Integrable manifolds and commuting transfer matrices of the odd eight-vertex model

The odd eight-vertex model — a two-state vertex model on the square lattice whose allowed vertices each carry an odd number of incoming and outgoing arrows — was introduced by Wu and Kunz, who computed its thermodynamic-limit free energy on the free-fermion manifold but left its finite-lattice integrability uncharacterized. Martins later showed that with arrow-inversion-symmetric weights the odd model is integrable on the same elliptic curve as Baxter's even eight-vertex model and uncovered a sheaf of four R-matrices (even-even, odd-odd, and mixed even-odd) satisfying Yang-Baxter equations that resemble intertwiner relations of equidimensional representations. The paper's concluding remarks explicitly pose two open questions that complete this integrability picture: whether the asymmetric odd model, in particular under the free-fermion condition, admits any integrable manifold at finite lattice size; and a proof for arbitrary lattice size that the staggered transfer-matrix products $T_1(v)T_2(v)$ commute for distinct weight values on the even-model free-fermion manifolds supplemented by Krinsky's three restrictions (verified numerically only for one and two sites). No later contribution resolving either question is known.

Background

A two-state vertex model on the square lattice places an arrow (equivalently a two-valued spin) on every edge, and assigns an energy weight to each allowed arrow configuration around a vertex. The sixteen possible configurations split into two families of eight: the even eight-vertex model, in which an even number of arrows point into each vertex, and the odd eight-vertex model, in which the number of incoming (equivalently outgoing) arrows is odd. The even model, with weights $w_1,\dots,w_8$, contains Baxter's celebrated eight-vertex model: when the weights are symmetric under inversion of all arrows at a vertex ($w_1=w_2=a$, $w_3=w_4=b$, $w_5=w_6=c$, $w_7=w_8=d$), Baxter showed the transfer matrices commute — the operational meaning of finite-size integrability here — provided the weights lie on the intersection of quadrics $(ab-cd)/(ab+cd)=\Gamma$, $(a^2+b^2-c^2-d^2)/(2(ab+cd))=\Delta$, an algebraic curve parametrized by Jacobi theta functions. The transfer matrix is built from a Lax operator, a $4\times 4$ matrix acting on an auxiliary and a quantum space, and commutation of transfer matrices across the lattice follows from a Yang–Baxter equation satisfied by an R-matrix intertwining two Lax operators. For generic (asymmetric) even weights, Fan and Wu showed the model is exactly solvable in the thermodynamic limit under the free-fermion condition $w_1w_2+w_3w_4-w_5w_6-w_7w_8=0$, whose partition function reduces to a Pfaffian (free-fermion) computation; Krinsky then found that commuting transfer matrices at finite lattice size require three extra algebraic restrictions on the weights beyond the free-fermion condition, namely that the invariants $\Delta_1=w_6w_8/(w_5w_7)$, $\Delta_2=(w_1w_4+w_2w_3)/(w_5w_7)$ and $\Delta_3=(w_1^2+w_4^2-w_2^2-w_3^2)/(w_5w_7)$ take fixed values (Eq. (23) of arXiv:1711.03131). The odd model, with weights $v_1,\dots,v_8$ and encompassing the classical Ashkin-Teller model (a lattice model of two Ising spin chains coupled on the same sites), was named and studied by Wu and Kunz: they showed its partition function equals that of a staggered (checkerboard-alternating) even eight-vertex model, and computed its free energy in the thermodynamic limit under the analogous free-fermion condition $v_1v_2+v_3v_4-v_5v_6-v_7v_8=0$ — but the existence of commuting transfer matrices at finite lattice size remained unaddressed. In 2018 Martins closed part of this gap for symmetric weights: the symmetric odd model is integrable on exactly the same elliptic curve as the even model, although its R-matrix takes a form different from its Lax operator, and the even and odd models together generate a sheaf of four R-matrices — $R(\mathrm{ev},\mathrm{ev})$, $R(\mathrm{od},\mathrm{od})$, $R(\mathrm{ev},\mathrm{od})$, $R(\mathrm{od},\mathrm{ev})$ — satisfying a closed family of Yang–Baxter relations on that curve, structurally similar to intertwiner relations — an intertwiner being a linear map between two representations of an algebra that commutes with the algebra's action — between distinct equidimensional representations of a single algebra. The staggering equivalence also converts the even model into an alternating odd model with partition function $Z=\mathrm{Tr}[(T_1T_2)^N]$ built from two alternating transfer matrices $T_1,T_2$, and numerical checks at $N=1,2$ suggest that the products $T_1(v)T_2(v)$ commute for distinct weight values $v$ on the even-model manifolds defined by the free-fermion condition together with Krinsky's three restrictions — not on Baxter's symmetric elliptic curve, where commutativity is already proven — even though $T_1$ and $T_2$ do not commute individually. What remains open, as explicitly stated in that paper's concluding remarks, is the completion of this picture for asymmetric weights: whether the asymmetric odd model admits any finite-size integrable manifold, and an algebraic proof for arbitrary lattice size of the suggested commutation of the staggered products.

Problem Statement

Complete the finite-lattice integrability characterization of the odd eight-vertex model, the problem left open by Martins (2018) after his solution of the arrow-symmetric case. The common objective uniting the requested results is to determine when the odd eight-vertex model with asymmetric weights possesses commuting transfer matrices or an equivalent Yang–Baxter structure at finite lattice size. Concretely: (1) determine whether the asymmetric odd eight-vertex model whose weights satisfy the free-fermion condition $v_1v_2+v_3v_4-v_5v_6-v_7v_8=0$ admits any integrable manifold of weights at finite lattice size, that is, a manifold on which the transfer matrices of the odd model itself commute pairwise for distinct weight values (asymmetric here meaning that the weights are not invariant under arrow inversion, so the arrow-inversion-symmetric manifold on which the odd model is already known to be integrable does not count) — the Krinsky-type constraints with $w_i\mapsto v_i$ are already excluded as such a manifold by small-size commutator computations on those transfer matrices, so a positive answer requires a genuinely different manifold, and a negative answer proves that this is a model exactly solvable in the thermodynamic limit without finite-size commutativity; this is a question about the odd model's own transfer matrices, not the products $T_1T_2$ of the two staggered transfer matrices treated in (2): the manifold excluded in (1) is the manifold on which (2) asks for those products to commute, so the two questions concern different operators on the same manifold of weights and a proof in (2) would not by itself settle (1); and (2) prove algebraically, for arbitrary lattice size $N$, that the staggered transfer-matrix products commute — that is, $[T_1(v)T_2(v),\,T_1(v')T_2(v')]=0$ for distinct weight values $v\neq v'$ — when the weights lie on the even-model manifolds, i.e. the free-fermion condition $w_1w_2+w_3w_4-w_5w_6-w_7w_8=0$ together with Krinsky's three restrictions fixing the invariants $\Delta_1=w_6w_8/(w_5w_7)$, $\Delta_2=(w_1w_4+w_2w_3)/(w_5w_7)$, $\Delta_3=(w_1^2+w_4^2-w_2^2-w_3^2)/(w_5w_7)$ (Eqs. (22,23) of arXiv:1711.03131) with $w_i=v_i$, where $T_1,T_2$ are the alternating transfer matrices built from the two staggered Lax operators of Eq. (26) of the source and do not commute individually; this commutation is suggested by the numerical verification for $N=1,2$ with two distinct random weight sets. The sheaf of four R-matrices on the symmetric elliptic curve and its resemblance to intertwiner relations of equidimensional representations of a single algebra motivate a structural interpretation of these results, but the source's concluding remarks pose (1) and (2) as the open questions.

References

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