Conjecture 5.1 of Hagendorf and Liénardy: an eigenvalue formula for the inhomogeneous supersymmetric eight-vertex model on a strip

The transfer matrix of the supersymmetric eight-vertex model (crossing parameter $\eta=\pi/3$) on a strip with open boundary conditions and appropriately chosen boundary $K$-matrices has the remarkably simple non-degenerate eigenvalue $\Lambda_L=(a+b)^{2L}\,\mathrm{tr}(K^+K^-)$ in the homogeneous case; this was proven by Hagendorf and Liénardy (2020) through the lattice supersymmetry of an associated open XYZ spin chain. Their Conjecture 5.1 predicts the inhomogeneous generalisation: with arbitrary inhomogeneity parameters $u_1,\dots,u_L$ on the vertical lines, the transfer matrix should possess the eigenvalue $\mathrm{tr}(K^+(u)K^-(u))\prod_j(a(u+u_j)+b(u+u_j))(a(u-u_j)+b(u-u_j))$. The conjecture was verified numerically for small strip widths in the trigonometric limit $p\to0$, and that limit has since been proven via boundary quantum Knizhnik-Zamolodchikov equations, but the conjecture for generic elliptic nome $0<p<1$ remains open, as does its periodic-boundary analogue. The open core is a proof (or disproof) of the eigenvalue formula in the elliptic regime.

Background

The square-lattice eight-vertex model is a classical statistical-mechanical model of arrows on lattice edges in which each vertex carries one of eight allowed arrow configurations; its local Boltzmann weights $a,b,c,d$ are encoded in an $R$-matrix acting on the tensor product of two two-dimensional vector spaces. In the standard Jacobi-theta-function parameterisation the weights depend on a spectral parameter $u$, a crossing parameter $\eta$ and an elliptic nome $p$ (the deformation parameter of the elliptic regime: $p=0$ is the trigonometric limit, and $0<p<1$ gives real weights) through

$$a(u)=\rho\,\vartheta_4(2\eta,p^2)\,\vartheta_4(u,p^2)\,\vartheta_1(u+2\eta,p^2),$$ $$b(u)=\rho\,\vartheta_4(2\eta,p^2)\,\vartheta_1(u,p^2)\,\vartheta_4(u+2\eta,p^2),$$ $$c(u)=\rho\,\vartheta_1(2\eta,p^2)\,\vartheta_4(u,p^2)\,\vartheta_4(u+2\eta,p^2),\qquad d(u)=\rho\,\vartheta_1(2\eta,p^2)\,\vartheta_1(u,p^2)\,\vartheta_1(u+2\eta,p^2),$$

where $\rho$ is a normalisation constant and $\vartheta_i$ denotes the Jacobi theta functions. The transfer matrix propagates a row of $L$ vertical lines; assigning distinct spectral parameters $u_1,\dots,u_L$ to the vertical lines gives the inhomogeneous transfer matrix, the natural generating object for exact correlation functions. Setting all $u_j$ equal recovers the homogeneous model.

On a strip, i.e. with open rather than periodic horizontal boundary conditions, Sklyanin's construction builds the transfer matrix by letting the $R$-matrix act on an auxiliary two-dimensional space (indexed $0$) attached to the row and taking the partial trace $\mathrm{tr}_0$ over it, together with two boundary $K$-matrices, $K^-(u)=K(u)$ and $K^+(u)=K(u+2\eta)$, where $K(u)$ solves the reflection equation — the boundary counterpart of the Yang-Baxter equation that guarantees commuting transfer matrices. The same construction yields the Hamiltonian of an open XYZ spin chain — a quantum spin-$1/2$ chain with anisotropic couplings $J_\alpha\sigma^\alpha\otimes\sigma^\alpha$ (with $\sigma^\alpha$ the Pauli matrices) and boundary magnetic fields — which commutes with the transfer matrix.

