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 non-additive 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.

Highlights

  • Over the past decades we have witnessed the importance played by vertex models in the development of the theory of integrable systems in two spatial dimensions [1]

  • In this paper we have investigated the Yang-Baxter algebra for three-state vertex model whose statistical configurations are invariant by the U(1) invariance but break in an explicit way the paritytime reversal symmetry

  • We argued that the assumption of unitarity of the respective R-matrix imposes us that the functional equations derived from the Yang-Baxter algebra are anti-symmetrical on the exchange of the Boltzmann weights of distinct Lax operators

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Summary

Introduction

Over the past decades we have witnessed the importance played by vertex models in the development of the theory of integrable systems in two spatial dimensions [1]. We investigate this question in the most simple case where non-rational weights can not be rule out: the three-state U(1) vertex model Another relevant motivation to study these kind of systems comes from the existence of concrete exactly solvable spin-1 quantum chains discovered by Alcaraz and Bariev within the coordinate Bethe ansatz method [12]. The fact that their Hamiltonian for general couplings can not be derived in terms of an additive R-matrix suggests that non rational three-state vertex models should exist.

Integrability Conditions
Three-state Vertex Model
The Functional Relations
Two Terms Equations
Three Terms Equations
Main Branch
Special Branch
Four Terms Equations
The weight g
Weights h andh
The Manifolds Geometry
The Main Branch
The Special Branch
The Integrable Lattice Models
Lax operator and R-matrix
Spin Chain Hamiltonians
Conclusions
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