The braid group representations (BGRs) corresponding to general eight–vertex model are obtained for the case in which the total spins are not conserved. A Yang–Baxterization scheme is applied to obtain from these BGRs.
Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.
It is shown that Baxter's method when applied to the XXZ hamiltonian gives a set of eigenvectors and wavenumbers different from those of the familiar Bethe solution. In this limiting case of Baxter's results for the eight-vertex model the author finds that one of Baxter's parameters corresponds to the freedom of making arbitrary rotations about the z axis. It is suggested that for the general XXZ hamiltonian there may be a generalization of the ordinary z component of spin which would be conserved in addition to Baxter's quantum number.