Finite-lattice integrability of the free-fermion odd eight-vertex model: commuting transfer matrices for the asymmetric and staggered cases

The odd eight-vertex model — the square-lattice two-state vertex model whose allowed vertices carry an odd number of arrows into each site — has an exactly known thermodynamic free energy in its free-fermion regime (Wu and Kunz, 2004) but no known commuting-transfer-matrix structure on finite lattices. Martins (2018) proved that for weights symmetric under arrow reversal the odd model is integrable on the same elliptic manifold as Baxter's even eight-vertex model, and singled out two open questions in the free-fermion sector: (1) whether the asymmetric odd model, with weights satisfying the free-fermion condition, admits any finite-lattice integrable manifold at all — the naive transcription of Krinsky's free-fermion constraints is already ruled out by two-site commutator computations — and (2) whether the product $T_{1}T_{2}$ of the two staggered odd-model transfer matrices commutes for arbitrary lattice size on the free-fermion-plus-Krinsky manifolds, as suggested by numerical verification for $N=1,2$. The problem posed here is to settle both questions, constructing explicit R-matrices and commuting families or ruling them out, and thereby to decide whether the free-fermion odd eight-vertex model is integrable at finite size or is a genuine example of a lattice model exactly solvable only in the thermodynamic limit.

Background

Two-dimensional vertex models are statistical-mechanical systems whose degrees of freedom are arrows placed on the edges of a square lattice, with a Boltzmann weight assigned to each allowed configuration of the four arrows meeting at a vertex. Restricting to the eight vertex configurations in which an even number of arrows points into each site gives the classical even eight-vertex model (Fan and Wu 1970; Baxter 1972). Its transfer matrix $T=\mathrm{Tr}_{0}[L_{01}L_{02}\cdots L_{0N}]$ is the trace, over a two-dimensional auxiliary space, of a product of local $4\times4$ Lax operators $L_{0j}$, one per column of a periodic $N\times N$ lattice. Baxter solved the model by showing that, when the weights are symmetric under reversal of all arrows ($w_{1}=w_{2}=a$, $w_{3}=w_{4}=b$, $w_{5}=w_{6}=c$, $w_{7}=w_{8}=d$) and lie on the intersection of two quadrics

$$\frac{ab-cd}{ab+cd}=\Gamma,\qquad \frac{a^{2}+b^{2}-c^{2}-d^{2}}{2(ab+cd)}=\Delta,$$

the transfer matrices form a one-parameter commuting family. Such a family is the definition of finite-lattice integrability for a vertex model; it is normally established by exhibiting an $R$-matrix that intertwines two Lax operators with different spectral parameters and satisfies the Yang–Baxter equation, an associativity condition on the intertwiners whose solutions are the algebraic core of exact solvability.

The complementary eight vertex configurations — those with an odd number of arrows into the site — define the odd eight-vertex model, introduced by Wu and Kunz (2004); it includes Ashkin–Teller-type interactions (a two-spin generalization of the Ising model) and is not covered by Baxter's solution. Wu and Kunz showed that the partition function of the odd model equals that of a staggered even eight-vertex model, i.e. an even model whose two sublattices carry different weights, and used this equivalence to compute the thermodynamic free energy whenever the eight weights $v_{1},\dots,v_{8}$ satisfy the free-fermion condition

$$v_{1}v_{2}+v_{3}v_{4}-v_{5}v_{6}-v_{7}v_{8}=0,$$

the analog of the condition under which the even model's free energy has been computable by Pfaffian (determinantal free-fermion) methods since Fan and Wu. Free-fermion solvability in the thermodynamic limit does not, however, imply integrability on finite lattices: for the even model, Krinsky (1972) showed that commuting transfer matrices require, besides the free-fermion condition, additional algebraic constraints on the weights (three in the general eight-weight case),

$$\frac{w_{6}w_{8}}{w_{5}w_{7}}=\Delta_{1},\qquad \frac{w_{1}w_{4}+w_{2}w_{3}}{w_{5}w_{7}}=\Delta_{2},\qquad \frac{w_{1}^{2}+w_{4}^{2}-w_{2}^{2}-w_{3}^{2}}{w_{5}w_{7}}=\Delta_{3},$$

with free parameters $\Delta_{j}$. For the asymmetric even model (arbitrary weights) an integrable manifold was later obtained by Galleas and Martins (2002), so asymmetry alone is not an obstruction to integrability in the even case.

