Doubly degenerate eigenvalue of the inhomogeneous supersymmetric eight-vertex transfer matrix

The square-lattice eight-vertex model whose vertex weights $a,b,c,d$ satisfy the supersymmetry condition $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ is known as the supersymmetric eight-vertex model. For an odd number $L=2n+1$ of vertical lines with periodic boundary conditions, its homogeneous transfer matrix possesses the doubly degenerate eigenvalue $\Theta_n=(a+b)^{2n+1}$ — Stroganov's 2001 conjecture, proved by Hagendorf and Liénardy in 2018 using the lattice supersymmetry of a related XYZ spin chain. Razumov and Stroganov conjectured in 2010 that the transfer matrix of the inhomogeneous model, in which each vertical line carries its own spectral parameter while the weights locally satisfy the same condition, possesses a doubly degenerate eigenvalue given by a product of one-site factors that reduces to $(a+b)^{2n+1}$ in the homogeneous limit. Zinn-Justin reformulated this as a single conjecture about the solution of the inhomogeneous eigenvalue problem, and all known sum rules and component formulae for the corresponding eigenvectors are derived from it conditionally. The problem is to prove, or refute, this inhomogeneous eigenvalue conjecture.

Background

The eight-vertex model is a statistical-mechanical lattice model in which arrows live on the edges of the square lattice and each vertex carries one of the eight arrow configurations compatible with the rule that an even number of arrows point into each vertex; the four equivalence classes of configurations receive weights $a,b,c,d$. The model is encoded in the R-matrix, an operator $R: \mathbb{C}^2\otimes\mathbb{C}^2\to\mathbb{C}^2\otimes\mathbb{C}^2$ with matrix elements given by the weights. The row transfer matrix on a cylinder with $L$ vertical lines and periodic boundary conditions along the horizontal direction is $T=\mathrm{tr}_0(R_{0L}R_{0,L-1}\cdots R_{01})$, an operator on $(\mathbb{C}^2)^{\otimes L}$. In the inhomogeneous model each vertical line $j$ carries an inhomogeneity parameter $u_j$ and the weights at that line depend on the difference $u-u_j$, where $u$ is the spectral parameter. It is standard to write the weights in terms of Jacobi theta functions $\vartheta_1,\vartheta_4$ (odd and even theta functions with elliptic nome $p$): $a(u)=\rho\,\vartheta_4(2\eta,p^2)\vartheta_1(u+2\eta,p^2)\vartheta_4(u,p^2)$, $b(u)=\rho\,\vartheta_4(2\eta,p^2)\vartheta_4(u+2\eta,p^2)\vartheta_1(u,p^2)$, with analogous expressions for $c$ and $d$; here $\eta$ is the crossing parameter. The R-matrix satisfies the Yang–Baxter equation $R_{12}(u-v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u-v)$, the algebraic consistency condition of the lattice model, which makes transfer matrices with different spectral parameters commute, so they can be diagonalised simultaneously.

At the supersymmetric point the weights obey $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$, which corresponds to the crossing parameter $\eta=\pi/3$ in the theta-function parameterisation. Through Baxter's correspondence between the eight-vertex transfer matrix and the XYZ spin chain, this case is related to the XYZ Hamiltonian $H_{\mathrm{XYZ}}=-\frac12\sum_j(J_x\sigma_j^x\sigma_{j+1}^x+J_y\sigma_j^y\sigma_{j+1}^y+J_z\sigma_j^z\sigma_{j+1}^z)$ with $J_x=1+\zeta$, $J_y=1-\zeta$, $J_z=\frac12(\zeta^2-1)$ and $\zeta=cd/ab$, where $\sigma_j^a$ are Pauli matrices. Fendley and Hagendorf showed that this Hamiltonian has a dynamical lattice supersymmetry: on the subspace of alternate-cyclic states (a sector of the translation operator with alternating momentum) its restriction is, up to an additive constant, the anticommutator $\{Q,Q^\dagger\}$ of nilpotent supercharges $Q$ and $Q^\dagger$ that map between chains whose lengths differ by one. For odd $L=2n+1$ this implies the existence of a two-dimensional space of supersymmetry singlets — zero-energy ground states annihilated by both supercharges.

