Integrability of the odd eight-vertex model with asymmetric weights: free-fermion manifolds and commutation of staggered transfer-matrix products

The odd eight-vertex model, in which each vertex of the square lattice carries one of the eight arrow configurations with an odd number of incoming arrows, was introduced by Wu and Kunz, who showed it equivalent to a staggered even eight-vertex model and solved it in the thermodynamic limit at the free-fermion condition. Martins (J. Phys. A 51, 225002 (2018)) proved that with arrow-inversion-symmetric weights the odd model is integrable exactly on Baxter's elliptic curve, and uncovered a sheaf of four even/odd R-matrices satisfying a family of Yang-Baxter relations. The concluding remarks of that paper pose two concrete problems that remain unanswered: (1) whether the asymmetric odd eight-vertex model with free-fermion weights admits any finite-lattice algebraic manifold with commuting transfer matrices — the naive substitution of Krinsky's even-model constraints is excluded by two-site commutator computations, but other manifolds are not — or whether the model is exactly solvable in the thermodynamic limit while lacking finite-size commuting transfer matrices; and (2) whether the products of the two staggered transfer matrices commute for arbitrary lattice size for weights on the free-fermion and Krinsky manifolds, verified numerically only for lattices of one and two sites.

Background

A two-state vertex model on the square lattice assigns arrows to edges and a Boltzmann weight to each allowed arrow configuration around a vertex. The sixteen possible arrow configurations split into two families of eight: the standard (even) eight-vertex model of Baxter, in which each vertex has an even number of arrows pointing in, and the odd eight-vertex model, named by Wu and Kunz, in which each vertex has an odd number of incoming arrows. The row-to-row transfer matrix is the product of local vertex (Lax) operators along a row; a one-parameter family of commuting transfer matrices is the algebraic hallmark of integrability, since it yields an extensive set of conserved quantities and, through an R-matrix satisfying the Yang-Baxter equation (an intertwining relation between Lax operators at different spectral parameters), exact finite-size spectra. Baxter solved the symmetric even model, whose weights are invariant under inverting all arrows at a vertex ($w_1=w_2=a$, $w_3=w_4=b$, $w_5=w_6=c$, $w_7=w_8=d$), by showing that a commuting family exists when the weights lie on the intersection of quadrics $(ab-cd)/(ab+cd)=\Gamma$, $(a^2+b^2-c^2-d^2)/(2(ab+cd))=\Delta$, parametrized by theta functions of a spectral variable. For asymmetric even weights the Yang-Baxter equations were later derived by Galleas and Martins. Separately, the free-fermion condition $w_1w_2+w_3w_4-w_5w_6-w_7w_8=0$ makes the thermodynamic free energy computable (Fan and Wu); Krinsky found that commuting transfer matrices for the free-fermion even model require three further constraints, $w_6w_8/(w_5w_7)=\Delta_1$, $(w_1w_4+w_2w_3)/(w_5w_7)=\Delta_2$ and $(w_1^2+w_4^2-w_2^2-w_3^2)/(w_5w_7)=\Delta_3$. Wu and Kunz proved that the odd model on an $N\times N$ torus (its partition function vanishes for odd $N$) is equivalent to a staggered even model on a $2N\times 2N$ lattice with alternating sublattice weights $\omega^X=(v_1,\dots,v_8)$ and $\omega^Y=(v_3,v_4,v_1,v_2,v_8,v_7,v_6,v_5)$, and computed its free energy at the free-fermion condition via the Pfaffian (antisymmetric-determinant) solution of the staggered model (Hsue, Lin and Wu). Assis later enlarged the set of mappings among the sixteen-vertex model and its even and odd eight-vertex subcases. Martins then showed, for symmetric odd weights, that the odd model is integrable on exactly the same Baxter curve as the even model, that its R-matrix differs from its Lax operator, and that the two models jointly produce a sheaf of four R-matrices $R^{(\alpha,\beta)}$, $\alpha,\beta\in\{\mathrm{ev},\mathrm{od}\}$, with theta-function entries satisfying a family of Yang-Baxter relations (the exchange ev$\leftrightarrow$od being the weight swap $a\leftrightarrow c$, $b\leftrightarrow d$), resembling intertwiners of equidimensional representations of a single, unidentified algebra. The concluding remarks of that paper single out two problems for the asymmetric odd model, which this record states: the free-fermion manifold decision problem and the staggered commutation problem. A full solution would naturally also illuminate the broader classification of integrable manifolds of the asymmetric odd model and the representation-theoretic identification of the even/odd R-matrix sheaf, which the source motivates in its abstract and introduction without posing them as explicit deliverables.

