The inhomogeneous transfer-matrix eigenvalue conjectures for the supersymmetric eight-vertex model (open strip and periodic)

At its supersymmetric point (crossing parameter $\eta=\pi/3$), the elliptic eight-vertex model with open or periodic boundaries possesses a remarkably simple homogeneous transfer-matrix eigenvalue, proven using lattice supersymmetry: $\Lambda_L=(a+b)^{2L}\operatorname{tr}(K^+K^-)$ on the strip and $\Theta_n=(a+b)^{2n+1}$ for odd periodic width. The inhomogeneous generalisations — Conjecture 5.1 of Hagendorf–Liénardy (2020) for the strip, and the conjecture of Razumov–Stroganov (2010) and Zinn-Justin (2013) for the periodic case — remain unproven in the genuine elliptic regime: only the trigonometric (six-vertex, $p\to0$) limit of the strip conjecture is settled, via contour-integral solutions of the boundary quantum Knizhnik–Zamolodchikov equations. The problem asks for a rigorous proof, or refutation, of both product-form inhomogeneous eigenvalue formulas, for all strip widths and all inhomogeneity parameters. A proof would unlock rigorous eigenvector analysis of the supersymmetric eight-vertex model, convert conditional sum-rule results into theorems, and connect the elliptic eigenvectors to combinatorics and Painlevé VI.

Background

The square-lattice eight-vertex model places arrows on lattice edges and assigns each vertex a Boltzmann weight selected from four values $a,b,c,d$ according to the configuration of its four incident edges (eight configurations, paired by global arrow reversal). Its transfer matrix — the operator implementing the addition of one lattice row, obtained by contracting $R$-matrices over a two-dimensional auxiliary space — generates the partition function, so the model is solved through the spectrum of this matrix. In the inhomogeneous version, each of the $L$ vertical lines carries its own spectral parameter $u_j$, and the transfer matrix becomes a function $\mathcal T(u|u_1,\dots,u_L)$ of the row spectral parameter $u$ and the column inhomogeneities $u_j$. Inhomogeneity is a tool rather than a complication: the dependence of eigenvectors on the $u_j$ carries the information needed for scalar products, sum rules, correlation functions, and combinatorial applications.

The genuine eight-vertex model is elliptic. In the standard Jacobi-theta parameterisation (nome $p$, $0<p<1$, overall constant $\rho$, crossing parameter $\eta$), $a(u)=\rho\,\vartheta_4(2\eta,p^2)\vartheta_4(u,p^2)\vartheta_1(u+2\eta,p^2),$ $b(u)=\rho\,\vartheta_4(2\eta,p^2)\vartheta_1(u,p^2)\vartheta_4(u+2\eta,p^2),$ $c(u)=\rho\,\vartheta_1(2\eta,p^2)\vartheta_4(u,p^2)\vartheta_4(u+2\eta,p^2),$ $d(u)=\rho\,\vartheta_1(2\eta,p^2)\vartheta_1(u,p^2)\vartheta_1(u+2\eta,p^2),$ with $\vartheta_i$ Jacobi theta functions. For $\eta=\pi/3$ the weights obey $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ — the supersymmetric (or combinatorial) line. The name comes from lattice supersymmetry: the logarithmic derivative of the transfer matrix at $u=0$ is the Hamiltonian of an anisotropic XYZ spin-$1/2$ chain (the fully anisotropic Heisenberg chain), and at $\eta=\pi/3$ this Hamiltonian admits a nilpotent local supercharge $\mathcal Q$ with $\mathcal Q^2=0$ and $H\propto\{\mathcal Q,\mathcal Q^\dagger\}$, so that its ground states are supersymmetry singlets (vectors annihilated by $\mathcal Q$ and $\mathcal Q^\dagger$, computable as the cohomology of $\mathcal Q$).