The supersymmetric eight-vertex model is the special case in which the weights obey $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$, equivalent to $\eta=\pi/3$ in the theta parameterisation. The name refers to lattice supersymmetry: up to a shift and rescaling, the associated XYZ Hamiltonian is the anticommutator of a nilpotent supercharge $\mathcal Q$ and its adjoint, and its ground states are supersymmetry singlets, i.e. states annihilated by both $\mathcal Q$ and $\mathcal Q^\dagger$.

For periodic boundary conditions and an odd number $L=2n+1$ of vertical lines, Stroganov conjectured in 2001 that the homogeneous transfer matrix of this model possesses the remarkably simple, doubly degenerate eigenvalue $\Theta_n=(a+b)^{2n+1}$; Hagendorf and Liénardy proved this in 2018 using the lattice supersymmetry. This circle of ideas connects the model to the enumerative combinatorics of alternating sign matrices (square arrays with entries $0,\pm1$ whose non-zero entries alternate in sign along every row and column and whose rows and columns each sum to $1$) and to tau functions of the Painlevé VI differential equation (special functions whose logarithmic derivatives encode solutions of that equation). For the strip, the same authors proved in 2020 that, for anisotropy $0<\zeta<1$ (with $\zeta=cd/ab$) and boundary parameter $y$ equal to the unique real root in $0<y<1$ of $\zeta(1+y^4)-(3-\zeta^2)y^2=0$ — parameterised as $y=\vartheta_1(t,p^2)/\vartheta_4(t,p^2)$ at $t=\pi/6$ — the homogeneous transfer matrix has the non-degenerate eigenvalue $\Lambda_L=(a+b)^{2L}\,\mathrm{tr}(K^+K^-)$, which is moreover the largest eigenvalue for positive weights. That proof is intrinsically homogeneous: the commutation relation between the supercharge and the transfer matrix does not extend directly to inhomogeneous transfer matrices, so the inhomogeneous generalisation could only be conjectured. The trigonometric limit $p\to0$ of this inhomogeneous conjecture was later proven by the same authors (2021) using the boundary quantum Knizhnik-Zamolodchikov equations, a system of functional equations for correlation functions of integrable lattice models with boundaries; no elliptic counterpart of that construction of comparable usability is known. For the periodic case, Razumov and Stroganov (2010) and Zinn-Justin (2013) had earlier conjectured the analogous inhomogeneous eigenvalue $\Theta_n(u|u_1,\dots,u_{2n+1})=\prod_{i=1}^{2n+1}r(u_i-u)$ with $r(u)=a(u)+b(u)$, together with a pseudo-periodicity characterisation of the corresponding eigenvector; that conjecture is likewise still open. Both conjectures are of interest because the inhomogeneous models allow one to investigate the properties of the corresponding eigenvectors rigorously, which underpins the links to combinatorics and Painlevé equations.

Problem Statement

Prove or disprove Conjecture 5.1 of Hagendorf and Liénardy (J. Stat. Mech. 2020, 033104). Let $\eta=\pi/3$, let the vertex weights be given by the theta parameterisation stated in the background, and let

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

be the general solution of the reflection equation used in that paper, with coefficients

$$\mu_1=\frac{\vartheta_4(\eta,p)}{\vartheta_1(\eta,p)}\frac{2\,\mathrm{Re}\,y}{1+|y|^2},\qquad \mu_2=\frac{\vartheta_3(\eta,p)}{\vartheta_1(\eta,p)}\frac{2\,\mathrm{Im}\,y}{1+|y|^2},\qquad \mu_3=\frac{\vartheta_2(\eta,p)}{\vartheta_1(\eta,p)}\frac{1-|y|^2}{1+|y|^2},$$

evaluated at $t=\pi/6$, that is at $y=\vartheta_1(\pi/6,p^2)/\vartheta_4(\pi/6,p^2)$, which is the unique real solution in $0<y<1$ of $\zeta(1+y^4)-(3-\zeta^2)y^2=0$ with $\zeta=(\vartheta_1(2\pi/3,p^2)/\vartheta_4(2\pi/3,p^2))^2$. (Following the conventions of the paper, the $K$-matrix and the coefficients $\mu_\alpha$ are written with nome $p$, whereas the vertex weights and the parameterisation of $y$ use nome $p^2$.) The conjecture asserts that the inhomogeneous transfer matrix of the supersymmetric eight-vertex model on the strip with $L\geqslant1$ vertical lines,

