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Modern Physics Letters A

QUANTUM ALGEBRA STRUCTURE OF EXACTLY SOLUBLE QUANTUM SPIN CHAINS

Luca Mezincescu, Rafael I. Nepomechie

Published Sep 7, 1991• 65 citations
Publisher

Abstract

We consider a large class of quantum spin chains, whose Hamiltonians commute with generators of a quantum algebra and which are integrable. We argue that the corresponding transfer matrices also commute with the quantum algebra. For the spin [Formula: see text] chain, we show that the Bethe states are highest weight states of Uq[ su (2)].

DOI
10.1142/s0217732391002931

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We consider a large class of quantum spin chains, whose Hamiltonians commute with generators of a quantum algebra and which are integrable. We argue that the corresponding transfer matrices also commute with the quantum algebra. For the spin [Formula: see text] chain, we show that the Bethe states are highest weight states of Uq[ su (2)].
We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.
We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.
The authors discuss the XXZ spin chain with certain boundary terms and the representation of quantum group SU q (2). It is shown that Bethe ansatz states are highest-weight states of SU q (2). With a generic q they construct the irreducible representations of the quantum group. For q being a root of unity, they show that there are new Bethe ansatz states, which coincide with null states of SU q (2). By taking certain limits the authors can derive the state mod b), which is necessary for constructing the indecomposable but reducible representations of SU q (2) and for the completeness of the state space. In this case the Hamiltonian may not be completely diagonalized.