On this line, Stroganov conjectured in 2001 that the homogeneous periodic transfer matrix with an odd number $L=2n+1$ of vertical lines possesses a doubly-degenerate eigenvalue $\Theta_n=(a+b)^{2n+1}$; this was proven in 2018 (Hagendorf–Liénardy, Part I of the series), together with the statement that $\Theta_n$ is the largest eigenvalue for positive weights, using lattice supersymmetry and Perron–Frobenius arguments (the classical theorem that a matrix with strictly positive entries has a unique eigenvalue of largest modulus). For open (strip) boundaries, the transfer matrix is defined by Sklyanin's construction from a $K$-matrix solving the reflection equation (the boundary analogue of the Yang–Baxter consistency equation): $\mathcal T=\operatorname{tr}_0(K^+_0U_{0,[1,L]}K^-_0\bar U_{0,[1,L]})$ with $K^-(u)=K(u)$, $K^+(u)=K(u+2\eta)$. The relevant $K$-matrix is $K(u)=1+\sum_{\alpha=1}^3\frac{\vartheta_1(u,p)}{\vartheta_{5-\alpha}(u,p)}\mu_\alpha\sigma^\alpha$ (with $\sigma^\alpha$ Pauli matrices), whose coefficients are fixed by a complex boundary parameter $y=\vartheta_1(t,p^2)/\vartheta_4(t,p^2)$ through $\mu_1=\frac{\vartheta_4(\eta,p)}{\vartheta_1(\eta,p)}\frac{2\operatorname{Re}y}{1+|y|^2},\quad\mu_2=\frac{\vartheta_3(\eta,p)}{\vartheta_1(\eta,p)}\frac{2\operatorname{Im}y}{1+|y|^2},\quad\mu_3=\frac{\vartheta_2(\eta,p)}{\vartheta_1(\eta,p)}\frac{1-|y|^2}{1+|y|^2}.$ For the choice $t=\pi/6$, Hagendorf–Liénardy (2020) proved that the homogeneous strip transfer matrix possesses the non-degenerate eigenvalue $\Lambda_L=(a+b)^{2L}\operatorname{tr}(K^+K^-)$, again the largest for positive weights.

What is missing is the inhomogeneous picture. This is precisely where the periodic case has proven its value: the inhomogeneous eigenvector (whose existence and analyticity properties were conjectured by Zinn-Justin in 2013) is the object from which the squared norm and sum rules of the XYZ ground state were derived, feeding a research program that connects the model to the sixth Painlevé equation (one of the six classical nonlinear second-order ordinary differential equations whose transcendental solutions, through their tau functions, encode the model's correlation functions) and to the enumeration of alternating sign matrices (square $0,\pm1$ matrices whose nonzero entries alternate in sign along each row and column). Brasseur–Hagendorf (2021) derived a large family of such sum rules conditionally on the same conjecture. Two inhomogeneous eigenvalue formulas are conjectural. Periodic (Razumov–Stroganov 2010; Zinn-Justin 2013): the inhomogeneous transfer matrix $T(u|u_1,\dots,u_{2n+1})$ possesses the doubly-degenerate eigenvalue $\Theta_n=\prod_{i=1}^{2n+1}r(u_i-u)$ with $r(u):=a(u)+b(u)$; in the normalisation $\rho=1$ of the theta parameterisation above, $r(u)=\vartheta_4(0,p^2)\vartheta_1(u+\eta,p^2)\vartheta_4(u+\eta,p^2)$. Open strip (Conjecture 5.1 of the 2020 paper, at $t=\pi/6$): $\mathcal T(u|u_1,\dots,u_L)$ possesses the eigenvalue $\Lambda_L=\operatorname{tr}(K^+(u)K^-(u))\prod_{j=1}^L(a(u+u_j)+b(u+u_j))(a(u-u_j)+b(u-u_j))$.

Both formulas reduce to the proven homogeneous ones at $u_1=\dots=u_L=0$; the strip formula was verified numerically for small $L$ only in the trigonometric limit $p\to0$, in which the model degenerates to the six-vertex model. That limit case of the strip conjecture has since been proven: Hagendorf–Liénardy (2021) constructed multiple-contour-integral solutions of the boundary quantum Knizhnik–Zamolodchikov equations (a system of difference equations for vectors depending on the inhomogeneities, associated with the six-vertex $R$-matrix and reflecting boundaries) and showed that their eigenvalue at the combinatorial point proves the trigonometric limit of the conjectured expression. However, no elliptic counterpart of that framework is known, and the lattice-supersymmetry method of the homogeneous proofs uses homogeneity in an essential way. The source paper closes by stating that both the strip conjecture and its periodic analogue remain to be proven; their proof would allow rigorous investigation of the corresponding eigenvectors.

