendorf and Liénardy proved Stroganov's conjecture in Part I of their work (2018) using the supersymmetry, and showed that $\Theta_n$ is the largest eigenvalue for positive weights.

In Part II (2020) the same authors extended the analysis to open boundary conditions. For $K^\pm$ built from a specific non-diagonal $K$ -matrix (containing all three Pauli matrices) with its boundary parameter fixed at $t=\pi/6$ , they proved that the homogeneous transfer matrix possesses the non-degenerate eigenvalue $\Lambda_L=(a+b)^{2L}\operatorname{tr}(K^+K^-)$ , that its eigenspace is the space of supersymmetry singlets of the associated open XYZ chain, and that for positive weights it is the largest eigenvalue, thereby fixing the free energy per site $f=-\ln(a+b)$ and the boundary free energy. Their proof rests on a commutation relation between the supercharge and the transfer matrix that is special to the homogeneous point.

The open question is the inhomogeneous generalisation, stated as Conjecture 5.1 of that paper: for $\eta=\pi/3$ and the same $K$ -matrices, the inhomogeneous transfer matrix $T(u|u_1,\dots,u_L)$ is conjectured to possess the eigenvalue obtained by replacing each homogeneous factor $(a+b)^2$ by $(a(u+u_j)+b(u+u_j))(a(u-u_j)+b(u-u_j))$ , multiplied by $\operatorname{tr}(K^+(u)K^-(u))$ . The conjecture was verified numerically for small $L$ in the trigonometric limit $p\to0$ and checked for compatibility with the functional equations obeyed by the transfer matrix. An analogous conjecture for the periodic inhomogeneous model is due to Razumov and Stroganov (2010) and Zinn-Justin (2013); Zinn-Justin used it as input to derive a sum rule for the ground-state components that had been observed by Bazhanov and Mangazeev, and Brasseur and Hagendorf developed this conditional framework further. Proving these conjectures is the stated route to a rigorous investigation of the eigenvectors of the supersymmetric eight-vertex model, with expected payoffs in alternating-sign-matrix combinatorics, exact finite-size correlation functions, and finite-chain Painlevé VI theory.

Problem Statement

Prove Conjecture 5.1 of Hagendorf–Liénardy (2020). Let $\eta=\pi/3$ and let $K(u)$ be the $K$ -matrix

$K(u)=\mathbf{1}+\sum_{\alpha=1}^{3}\frac{\vartheta_1(u,p)}{\vartheta_{5-\alpha}(u,p)}\,\mu_\alpha\,\sigma^\alpha$

with the coefficients $\mu_\alpha$ given by eq. (111) of that paper, evaluated at the boundary parameter $t=\pi/6$ . (In the source paper's conventions, the $K$ -matrix (101) and the coefficients (111) are expressed through Jacobi theta functions of nome $p$ , whereas the vertex weights $a(u),b(u),c(u),d(u)$ of eq. (96) are theta functions of nome $p^2$ ; both conventions are retained here exactly as in the source.) Show that, for every $L\geqslant1$ and arbitrary inhomogeneity parameters $u_1,\dots,u_L$ , the double-row transfer matrix

$T(u|u_1,\dots,u_L)=\operatorname{tr}_0\,K_0^+(u)\,U_{0,[1,L]}(u|u_1,\dots,u_L)\,K_0^-(u)\,\bar U_{0,[1,L]}(u|u_1,\dots,u_L)$

with $K^-(u)=K(u)$ , $K^+(u)=K(u+2\eta)$ , $U_{0,[1,L]}=R_{0L}(u+u_L)\cdots R_{01}(u+u_1)$ , $\bar U_{0,[1,L]}=R_{01}(u-u_1)\cdots R_{0L}(u-u_L)$ , and $R(u)$ the eight-vertex $R$ -matrix with weights parametrised as in eq. (96) of the source paper, possesses the eigenvalue

$\Lambda_L=\operatorname{tr}\bigl(K^+(u)K^-(u)\bigr)\prod_{j=1}^{L}\bigl(a(u+u_j)+b(u+u_j)\bigr)\bigl(a(u-u_j)+b(u-u_j)\bigr).$

The proof must hold in the elliptic regime of generic nome $0<p<1$ ; answers restricted to the trigonometric limit $p\to0$ (already proven via boundary quantum Knizhnik–Zamolodchikov equations) or to the homogeneous point $u_1=\cdots=u_L=0$ (already proven in the source paper) do not discharge the conjecture. The proof may proceed by any rigorous technique — for example an explicit construction of the eigenvector with $T\Psi=\Lambda_L\Psi$ , functional $TQ$ -type relations, separation of variables, elliptic extensions of the boundary quantum Knizhnik–Zamolodchikov equations, or supersymmetry arguments. A natural strengthening, envisaged but not required by the source, is the characterisation of the corresponding eigenspace, expected to generalise the space of supersymmetry singlets of the associated open XYZ spin chain; the required result is the eigenvalue formula itself.

References

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