The square-lattice eight-vertex model whose vertex weights obey — the supersymmetric eight-vertex model — has, on a cylinder with an odd number of vertical lines and periodic boundary conditions, a doubly degenerate transfer-matrix eigenvalue . Conjectured by Stroganov in 2001, this was proved by Hagendorf and Liénardy in 2018 using the lattice supersymmetry of a related XYZ spin chain. The inhomogeneous counterpart remains open: Razumov and Stroganov (2010) conjectured that the transfer matrix with arbitrary inhomogeneity parameters on the vertical lines has the doubly degenerate eigenvalue (where in the theta-function parametrization) reducing to in the homogeneous limit, and Zinn-Justin (2012) conjectured that the corresponding eigenvectors have components that are theta functions of degree in each inhomogeneity with prescribed pseudo-periodicity. The conjecture underpins a web of conditional results — sum rules and special eigenvector components assume it directly, while the exact finite-size correlation functions of Hagendorf and Rosengren rest on a related unproven Q-operator analyticity assumption for which the conjecture would supply the missing ingredient — and has been verified by computer for the small odd sizes , but no proof is known. The problem is to prove the conjecture rigorously, or to refute it by an explicit certified counterexample.
Background
The eight-vertex model of statistical mechanics assigns arrows to the edges of a square lattice; only eight arrow patterns are allowed around each vertex, with Boltzmann weights , , , (each weight serves two patterns related by reversing all four arrows). Baxter solved the model by the transfer-matrix method: the row-to-row transfer matrix is built from a local -matrix acting on two copies of the local arrow space . In the standard elliptic parametrization the weights depend on a spectral parameter , a crossing parameter and an elliptic nome with , through Jacobi theta functions (the standard doubly quasi-periodic entire functions of the argument and the nome ):
These parametrizations are fixed up to an overall normalisation constant (set to here), which drops out of the supersymmetric condition and of all eigenvalue statements. The inhomogeneous model attaches a spectral parameter (an 'inhomogeneity') to each of the vertical lines, and its transfer matrix acts on as
where the trace runs over the auxiliary space ; setting gives the homogeneous model. The local -matrix is Baxter's standard eight-vertex -matrix: in the basis of ,
and denotes its action on .
At the 'supersymmetric point' the weights obey the relation , and the model is called the supersymmetric eight-vertex model. In the homogeneous case the transfer matrix commutes with the Hamiltonian of the XYZ spin chain (the anisotropic Heisenberg chain, i.e. a chain of spin-'s with nearest-neighbour couplings along the three Pauli directions) on the coupling line , which carries a non-local lattice supersymmetry: nilpotent supercharges that map a chain of sites to one of sites (Fendley and Hagendorf, 2012). Stroganov, who had earlier found an analogous odd-length phenomenon for the XXZ chain at anisotropy , conjectured in 2001 that for an odd number of vertical lines with periodic boundary conditions the transfer matrix of the homogeneous supersymmetric eight-vertex model has a doubly degenerate eigenvalue ; 'doubly degenerate' means that the eigenspace is two-dimensional, split by the spin-reversal operator (which reverses all arrows/spins) into eigenvectors with eigenvalues and . This was proved by Hagendorf and Liénardy (2018), who showed moreover that the corresponding eigenvectors are the ground states of the supersymmetric XYZ chain and that is the largest eigenvalue for positive weights.