Martins (2018) closed the symmetric case of the odd model: for weights symmetric under arrow reversal, the odd model has commuting transfer matrices on Baxter's very manifold above. The odd Lax operator is related to the even one by a local Pauli-matrix transformation, $L^{(\mathrm{od})}=(\sigma^{x}\otimes\sigma^{x})L^{(\mathrm{ev})}(\sigma^{x}\otimes I_{2})$, but the odd R-matrix differs structurally from the odd Lax operator — its entries are the weights in a permuted order — and the even and odd Lax operators together with two mixed intertwiners form a four-element family of R-matrices on Baxter's elliptic theta-function curve. The paper's concluding remarks then leave the free-fermion sector of the odd model open in two precise directions: the asymmetric case, where the direct transcription of Krinsky's constraints with $w_i\mapsto v_i$ fails (two-site commutators are nonzero), and the staggered case, where the natural two-row transfer matrix of the staggered odd model is conjectured — on the basis of exact numerics for $N=1$ and $N=2$ — to commute on the free-fermion-plus-Krinsky manifolds. No finite-size commuting-transfer-matrix structure is known for the odd model beyond the symmetric manifold, and no later literature has addressed either question.

Problem Statement

Decide whether the odd eight-vertex model admits finite-lattice integrability — a family of commuting transfer matrices — in its free-fermion sector, by settling the two questions posed in the concluding remarks of Martins (2018):

(1) Asymmetric odd model. Does there exist an algebraic manifold of weights (a one-parameter family in the sense of the Background definition of finite-lattice integrability, so that degenerate zero-dimensional weight sets do not qualify), containing or refining the free-fermion condition $v_{1}v_{2}+v_{3}v_{4}-v_{5}v_{6}-v_{7}v_{8}=0$, on which the transfer matrices $T^{(\mathrm{od})}(v_{1},\dots,v_{8})=\mathrm{Tr}_{0}[L^{(\mathrm{od})}_{01}\cdots L^{(\mathrm{od})}_{0N}]$ of the generally asymmetric odd eight-vertex model commute pairwise — $[T^{(\mathrm{od})}(v'),T^{(\mathrm{od})}(v'')]=0$ for distinct weight points $v',v''$ on the manifold — for arbitrary finite $N$? Here $L^{(\mathrm{od})}(v_{1},\dots,v_{8})$ is the $4\times4$ odd-model Lax operator, given explicitly by the matrix $L(v_{1},\dots,v_{8})$ displayed in part (2) below. The direct transcription of Krinsky's even-model constraints with $w_i\mapsto v_i$ is excluded by two-site commutator computations, but no other manifold is. A negative resolution would establish the free-fermion odd eight-vertex model as a clean example of a model exactly solvable in the thermodynamic limit (Wu-Kunz) yet not integrable at finite size.

(2) Staggered odd model. On a $2N\times2N$ torus, let $L$ and $\mathbf{L}$ be the two alternating Lax operators built from the odd weights and their staggered sublattice permutation,

$$L(v_{1},\dots,v_{8})=\begin{bmatrix}0&v_{1}&v_{7}&0\\ v_{3}&0&0&v_{6}\\ v_{5}&0&0&v_{4}\\ 0&v_{8}&v_{2}&0\end{bmatrix},\qquad \mathbf{L}(v_{1},\dots,v_{8})=\begin{bmatrix}0&v_{3}&v_{6}&0\\ v_{1}&0&0&v_{7}\\ v_{8}&0&0&v_{2}\\ 0&v_{5}&v_{4}&0\end{bmatrix},$$

and $T_{1}=\mathrm{Tr}_{0}[L_{01}\mathbf{L}_{02}\cdots L_{0,2N-1}\mathbf{L}_{0,2N}]$, $T_{2}=\mathrm{Tr}_{0}[\mathbf{L}_{01}L_{02}\cdots\mathbf{L}_{0,2N-1}L_{0,2N}]$ the two alternating transfer matrices, so that the staggered odd-model partition function is $\mathrm{Tr}[(T_{1}T_{2})^{N}]$. Prove or refute that

$$[T_{1}(v')T_{2}(v'),\,T_{1}(v'')T_{2}(v'')]=0$$

for arbitrary $N$ whenever the weights lie on the manifold defined by the free-fermion condition together with the three Krinsky-type constraints ($w_i=v_i$)

$$\frac{v_{6}v_{8}}{v_{5}v_{7}}=\Delta_{1},\qquad \frac{v_{1}v_{4}+v_{2}v_{3}}{v_{5}v_{7}}=\Delta_{2},\qquad \frac{v_{1}^{2}+v_{4}^{2}-v_{2}^{2}-v_{3}^{2}}{v_{5}v_{7}}=\Delta_{3},$$

where $v'$ and $v''$ are two points on the same manifold — that is, with the same invariants $\Delta_{j}$, since in the even-model convention a commuting family lives on one fixed Krinsky manifold — and the $\Delta_{j}$ themselves are free parameters. Note that $T_{1}$ and $T_{2}$ individually do not commute on these manifolds, which is why the double-row product is the object of interest.

The two parts share one objective — determining the finite-size integrability structure of the free-fermion odd eight-vertex model.

References

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