Commuting transfer matrices for the asymmetric and staggered odd eight-vertex model

The odd eight-vertex model — the square-lattice vertex model whose allowed configurations are the eight arrow patterns with an odd number of arrows pointing into each vertex — has an exactly computable free energy whenever its weights satisfy the free-fermion condition, by Wu and Kunz's map to a staggered even eight-vertex model solved with Pfaffians. Yet no Yang–Baxter structure is known for it outside the symmetric-weight subspace, where Martins (2018) exhibited a family of R-matrices on Baxter's elliptic curve. The integrability questions left open are: (1) whether any algebraic manifold of asymmetric free-fermion weights, outside the symmetric-weight subspace already solved by Martins and other than the naive transplant of Krinsky's even-model constraints (which two-site commutator computations already exclude), yields commuting transfer matrices for finite lattices; and (2) an algebraic proof, for arbitrary lattice size, that in the staggered odd model with weights on the free-fermion and Krinsky manifolds the products $T_1T_2$ of the two alternating transfer matrices form a commuting family — a confirmation that may be pursued through the Yang–Baxter equations for the tensor product of the two distinct Lax operators.

Background

In a two-dimensional vertex model each edge of the square lattice carries an arrow (a two-valued spin), and the statistical weight of a lattice face (a "vertex") depends only on the arrow configuration on its four incident edges. Sixteen configurations are possible. The classical eight-vertex model keeps the eight configurations with an even number of arrows pointing into the vertex; Baxter solved it for symmetric weights (configurations related by reversing all four arrows share a weight) on a two-parameter algebraic manifold — an intersection of quadrics — where the row-to-row transfer matrices commute. The odd eight-vertex model introduced by Wu and Kunz instead keeps the complementary eight configurations, those with an odd number of arrows pointing into the vertex, with weights $v_1,\dots,v_8$.

The transfer matrix $T$ of a periodic lattice row is the trace, over an auxiliary two-dimensional space, of an ordered product of Lax operators $L_{0j}$: $4\times4$ matrices acting on the auxiliary space and the two-dimensional quantum space of site $j$, whose nonzero entries are the vertex weights. Integrability in the Yang–Baxter sense means that the transfer matrices form a commuting family, $[T(\omega'),T(\omega'')]=0$ for any two allowed weight sets; this follows from an intertwiner (R-matrix) satisfying the Yang–Baxter equation $R_{12}(\omega',\omega'')L_{13}(\omega')L_{23}(\omega'')=L_{23}(\omega'')L_{13}(\omega')R_{12}(\omega',\omega'')$, and it unlocks exact finite-size spectra and Bethe-ansatz methods (an exact diagonalization technique for such transfer matrices).

When the weights satisfy the free-fermion condition

$$w_1w_2+w_3w_4-w_5w_6-w_7w_8=0$$

the even eight-vertex model is solvable by Pfaffian methods (Fan and Wu) — the partition function reduces to a Pfaffian, the alternating-sign analogue of a determinant, for free-fermion (Gaussian) weights — so the thermodynamic-limit free energy is exactly computable, but the transfer matrices need not commute. Krinsky observed that, within the free-fermion class, commuting transfer matrices for finite lattices require three further algebraic constraints on the weights; adapted by Martins to eight arbitrary weights (his Eqs. (22), (23)) these read

$$\frac{w_6w_8}{w_5w_7}=\Delta_1,\qquad \frac{w_1w_4+w_2w_3}{w_5w_7}=\Delta_2,\qquad \frac{w_1^2+w_4^2-w_2^2-w_3^2}{w_5w_7}=\Delta_3,$$

with free parameters $\Delta_1,\Delta_2,\Delta_3$. Hsue, Lin and Wu extended Pfaffian solvability to staggered (two-sublattice) eight-vertex models. Wu and Kunz then showed that the odd model with free-fermion weights maps onto a staggered even model of exactly this type, so its free energy is also exactly computable; the correspondence identifies the odd model with weights $v_1,\dots,v_8$ with the staggered even model whose two sublattice weight sets are $\omega_X=(v_1,\dots,v_8)$ and $\omega_Y=(v_3,v_4,v_1,v_2,v_8,v_7,v_6,v_5)$, and conversely.

For symmetric weights the odd model was settled by Martins: the odd and even symmetric Lax operators are related by a weight-independent local transformation built from Pauli matrices, so both models have commuting transfer matrices on Baxter's manifold, and the corresponding R-matrix — first found numerically from a general $16\times16$ ansatz, then in closed form — has a structure distinct from that of the Lax operator; altogether four types of R-matrices satisfy Yang–Baxter relations on the same elliptic curve — Baxter's two-parameter manifold, a genus-one algebraic curve — with the even/odd choice playing the role of a representation index. This settles integrability only on the symmetric-weight subspace.