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

with $K^-(u)=K(u)$, $K^+(u)=K(u+2\eta)$, $U_{0,[1,L]}=R_{0L}(u+u_L)\cdots R_{01}(u+u_1)$ and $\bar U_{0,[1,L]}=R_{01}(u-u_1)\cdots R_{0L}(u-u_L)$, possesses, for arbitrary inhomogeneity parameters $u_1,\dots,u_L$, the eigenvalue

$$\Lambda_L=\mathrm{tr}\left(K^+(u)K^-(u)\right)\prod_{j=1}^{L}\left(a(u+u_j)+b(u+u_j)\right)\left(a(u-u_j)+b(u-u_j)\right).$$

The open content is the elliptic regime $0<p<1$: the trigonometric limit $p\to0$ of the conjectured eigenvalue has already been proven (Hagendorf-Liénardy 2021, via boundary quantum Knizhnik-Zamolodchikov equations), and the conjecture was verified numerically for small $L$ in that limit. A complete resolution is a rigorous proof covering the full elliptic parameter range; a disproof is an explicit instance $(L,p,u_1,\dots,u_L)$ with $0<p<1$ for which $\mathcal T$ fails to have the eigenvalue $\Lambda_L$. The closely analogous periodic-boundary conjecture of Razumov-Stroganov and Zinn-Justin is a separate open problem and is not part of this problem's completion boundary, though a method proving one would plausibly settle the other.

References

1. Yu. G. Stroganov, 'The importance of being odd', J. Phys. A: Math. Gen. 34 (2001) L179-L185. DOI: 10.1088/0305-4470/34/13/104; arXiv:cond-mat/0012035, https://arxiv.org/abs/cond-mat/0012035
2. A. V. Razumov and Yu. G. Stroganov, 'A possible combinatorial point for the XYZ spin chain', Theor. Math. Phys. 164 (2010) 977-991. DOI: 10.1007/s11232-010-0078-3; arXiv:0911.5030, https://arxiv.org/abs/0911.5030
3. P. Zinn-Justin, 'Sum rule for the eight-vertex model on its combinatorial line', in Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statistics, Springer, London (2013), pp. 599-637. DOI: 10.1007/978-1-4471-4863-0_26; arXiv:1202.4420, https://arxiv.org/abs/1202.4420
4. E. K. Sklyanin, 'Boundary conditions for integrable quantum systems', J. Phys. A: Math. Gen. 21 (1988) 2375-2389. DOI: 10.1088/0305-4470/21/10/015, https://doi.org/10.1088/0305-4470/21/10/015
5. C. Hagendorf and 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
6. C. Hagendorf and J. 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
7. C. Hagendorf and J. Liénardy, 'The open XXZ chain at $\Delta=-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations', J. Stat. Mech. (2021) 013104. DOI: 10.1088/1742-5468/abd028; arXiv:2008.03220, https://arxiv.org/abs/2008.03220
8. S. Brasseur and C. Hagendorf, 'Sum rules for the supersymmetric eight-vertex model', J. Stat. Mech. (2021) 023102. DOI: 10.1088/1742-5468/abda28; arXiv:2009.14077, https://arxiv.org/abs/2009.14077
9. C. Hagendorf and H. Rosengren, 'Nearest-neighbour correlation functions for the supersymmetric XYZ spin chain and Painlevé VI', Commun. Math. Phys. 405, article 97 (2024). DOI: 10.1007/s00220-024-04977-w; arXiv:2208.13533, https://arxiv.org/abs/2208.13533
10. J. Liénardy and C. W. Hagendorf, 'The open XXZ chain at $\Delta=-1/2$ and totally-symmetric alternating sign matrices', arXiv:2509.09209 (2025), https://arxiv.org/abs/2509.09209