Ground state of the odd-length XYZ spin chain at the combinatorial point: the inhomogeneous eigenvalue conjecture and the polynomial-combinatorial structure of its components

At the combinatorial (supersymmetric) point of the periodic eight-vertex model - the XYZ spin chain of odd length N with couplings $J_x=1+\alpha$, $J_y=1-\alpha$, $J_z=(\alpha^2-1)/2$ on the line $J_xJ_y+J_yJ_z+J_zJ_x=0$ - the transfer matrix is now known (Hagendorf and Liénardy, 2018) to possess the doubly degenerate eigenvalue $[a(v)+b(v)]^N$ whose eigenspace is the ground-state space of the chain. What remains open, and what the 2009 conjectures of Razumov and Stroganov target, is the explicit structure of this ground state: the inhomogeneous transfer-matrix eigenvalue and its theta-function eigenvector; the polynomiality of the components in $\alpha$ with positive integer coefficients, with degree $(N^2-1)/8$ for the all-down component; the linear and quadratic sum rules, together with the Bazhanov-Mangazeev Painlevé VI tau-function formula for the squared norm; and the identification of extreme coefficients with squares of vertically-symmetric alternating-sign-matrix and cyclically-symmetric transpose-complement plane-partition enumeration numbers. The problem asks to prove or refute these conjectures, with the anticipated route being the missing analytic solution of Baxter's TQ-equation at the combinatorial point, where the standard Q-operator construction degenerates.

Background

The eight-vertex model of Baxter is a two-dimensional lattice model in which arrows are placed on the edges of a square lattice, a Boltzmann weight being assigned to each of the eight allowed vertex configurations; the four weights are conventionally denoted $a$, $b$, $c$, $d$. Its row-to-row transfer matrix $T(v)$ depends on a spectral parameter $v$, and the family $\{T(v)\}$ commutes with itself. Sutherland and Baxter showed that the Hamiltonian of the one-dimensional anisotropic spin-1/2 chain (the XYZ chain)

$$H_{XYZ}=-\frac{1}{2}\sum_{j=1}^{N}\left(J_x\sigma_j^x\sigma_{j+1}^x+J_y\sigma_j^y\sigma_{j+1}^y+J_z\sigma_j^z\sigma_{j+1}^z\right)$$

(with Pauli matrices $\sigma_j^{x,y,z}$ and periodic boundary conditions) is proportional to the logarithmic derivative of $T(v)$ at a special value of $v$, so that eigenvectors of $T(v)$ are eigenvectors of the chain. Unlike the six-vertex/XXZ case, the eight-vertex model cannot be solved by the coordinate Bethe ansatz; Baxter instead introduced the functional TQ-relation, in which an auxiliary object $Q(u)$ intertwines with $T(u)$, and used it to compute the partition function.

On the line of couplings

$$J_xJ_y+J_yJ_z+J_zJ_x=0,$$

which in eight-vertex language corresponds to the crossing parameter taking the values $\eta=\pm 2K/3$ or $\eta=\pm 4K/3$ ($K$ the complete elliptic integral of the first kind; the crossing parameter $\eta$ is the eight-vertex analogue of the chain anisotropy, entering the Boltzmann weights as a shift of the spectral argument), equivalently to the vertex-weight constraint $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ - often called the combinatorial line, or the supersymmetric eight-vertex model - the couplings are parameterized by a single real quantity: $J_x=1+\alpha$, $J_y=1-\alpha$, $J_z=(\alpha^2-1)/2$, with $0\le\alpha\le 1$. Baxter observed that on this line the ground-state energy per site has the simple value $-(J_x+J_y+J_z)/2$ in the thermodynamic limit, and that the inversion relations for the transfer-matrix eigenvalues admit the simple solution $T(v)=[a(v)+b(v)]^N$ even for finite chains. Stroganov found by exact diagonalization that the corresponding eigenvector exists precisely for odd chain lengths, and conjectured that in that case it is the (doubly degenerate) ground state. Fendley and Hagendorf later showed that the XYZ chain on this line carries a non-local lattice supersymmetry: a nilpotent supercharge whose anticommutator with its adjoint reproduces the Hamiltonian on a subsector, which explains the special spectral structure.