Beyond the symmetric case almost nothing is known. Assis surveyed the sixteen-vertex model and its even and odd eight-vertex subcases using weak-graph (duality) transformations, found no new integrable points, and gave instead a Pfaffian-type finite-lattice solution of the odd free-fermion model — solvability without commuting transfer matrices. Melotti constructed explicit checkerboard (staggered) free-fermion solutions $R$ and $\bar{R}$ of a staggered Yang–Baxter equation with elliptic weights and Z-invariance (invariance under star-triangle rearrangements of the lattice), recovering Baxter's free-fermion checkerboard solution; but his weights are zero-field (complementary configurations have equal weight), he states that no complete parametrization of checkerboard Yang–Baxter solutions is known, and his framework does not reach the asymmetric odd-model manifolds at issue here.

For the staggered question, the odd model on a $2N\times2N$ lattice with sublattice weights $\omega_X$ and $\omega_Y$ as above has partition function $Z=\mathrm{Tr}[(T_1T_2)^N]$, where $T_1$ and $T_2$ alternate two distinct Lax operators encoding $\omega_X$ and $\omega_Y$:

$$L(v)=\begin{pmatrix}0&v_1&v_7&0\\ v_3&0&0&v_6\\ v_5&0&0&v_4\\ 0&v_8&v_2&0\end{pmatrix},\qquad \mathbf{L}(v)=\begin{pmatrix}0&v_3&v_6&0\\ v_1&0&0&v_7\\ v_8&0&0&v_2\\ 0&v_5&v_4&0\end{pmatrix}.$$

Numerical tests for $N=1,2$ indicate that when the weights lie on the manifolds (22) and (23) — the free-fermion condition plus the Krinsky constraints — the products $T_1T_2$ form a commuting family, even though $T_1$ and $T_2$ individually do not commute. The phenomenon of commuting products of non-commuting transfer matrices is what the second question below asks to establish.

Problem Statement

The overarching objective is to establish whether the odd eight-vertex model admits Yang-Baxter integrability — commuting transfer matrices on an explicit algebraic manifold of weights — outside the symmetric-weight subspace already solved by Martins. Two concrete questions are open, and both are required for a full resolution: question (1) addresses the overarching objective directly and may be resolved either by exhibiting such a manifold or by proving that none exists; question (2) is an additional required deliverable for the staggered model.

(1) Free-fermion manifold for the asymmetric odd model. Let $v_1,\dots,v_8$ be unrestricted (asymmetric) weights of the odd eight-vertex model satisfying the free-fermion condition

$$v_1v_2+v_3v_4-v_5v_6-v_7v_8=0.$$

Does there exist a positive-dimensional algebraic manifold of these weights, not contained in the symmetric-weight subspace already solved by Martins, on which the transfer matrices commute for finite lattice sizes but are not merely proportional to one another (in particular, the manifold must not be an overall-rescaling orbit of a single weight set)? The manifold obtained by transplanting Krinsky's even-model constraints with $w_i$ replaced by $v_i$, namely

$$\frac{v_6v_8}{v_5v_7}=\Delta_1,\qquad \frac{v_1v_4+v_2v_3}{v_5v_7}=\Delta_2,\qquad \frac{v_1^2+v_4^2-v_2^2-v_3^2}{v_5v_7}=\Delta_3,$$

is excluded: direct computation of transfer-matrix commutators already for two sites rules it out. The question is whether any other constraint set works; a negative answer would exhibit a natural model whose thermodynamic free energy is exactly computable while no commuting finite-size transfer matrices exist.

(2) Commuting product family in the staggered odd model. Consider the staggered odd eight-vertex model on a $2N\times2N$ lattice with sublattice weights $\omega_X=(v_1,\dots,v_8)$ and $\omega_Y=(v_3,v_4,v_1,v_2,v_8,v_7,v_6,v_5)$, where the $v_i$ satisfy both the free-fermion condition and the Krinsky constraints above (the manifolds (22),(23) of the source). Its partition function is $Z=\mathrm{Tr}[(T_1T_2)^N]$, where $T_1$ and $T_2$ are the alternating products of the two distinct Lax operators $L$ and $\mathbf{L}$. Prove algebraically, and for arbitrary lattice size $N$, that for any two weight sets on the same manifold (22),(23) — that is, with equal $\Delta_1,\Delta_2,\Delta_3$ — the operators $T_1T_2$ commute among themselves, as verified numerically only for $N=1,2$. The source suggests pursuing this confirmation through the respective Yang-Baxter (intertwiner) equations for the tensor product of the two types of Lax operators. Any proof must respect the fact, stressed in the source, that $T_1$ and $T_2$ individually do not commute on these manifolds.

References

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