Stroganov conjectured in 2001 that, for odd $L=2n+1$, the transfer matrix of the homogeneous model possesses the remarkably simple doubly degenerate eigenvalue $\Theta_n=(a+b)^{2n+1}$ ('the importance of being odd'). Degeneracy here is necessarily even for odd $L$: the transfer matrix commutes both with spin reversal (the operator $\mathcal{R}=\sigma_1^x\sigma_2^x\cdots\sigma_L^x$ flipping all spins simultaneously) and with spin parity (the parity of the number of spins up), and these two symmetries anticommute with each other when $L$ is odd, which excludes non-degenerate eigenvalues. 'Doubly degenerate' thus means that the eigenspace is exactly two-dimensional, spanned by two vectors of opposite spin-reversal parity. Hagendorf and Liénardy proved this conjecture in 2018 by identifying the eigenspace of $\Theta_n$ with the space of supersymmetry singlets, using a commutation relation between the supercharge and the transfer matrix; for positive weights $\Theta_n$ is the largest eigenvalue.

The inhomogeneous generalisation is the present subject. Razumov and Stroganov argued from the inversion relations of the inhomogeneous transfer matrix at $\eta=\pm 2K/3$ (in their parameterisation, where $K$ is the complete elliptic integral; this is the same supersymmetric point as $\eta=\pi/3$ in the theta-function parameterisation above) that the transfer matrix $T(u|u_1,\dots,u_L)$ of the inhomogeneous model should possess the eigenvalue $\prod_{j=1}^{L}[a(u-u_j)+b(u-u_j)]$, which reduces to $(a+b)^L$ when all $u_j=0$; they supported this by analogy with the corresponding homogeneous eigenvalue relation, which they verified directly for small systems. In the normalisation used by Brasseur and Hagendorf the conjectured eigenvalue reads $\Theta_n(u|u_1,\dots,u_{2n+1})=\prod_{i=1}^{2n+1} r(u_i-u)$ with $r(u)=a(u)+b(u)=\vartheta_4(0,p^2)\vartheta_1(u+\eta,p^2)\vartheta_4(u+\eta,p^2)$ at $\eta=\pi/3$. Zinn-Justin reformulated the conjecture as an eigenvalue problem for the transfer matrix together with a spin-reversal constraint, and showed that a single conjecture about its solution — existence, up to normalisation, of a vector whose components are entire functions of each inhomogeneity $u_j$ with prescribed quasi-periodicity under $u_j\mapsto u_j+\pi$ and $u_j\mapsto u_j+2\pi\tau$ (where $\tau$ is the half-period ratio of the theta functions, a complex parameter with positive imaginary part, and $p=e^{\pi i\tau}$ is the elliptic nome) — implies a sum rule for the components of the homogeneous ground states $|\Psi_n\rangle$ and $|\bar\Psi_n\rangle$. All subsequent sum rules and explicit component formulae for the eigenspace, derived by Brasseur and Hagendorf via the elliptic Tsuchiya determinant (a determinantal expression for scalar products of the eight-vertex model), are proven only under this conjecture. The conjecture is therefore the single unproven ingredient on which the rigorous eigenvector and sum-rule programme for the supersymmetric eight-vertex model rests, and it is the natural inhomogeneous analogue of the homogeneous result proved in 2018.

Problem Statement

Prove, or refute, the following conjecture. Let $L=2n+1$ with $n>0$, so that $L$ is odd, and consider the inhomogeneous eight-vertex model on a cylinder with $L$ vertical lines and periodic boundary conditions along the horizontal direction, with vertex weights in the theta-function parameterisation at the supersymmetric crossing parameter $\eta=\pi/3$ (equivalently, weights locally satisfying $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ at every line), with inhomogeneity parameters $u_1,\dots,u_L$. Then the transfer matrix $T(u|u_1,\dots,u_L)$ possesses the doubly degenerate eigenvalue

