Integrable three-state vertex models with weights lying on genus five curves

Abstract

We investigate the Yang-Baxter algebra for U(1) invariant three-state vertex models whose Boltzmann weights configurations break explicitly the parity-time reversal symmetry. We uncover two families of regular Lax operators with nineteen non-null weights which ultimately sit on algebraic plane curves with genus five. We argue that these curves admit degree two morphisms onto elliptic curves and thus they are bielliptic. The associated R-matrices are nonadditive in the spectral parameters and it has been checked that they satisfy the Yang-Baxter equation. The respective integrable quantum spin-1 Hamiltonians are exhibited.

An integrable nineteen vertex model lying on a hypersurface

“…We would like to remark that the request of time reversal invariance forces us from the very beginning to be far away of the recent found integrable genus five manifold [7]. We also note that this symmetry is not that stringent since the vertex model space of parameters is reduced to still fourteen distinct weights.…”

“…In fact, the requirement that F j (ω , ω ) should satisfy the property (10) has been decisive to simplify cumbersome high degree polynomials expressions emerging in the analysis of a three-state vertex model [7].…”

“…For the technical details concerning this interpretation see [7] and in what follows we shall refer to such special solutions as divisors.…”

3-state Hamiltonians associated to solvable 33-vertex models

“…When H commutes with only one charge, one usually takes q with three different eigenvalues: h has then 19 non vanishing entries. Such solvable models have been classified and studied previously in [7][8][9][10][11]. They correspond to out-of-equilibrium models with only one species of particle, but where two particles can occupy the same site.…”

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)Vieira1, Santos2 2017Physics Letters A2030View full textAdd to dashboardCite

“…Among the ZFvm and IKvm, other nineteen vertex models were discovered since then, for instance, the supersymmetric vertex models associated with the sl (2|1) and osp (2|1) Lie superalgebras, whose R-matrix was presented in [38] and the corresponding reflection K-matrices were obtained in [33]; for the Bethe Ansätze of these models, see [36,39]. Finally, other more complex nineteen vertex models were also discovered in the last decade − see, for instance, [40][41][42][43][44][45].The supersymmetric nineteen vertex model which we consider here is not included in the list above. The Rmatrix associated with this model was constructed in [1] and it can be written (up to a normalizing factor and employing a different notation) as, …”