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 $\mathrm{\Omega }=\pm 1$ with $\mathrm{\Omega }$ 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 $(\mathrm{\Omega },\mathrm{{\rm Y}})=(\pm 1,\mp 1)$ where $\mathrm{{\rm Y}}$ is the eigenvalue of $V$. Therefore the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number $(\mathrm{\Omega },\mathrm{{\rm Y}})=(\pm 1,\mp 1)$. 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.