The case $\alpha=0$ is the XXZ chain at $\Delta=-1/2$, where the ground-state components are positive integers related to the enumeration of alternating-sign matrices (ASMs) - the $(0,\pm 1)$-matrices in which the nonzero entries alternate in sign along each row and each column, with all row and column sums equal to 1. That connection (the Razumov-Stroganov conjecture for the related loop model, proved by Cantini and Sportiello) is one of the celebrated links between integrable models and enumerative combinatorics. Razumov and Stroganov (2010) diagonalized $H(\alpha)$ exactly for all odd $N\le 19$ and found that at general $\alpha$ the ground-state components can be normalized to be polynomials in $\alpha$ with positive integer coefficients. They conjectured: the degree of the all-down component is $D_N=(N^2-1)/8$; its coefficient of $\alpha^m$ equals the square of the number $A_V(2m+1)$ of vertically-symmetric ASMs of order $2m+1$ - ASMs invariant under reflection about the vertical axis - (for $N=4m+1$), or the square of the number $N_8(2m)$ of cyclically-symmetric transpose-complement plane partitions in a $2m\times 2m\times 2m$ box (a plane partition is a pile of unit cubes inside a box, weakly decreasing along each coordinate axis; this is one of its symmetry classes; for $N=4m-1$) - both known product formulas, due to Kuperberg and to Mills, Robbins and Rumsey; the linear and quadratic sum rules $S_1(\alpha)=\sum_\mu\Psi_\mu(\alpha)$ and $S_2(\alpha)=\sum_\mu\Psi_\mu(\alpha)^2$ obey a divisibility $S_2/S_1=F(\alpha)/(\alpha+3)^m$ and exact identities under the fractional-linear map $\alpha\mapsto(\alpha+3)/(\alpha-1)$; and the whole structure should follow from an inhomogeneous transfer-matrix eigenvalue $\prod_j[a(v-v_j)+b(v-v_j)]$. Independently, Bazhanov and Mangazeev (2010) conjectured that the squared norm and special components are expressed through polynomials $s_n$, $\bar s_n$ defined by differential recurrences that identify them with tau-functions of the Painlevé VI equation (Painlevé VI is the sixth of the classical second-order nonlinear ordinary differential equations with the Painlevé property; its tau-functions are auxiliary functions from which the solutions are recovered by logarithmic derivatives; here built from Picard's algebraic solutions).

In 2018 Hagendorf and Liénardy proved the existence part: for every odd $N\ge 3$ and non-zero weights on the combinatorial line, $[a(v)+b(v)]^N$ is a doubly degenerate transfer-matrix eigenvalue whose eigenspace is spanned by the doubly degenerate ground states of $H(\alpha)$, of energy $-N(3+\alpha^2)/4$. What is still missing, and what this problem targets, is the explicit structure of that ground state: the inhomogeneous eigenvalue and its theta-function eigenvector, the polynomiality and positivity of the components, the sum rules, and the combinatorial identifications - in other words, the eight-vertex analogue of the six-vertex Razumov-Stroganov correspondence, and an eigenvector-level solution of Baxter's TQ-equation at $\eta=\pm 2K/3$, where the standard Q-operator construction degenerates.

Problem Statement

For the periodic XYZ spin-1/2 chain of odd length $N=2n+1$ on the combinatorial (supersymmetric) line $J_xJ_y+J_yJ_z+J_zJ_x=0$, written $J_x=1+\alpha$, $J_y=1-\alpha$, $J_z=(\alpha^2-1)/2$ (equivalently, the eight-vertex model with crossing parameter $\eta=\pm 2K/3$ or $\pm 4K/3$, or with vertex weights obeying $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$), the ground state is known to be doubly degenerate, of energy $-N(3+\alpha^2)/4$. The Hamiltonian commutes both with the parity of the number of up spins and with total spin reversal $I=\sigma_1^x\sigma_2^x\cdots\sigma_N^x$; for odd $N$ the two ground states lie in opposite parity sectors and are exchanged by $I$. Write $|\Psi(\alpha)\rangle$ for the ground state in the sector containing the all-down state (the sector with an even number of up spins), $|\bar\Psi(\alpha)\rangle=I|\Psi(\alpha)\rangle$ for the other, and $\Psi_{\mu_1\dots\mu_N}(\alpha)$, $\mu_j=\pm1$, for the components of $|\Psi(\alpha)\rangle$ in the basis of joint $\sigma_j^z$ eigenvectors ($\mu_j=-1$: spin down at site $j$); $\Psi_{-\cdots-}(\alpha)$ is the all-down component. Prove or refute the following conjectured explicit structure:

1. Inhomogeneous eigenvalue (Razumov-Stroganov Conjecture 3.3, in the sharper form of Zinn-Justin's conjecture): for every odd $N$, the transfer matrix $T(v\,|\,v_1,\dots,v_N)$ of the inhomogeneous eight-vertex model (the spectral parameter of the Boltzmann weights being shifted by site-dependent inhomogeneities $v_j$) has the doubly degenerate eigenvalue

$$\Theta(v)=\prod_{j=1}^{N}\bigl[a(v-v_j)+b(v-v_j)\bigr],$$

and the corresponding eigenvector (whose homogeneous limit $v_1=\dots=v_N$ lies in the two-dimensional ground-state space above) can be chosen so that its components $\Psi_\mu(v_1,\dots,v_N)$ are theta functions of degree $N-1$ and nome $p^2$ in each inhomogeneity $v_j$; that is, each component is holomorphic in each $v_j$ (generically non-vanishing, with no common factor) and obeys, for every $i$, the quasi-periodicity relations

$$\Psi_\mu(\dots,v_i+\pi,\dots)=\Bigl(\prod_{j\ne i}\mu_j\Bigr)\Psi_\mu(\dots,v_i,\dots),\qquad \Psi_\mu(\dots,v_i+2\pi\tau,\dots)=p^{-4n}z_i^{\,2n}\Bigl(\prod_{j\ne i}z_j^{-1}\Bigr)\Psi_\mu(\dots,v_i,\dots),$$

