We investigate the Temperley–Lieb–Jones algebraic structures in isomorphic higher spin chain models with nearest neighbour interactions and symmetry. The Temperley–Lieb–Jones algebraic constructions for such chains of spins from 1 to are presented. For the case of spin-1 we also give the Temperley–Lieb–Jones algebraic representations related to the anisomorphic spin chain with -deformed symmetry. Their related spin chain Hamiltonians with and symmetries as well as the corresponding solutions of Yang–Baxter equation are also discussed.
There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)
k anyons, well-known deformations of SU(2), in which only the first
k+1 angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin
S=1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter
k when considering su(2)
k anyonic theories with $k\ensuremath{\ge}5$, as well as for the special case of the su(2)
4 theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-
21 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.
We solve for the spectrum of quantum spin chains based on representations of the Temperley - Lieb algebra associated with the quantum groups for and . We employ a generalization of the coordinate Bethe ansatz developed previously for the deformed biquadratic spin-one chain. As expected, all of these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower-dimensional representations are contained entirely in the higher-dimensional ones. The Bethe states are highest-weight states of the quantum group, except in certain cases where the states have energy zero.