Problem Statement

Let $v_1,\dots,v_8$ be the weights of the odd eight-vertex model on an $N\times N$ square lattice with periodic boundary conditions, with $N$ even (the partition function vanishes for odd $N$). Via the Wu-Kunz mapping the model is equivalent to a staggered even eight-vertex model on a $2N\times 2N$ lattice with alternating sublattice weights $\omega^X=(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8)$ and $\omega^Y=(v_3,v_4,v_1,v_2,v_8,v_7,v_6,v_5)$.

(1) Free-fermion manifold decision problem. For asymmetric weights, that is weights outside the arrow-inversion-symmetric subspace $v_1=v_2$, $v_3=v_4$, $v_5=v_6$, $v_7=v_8$ already settled by the source, satisfying the free-fermion condition $v_1v_2+v_3v_4-v_5v_6-v_7v_8=0,$ determine whether there exists an algebraic manifold of weights on which the odd-model transfer matrices commute for finite lattice sizes, together with the corresponding R-matrix satisfying the Yang-Baxter equation. The direct substitution of Krinsky's even-model constraints $\frac{v_6v_8}{v_5v_7}=\Delta_1,\quad \frac{v_1v_4+v_2v_3}{v_5v_7}=\Delta_2,\quad \frac{v_1^2+v_4^2-v_2^2-v_3^2}{v_5v_7}=\Delta_3$ with free parameters $\Delta_1,\Delta_2,\Delta_3$ is excluded by two-site commutator computations. An answer must either exhibit such a manifold with its R-matrix, or prove that no such manifold exists, in which case the free-fermion odd eight-vertex model is an example of a model exactly solvable in the thermodynamic limit (through the staggered Pfaffian solution) without commuting transfer matrices at finite lattice sizes.

(2) Staggered commutation problem. The staggered row operators are built from two Lax operators acting on the tensor product of an auxiliary space (index 0) and a single lattice-site space, whose $4\times 4$ matrices, with rows and columns each running over the auxiliary index first and the site index second so that the $2\times 2$ blocks are labeled by the auxiliary index, are $L(v)=\begin{pmatrix}0&v_1&v_7&0\\ v_3&0&0&v_6\\ v_5&0&0&v_4\\ 0&v_8&v_2&0\end{pmatrix},\qquad \mathbf{L}(v)=\begin{pmatrix}0&v_3&v_6&0\\ v_1&0&0&v_7\\ v_8&0&0&v_2\\ 0&v_5&v_4&0\end{pmatrix},$ where $L(v)$ carries the sublattice-$X$ weights $\omega^X$ and $\mathbf{L}(v)$ the sublattice-$Y$ weights $\omega^Y$ (these are Eqs. (26)-(27) of the source). The two alternating transfer matrices along a $2N$-site row are $T_1(v)=\mathrm{Tr}_0\left[L_{01}(v)\,\mathbf{L}_{02}(v)\cdots L_{0,2N-1}(v)\,\mathbf{L}_{0,2N}(v)\right],\qquad T_2(v)=\mathrm{Tr}_0\left[\mathbf{L}_{01}(v)\,L_{02}(v)\cdots \mathbf{L}_{0,2N-1}(v)\,L_{0,2N}(v)\right],$ so that the staggered partition function is $Z_{\mathrm{8od}}^{\mathrm{stag}}(v_1,\dots,v_8)=\mathrm{Tr}\left[\left(T_1(v)T_2(v)\right)^{N}\right]$. For weights lying on the free-fermion condition together with Krinsky's constraints (with $w_i=v_i$), prove or refute that the products $T_1(v)T_2(v)$ form a commuting family for arbitrary lattice size, that is $[T_1(v)T_2(v),\,T_1(v')T_2(v')]=0$ for all weight pairs on the manifold. The source verifies this only for $N=1,2$ and stresses that the individual transfer matrices $T_1$ and $T_2$ do not commute for distinct weights. The suggested route, the Yang-Baxter equations for the tensor product of the two types of Lax operators, or any other method is admissible.

These two questions share one scientific objective: to determine on which weights of the odd eight-vertex model, beyond the arrow-inversion-symmetric Baxter curve already settled, finite-lattice transfer-matrix commutativity survives, and to establish the corresponding Yang-Baxter structure. A complete solution would also illuminate the broader questions the source motivates but does not pose as deliverables: the classification of all integrable manifolds of the possibly asymmetric odd model, and the identification of the algebra whose equidimensional representations are interchanged by the even/odd sheaf of R-matrices.

References

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