where $p=e^{i\pi\tau}$ and $z_j=e^{-2iv_j}$ (Zinn-Justin's conventions). Convention note: in the literature stating this conjecture the inhomogeneous eigenvector is fixed by diagonalizing the spin reversal ($F_*\Psi=(-1)^n\Psi$ with $F_*=\sigma_1^x\cdots\sigma_N^x=I$, in Zinn-Justin's notation; Brasseur and Hagendorf add an explicit normalization) rather than by the up-spin-parity convention used above for $|\Psi(\alpha)\rangle$; for odd $N$ the up-spin parity and the spin reversal anticommute, so no non-zero vector satisfies both conditions. The identification of the homogeneous-limit eigenvector's components, up to normalization, with those of the parity-sector vector $|\Psi(\alpha)\rangle$ used in items 2-4 (equivalently, with the normalized Razumov-Stroganov and Bazhanov-Mangazeev vectors) is not rigorously established in the literature and is part of what must be established. 2. Polynomial structure (Conjectures 4.2 and 4.3): in the normalization in which the all-down component is monic, $\Psi_{-\cdots-}(\alpha)=\alpha^{D_N}+\dots$, every component $\Psi_{\mu_1\dots\mu_N}(\alpha)$ is a polynomial in $\alpha$ with positive integer coefficients, and $D_N=(N^2-1)/8$. 3. Sum rules (Conjectures 5.1 and 5.2 with corollaries, together with the Bazhanov-Mangazeev norm conjecture): with $S_1(\alpha)=\sum_\mu\Psi_\mu(\alpha)$ and $S_2(\alpha)=\sum_\mu\Psi_\mu(\alpha)^2$ (the squared norm of $|\Psi(\alpha)\rangle$), one has for $N=4m\pm1$ a divisibility $S_2(\alpha)/S_1(\alpha)=F(\alpha)/(\alpha+3)^m$ with $F$ a polynomial; the identities $S_1(\alpha)=2^{-(N-3)(N-1)/8}(\alpha-1)^{D_N}\Psi_{-\cdots-}(\alpha')$ and $S_1(\alpha')=2^{(N-1)(N+5)/8}(\alpha-1)^{-D_N}\Psi_{-\cdots-}(\alpha)$ with $\alpha'=(\alpha+3)/(\alpha-1)$; and the Bazhanov-Mangazeev norm formula

$$S_2(\alpha)=\Bigl(\frac{4}{3}\Bigr)^{\!n}\alpha^{n(n+1)}\,s_n(\alpha^{-2})\,s_{-n-1}(\alpha^{-2}),$$

where the polynomials $s_n(z)$, $n\in\mathbb{Z}$, are defined by the differential recurrence

$$2z(z-1)(9z-1)^2\,\partial_z^2\log s_n(z)+2(3z-1)^2(9z-1)\,\partial_z\log s_n(z)+8(2n+1)^2\frac{s_{n+1}(z)s_{n-1}(z)}{s_n(z)^2}-\bigl[4(3n+1)(3n+2)+(9z-1)\,n(5n+3)\bigr]=0$$

with $s_0(z)=s_1(z)=1$ (this recurrence is what identifies the $s_n$ with tau-functions of the Painlevé VI equation constructed from Picard's algebraic solutions). 4. Combinatorial identifications (Conjecture 4.4): under the normalization of item 2, for $N=4m+1$ the expansion of the all-down component is $\Psi_{-\cdots-}(\alpha)=\alpha^{m(2m+1)}+\dots+A_V(2m+1)^2\alpha^m$, where $A_V(2m+1)$ is the number of vertically-symmetric (invariant under reflection about the vertical axis) alternating-sign matrices of order $2m+1$; for $N=4m-1$ it is $\Psi_{-\cdots-}(\alpha)=\alpha^{m(2m-1)}+\dots+N_8(2m)^2\alpha^m$, where $N_8(2m)$ is the number of cyclically-symmetric transpose-complement plane partitions in a $2m\times 2m\times 2m$ box; in both cases the coefficient of the lowest-order term $\alpha^m$ is the square of the corresponding enumeration number, and both $A_V(2m+1)$ and $N_8(2m)$ are given by explicit product formulas.

These four items are linked parts of one objective, the explicit solution of the ground state at the combinatorial point. In the existing literature, parts of items 3 and 4 (the Bazhanov-Mangazeev norm factorization, the fractional-linear sum-rule identity and the extreme-coefficient identifications) have been derived under the assumption of Zinn-Justin's theta-function conjecture, i.e. item 1; however, even granting item 1, the identification of the inhomogeneous eigenvector with the Razumov-Stroganov and Bazhanov-Mangazeev ground-state vectors remains non-rigorous, and the polynomiality and positivity of all components (item 2) remain open. Producing the missing analytic solution of Baxter's TQ-equations at $\eta=\pm 2K/3$ - equivalently, a Q-operator analytic at this point, where Baxter's standard construction degenerates - is a route suggested in the literature, but items 2-4 are not known to follow from it. A refutation of any of items 1-4 by an explicit counterexample resolves the corresponding part.

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