Completeness of ``Good'' Bethe Ansatz Solutions of a Quantum Group Invariant Heisenberg Model

Abstract

The $s{l}_q(2)$-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 path's on the $s{l}_q(2)$-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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• CONNECTIONS of the LIOUVILLE MODEL and XXZ SPIN CHAIN
• Irreversibility of the renormalization group flow in non-unitary quantum field theory
• Fusion Hierarchy and Finite-Size Corrections of $U_q[sl(2)]$ Invariant Vertex Models with Open Boundaries
• On zero-remainder conditions in the Bethe ansatz
• Completeness of ``Good'' Bethe Ansatz Solutions of a Quantum Group Invariant Heisenberg Model
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• Minimal Unitary Models and The Closed SU(2)-q Invariant Spin Chain
• Non-commutative Euclidean structures in compact spaces
• Non-Hermitian Parent Hamiltonian from Generalized Quantum Covariance Matrix