Spin chain Hamiltonians with affine Uqg symmetry

Akishi Kato

Tetsuo Deguchi, Chihiro Matsui

A. Avakyan, Tigran Hakobyan, A. Sedrakyan

We construct the family of spin-chain Hamiltonians, which have affine quantum group symmetry Uqĝ. Their eigenvalues coincide with the eigenvalues of the usual spin-chain Hamiltonians, but have the degeneracy of levels, corresponding to affine Uqĝ. The space of states of these spin-chains is formed by the tensor product of fully reducible representations. The fermionic representations of spin-chain Hamiltonians, which have affine quantum group symmerty, was consructed. They correspond to new extensions of Hubbard Hamiltonians. The exact ground state of some examples is presented, exhibiting superconducting behavior via η-pairing mechanism.

Anastasia Doikou

Solutions of the reflection equation for the Uq ( sln) case are obtained by implementing certain realisations of the affine Hecke algebra. The boundary non–local charges associated to nondiagonal solutions of the reflection equation are constructed systematically, and the symmetry of the corresponding open spin chain is investigated. We prove that the symmetry of the open spin chain is essentially a remnant of the Uq(gln) algebra.

M. Fannes, Bruno Nachtergaele, R. F. Werner

Anastasia Doikou, Paul Martin

Motivated by a study of the crossing symmetry of the asymmetric twin or ‘gemini’ representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang–Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley–Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the boundary Yang–Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin’s commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary—despite differing from this model in having reference states.) Using these multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.