Construction of the Temperley-Lieb algebra from bond algebra: From the transverse-field Ising to the XXZ

Abstract: We show that the Temperley-Lieb algebra is constructed from the generators of the transverse-field Ising-type bond algebra. It is also shown that, when we take the representation of the generators to the one-dimensional transverse-field Ising model, then the Temperley-Lieb algebra constructed on three consecutive bond generators becomes the XXZ model. Furthermore, we obtain several integrable spin systems from other representations for the bond generators.