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

Abstract

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

final version (5 pages, 2 figures)

References in corpus (1)

• Finite-size scaling and the toroidal partition function of the critical asymmetric six-vertex model

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