Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω = ±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, Υ) = (±1, ∓1) where Υ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, Υ) = (±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.