$\Theta_n(u|u_1,\dots,u_L)=\prod_{j=1}^{L}\bigl[a(u-u_j)+b(u-u_j)\bigr],$

in the normalisation of Razumov and Stroganov; equivalently, in the normalisation of Brasseur and Hagendorf, $\Theta_n(u|u_1,\dots,u_{2n+1})=\prod_{i=1}^{2n+1} r(u_i-u)$ with $r(u)=a(u)+b(u)=\vartheta_4(0,p^2)\vartheta_1(u+\eta,p^2)\vartheta_4(u+\eta,p^2)$ and $\eta=\pi/3$. 'Doubly degenerate' means that the eigenspace is exactly two-dimensional, spanned by two vectors of opposite spin-reversal parity, and in the homogeneous limit $u_1=\cdots=u_L=0$ the eigenvalue reduces to the proven value $\Theta_n=(a+b)^{2n+1}$. The requested outcome is a single coherent result establishing the eigenvalue formula and its degeneracy; the sum rules and component properties are consequences that need not be reproved. Zinn-Justin's strengthened eigenvector characterisation of the inhomogeneous eigenvalue problem (Conjecture 2.1 of Brasseur–Hagendorf), described in the Background, is a strictly stronger related proposition whose resolution would imply the eigenvalue formula; it is not part of the requested result. The statement is uniform over all $n>0$ and all admissible non-zero weights, so a refutation may take the form of an explicit counterexample with concrete $n$ and weight parameters.

References

1. Yu. Stroganov, 'The importance of being odd', J. Phys. A: Math. Gen. 34 (2001) L179–L185. DOI: 10.1088/0305-4470/34/13/104, https://doi.org/10.1088/0305-4470/34/13/104 (preprint 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, https://doi.org/10.1007/s11232-010-0078-3 (preprint arXiv:0911.5030, https://arxiv.org/abs/0911.5030)
3. V. V. Mangazeev and V. V. Bazhanov, 'The eight-vertex model and Painlevé VI equation II: eigenvector results', J. Phys. A: Math. Theor. 43 (2010) 085206. DOI: 10.1088/1751-8113/43/8/085206, https://doi.org/10.1088/1751-8113/43/8/085206 (preprint arXiv:0912.2163, https://arxiv.org/abs/0912.2163)
4. P. Zinn-Justin, 'Sum Rule for the Eight-Vertex Model on Its Combinatorial Line', in K. Iohara, S. Morier-Genoud, B. Rémy (eds.), Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat. 40, Springer, London (2013), pp. 599–637. DOI: 10.1007/978-1-4471-4863-0_26, https://doi.org/10.1007/978-1-4471-4863-0_26 (preprint arXiv:1202.4420, https://arxiv.org/abs/1202.4420)
5. P. Fendley and C. Hagendorf, 'Exact and simple results for the XYZ and strongly interacting fermion chains', J. Phys. A: Math. Theor. 43 (2010) 402004. DOI: 10.1088/1751-8113/43/40/402004
6. C. Hagendorf and P. Fendley, 'The eight-vertex model and lattice supersymmetry', J. Stat. Phys. 146 (2012) 1122–1155. DOI: 10.1007/s10955-012-0430-0, https://doi.org/10.1007/s10955-012-0430-0
7. 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, https://doi.org/10.1088/1742-5468/aab01d (preprint arXiv:1711.04397, https://arxiv.org/abs/1711.04397)
8. 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, https://doi.org/10.1088/1742-5468/ab7748 (preprint arXiv:1911.03348, https://arxiv.org/abs/1911.03348)
9. S. Brasseur and C. Hagendorf, 'Sum rules for the supersymmetric eight-vertex model', J. Stat. Mech. (2021) 023102. DOI: 10.1088/1742-5468/abda28, https://doi.org/10.1088/1742-5468/abda28 (preprint arXiv:2009.14077, https://arxiv.org/abs/2009.14077)
10. C. Hagendorf and H. Rosengren, 'Nearest-Neighbour Correlation Functions for the Supersymmetric XYZ Spin Chain and Painlevé VI', Commun. Math. Phys. 405 (2024), article 97. DOI: 10.1007/s00220-024-04977-w, https://doi.org/10.1007/s00220-024-04977-w (preprint arXiv:2208.13533, https://arxiv.org/abs/2208.13533)
11. L. Hietala, 'A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials $p_n$', SIGMA 18 (2022) 036. DOI: 10.3842/SIGMA.2022.036, https://doi.org/10.3842/SIGMA.2022.036 (preprint arXiv:2104.04651, https://arxiv.org/abs/2104.04651)
12. R. Świętek, M. Kliczkowski, L. Vidmar and M. Rigol, 'Eigenstate entanglement entropy in the integrable spin-1/2 XYZ model', Phys. Rev. E 109 (2024) 024117. DOI: 10.1103/PhysRevE.109.024117, https://doi.org/10.1103/PhysRevE.109.024117 (preprint arXiv:2311.10819, https://arxiv.org/abs/2311.10819)