Problem Statement

Prove or refute the two inhomogeneous eigenvalue conjectures for the supersymmetric eight-vertex model ($\eta=\pi/3$), which the source poses jointly ("Both these conjectures remain to be proven"). The two parts are the open and periodic faces of one objective — the inhomogeneous extension of the supersymmetric point's simple eigenvalue — and the problem is resolved when each conjecture is rigorously settled, positively or negatively; the two parts may be settled by different techniques, and a certified counterexample to either is an admissible resolution of that part.

(1) Open strip (primary; Conjecture 5.1 of the source). Let $L\geqslant1$, $0<p<1$, $\eta=\pi/3$, with vertex weights $a,b,c,d$ in the theta parameterisation given in the background. Let $K(u)=1+\sum_{\alpha=1}^3\frac{\vartheta_1(u,p)}{\vartheta_{5-\alpha}(u,p)}\mu_\alpha\sigma^\alpha$ with the coefficients $\mu_\alpha$ fixed by $y=\vartheta_1(t,p^2)/\vartheta_4(t,p^2)$ at $t=\pi/6$ through the displayed formulas, $K^-(u)=K(u)$, $K^+(u)=K(u+2\eta)$. Let $\mathcal T(u|u_1,\dots,u_L)=\operatorname{tr}_0\bigl(K^+_0(u)U_{0,[1,L]}K^-_0(u)\bar U_{0,[1,L]}\bigr),$ $U_{0,[1,L]}=R_{0L}(u+u_L)\cdots R_{01}(u+u_1),\qquad\bar U_{0,[1,L]}=R_{01}(u-u_1)\cdots R_{0L}(u-u_L),$ be the inhomogeneous strip transfer matrix. Establish or disprove that, for all $L\geqslant1$, all $0<p<1$, and all inhomogeneity parameters $u_1,\dots,u_L$ for which the parameterisation is defined (in particular all real inhomogeneities in the positive-weight regime), this transfer matrix possesses the eigenvalue $\Lambda_L=\operatorname{tr}\bigl(K^+(u)K^-(u)\bigr)\prod_{j=1}^L\bigl(a(u+u_j)+b(u+u_j)\bigr)\bigl(a(u-u_j)+b(u-u_j)\bigr).$

(2) Periodic twin (Razumov–Stroganov 2010; Zinn-Justin 2013). With the same weights, let the inhomogeneous periodic transfer matrix be $T(u|u_1,\dots,u_L)=\operatorname{tr}_0\bigl(R_{0L}(u_L-u)\cdots R_{02}(u_2-u)R_{01}(u_1-u)\bigr),$ with $L=2n+1$ vertical lines. Establish or disprove that, for all $n\geqslant0$, all $0<p<1$, and all inhomogeneities $u_1,\dots,u_{2n+1}$, this transfer matrix possesses the doubly-degenerate eigenvalue $\Theta_n=\prod_{i=1}^{2n+1}r(u_i-u),\qquad r(u):=a(u)+b(u),$ where, in the normalisation $\rho=1$ of the theta parameterisation above, $r(u)=\vartheta_4(0,p^2)\vartheta_1(u+\eta,p^2)\vartheta_4(u+\eta,p^2)$; for general $\rho$, both $T$ and $\Theta_n$ scale as $\rho^{2n+1}$, so the claim is independent of the value of $\rho$.

Known partial results that do not close the problem: the homogeneous cases (both proven) and the trigonometric limit $p\to0$ of part (1) (proven via boundary quantum Knizhnik–Zamolodchikov solutions); the conjectures concern the genuine elliptic regime $0<p<1$.

References

1. Christian Hagendorf and Jean Liénardy, "On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions", J. Stat. Mech. (2020) 033104. DOI: 10.1088/1742-5468/ab7748; arXiv:1911.03348; https://arxiv.org/abs/1911.03348 (source paper posing Conjecture 5.1)
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