The open conjecture at issue is the inhomogeneous counterpart. Razumov and Stroganov (2010) conjectured that the inhomogeneous transfer matrix with vertical lines has the doubly degenerate eigenvalue
which reduces to in the homogeneous limit . Zinn-Justin (2012, published 2013) conjectured the full eigenvector structure: the eigenvalue problem with spin-reversal parity has a solution whose components are theta functions of degree and nome in each inhomogeneity with prescribed pseudo-periodicity — equivalently, entire functions satisfying the pseudo-periodicity conditions
\Psi_L(\ldots,u_i+\pi,\ldots)=\Bigl(\prod_{j\neq i}\sigma^z_j\Bigr)\,\Psi_L(\ldots,u_i,\ldots),$$ with components generically non-zero and without a common factor. In the trigonometric limit $p\to0$ the model degenerates to the six-vertex model at $\Delta=-1/2$, where the analogous special eigenvector is characterized rigorously through polynomial solutions of the quantum Knizhnik–Zamolodchikov (qKZ) equation (Razumov, Stroganov and Zinn-Justin, 2007). No comparable rigorous characterization is known in the elliptic regime. The conjecture matters well beyond bookkeeping: because the conjectured eigenvector components are theta-regular in the inhomogeneities, a proof would unlock the chain of currently conditional results that assume it — the sum rule for the squared norm of the XYZ ground state (Zinn-Justin), the special eigenvector components conjectured by Bazhanov–Mangazeev and by Razumov–Stroganov, and the refined scalar-product and sum-rule identities (Brasseur and Hagendorf, 2021) — and would supply the inhomogeneous eigenvector formulas whose absence leaves the exact finite-size correlation functions of Hagendorf and Rosengren (2024) conditional on their related Q-operator analyticity assumption. ## Problem Statement Prove, or refute by an explicit certified counterexample, the inhomogeneous Stroganov–Razumov–Stroganov–Zinn-Justin conjecture for the periodic supersymmetric eight-vertex model: Let $L=2n+1$ with integer $n\geq 1$, let $p=e^{i\pi\tau}$ with $\operatorname{Im}\tau>0$ be an elliptic nome (the standard regime of positive weights is the real interval $p\in(0,1)$), let $\eta=\pi/3$, let $a(u),b(u),c(u),d(u)$ be the weights in the theta-function parametrization stated in the background, and let $T_L(u|u_1,\ldots,u_L)=\operatorname{tr}_0\bigl(R_{0,L}(u_L-u)\cdots R_{0,1}(u_1-u)\bigr)$ be the transfer matrix with periodic boundary conditions along the horizontal direction and inhomogeneities $u_1,\ldots,u_L$ on the $L$ vertical lines. (a) Show that, for all spectral parameters $u$ and all inhomogeneities $u_1,\ldots,u_L$, the transfer matrix possesses the eigenvalue $$\Theta_n(u|u_1,\ldots,u_L)=\prod_{i=1}^{L}r(u_i-u),\qquad r(u)=a(u)+b(u)=\vartheta_4(0,p^2)\,\vartheta_1(u+\eta,p^2)\,\vartheta_4(u+\eta,p^2),$$ and that this eigenvalue is doubly degenerate: its eigenspace is two-dimensional and spanned by two vectors $\Psi_L$ and $\bar\Psi_L$ that are eigenvectors of the spin-reversal operator $F=\prod_{j=1}^{L}\sigma^x_j$ with eigenvalues $(-1)^n$ and $(-1)^{n+1}$. (b) Show that the eigenvalue problem $T_L(u|u_1,\ldots,u_L)\Psi_L=\Theta_n(u|u_1,\ldots,u_L)\Psi_L$ with $F\Psi_L=(-1)^n\Psi_L$ admits a solution $\Psi_L(u_1,\ldots,u_L)$ whose components are theta functions of degree $2n$ and nome $p^2$ in each variable $u_i$ — equivalently, entire functions satisfying the pseudo-periodicity relations stated in the background — generically non-zero and without common factor. The claims are consistent with two known specializations: in the homogeneous limit $u_1=\cdots=u_L$, claim (a) reduces to the theorem of Hagendorf and Liénardy (2018) that $\Theta_n=(a+b)^{2n+1}$ is a doubly degenerate transfer-matrix eigenvalue, and in the trigonometric limit $p\to0$ the model degenerates to the six-vertex model at $\Delta=-1/2$, where the analogous special eigenvector is characterized rigorously via polynomial qKZ solutions. A refutation must exhibit an explicit parameter point ($n$, $p$, $u$, $u_1,\ldots,u_L$) together with a certified computation showing that claim (a) or (b) fails there. ## References 1. C. Hagendorf, J. Liénardy, On the transfer matrix of the supersymmetric eight-vertex model. I. Periodic boundary conditions, J. Stat. Mech. (2018) 033106, DOI 10.1088/1742-5468/aab01d, arXiv:1711.04397, https://arxiv.org/abs/1711.04397 2. C. Hagendorf, J. Liénardy, On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions, J. 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