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Completeness of “good” Bethe ansatz solutions of a quantum-group-invariant Heisenberg model

Authors
Journal
Nuclear Physics B
0550-3213
Publisher
Elsevier
Publication Date
Volume
430
Issue
3
Identifiers
DOI: 10.1016/0550-3213(94)90162-7
Disciplines
  • Physics

Abstract

Abstract The sl q (2) quantum-group-invariant spin- 1 2 XXZ Heisenberg model with open boundary conditions is investigated by means of the Bethe ansatz. As is well known, quantum groups for q equal to a root of unity possess a finite number of “good” representations with non-zero q-dimension and “bad” ones with vanishing q-dimension. Correspondingly, the state space of an invariant Heisenberg chain decomposes into “good” and “bad” states. A “good” state may be described by a path of only “good” representations. It is shown that the “good” states are given by all “good” Bethe ansatz solutions with roots restricted to the first periodicity strip, i.e. only positive-parity strings (in the language of Takahashi) are allowed. Applying Bethe's string-counting technique completeness of the “good” Bethe states is proven, i.e. the same number of states is found as the number of all restricted paths on the sl q (2) Bratteli diagram. It is the first time that a “completeness” proof for an anisotropic quantum-invariant reduced Heisenberg model is performed.

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