Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains

TL;DR

This paper uncovers special phantom excitations in integrable spin chains that lead to degeneracies and spin helix states, with implications for experimental observations in quantum magnetic systems.

Contribution

It introduces the concept of phantom Bethe roots in the XXZ Heisenberg chain, revealing their role in degeneracies and spin helix states across different boundary conditions.

Findings

Phantom Bethe roots cause degeneracies in the spectrum.

Spin helix states emerge from phantom excitations.

Phenomenon occurs at specific anisotropies and boundary conditions.

Abstract

We demonstrate the existence of special phantom excitations for open and periodically closed integrable systems at the example of the $XXZ$ Heisenberg spin chain. The phantom excitations do not contribute to the energy of the Bethe state and correspond to special solutions to the Bethe Ansatz equations with infinite "phantom" Bethe roots. The phantom Bethe roots lead to degeneracies between different magnetization sectors in the periodic case and to the appearance of spin helix states (SHS), i.e. periodically modulated states of chiral nature in both open and closed systems. For the periodic chain, phantom Bethe root (PBR) solutions appear for anisotropies $\De=\cosh\eta$ with $\exp(\eta)$ being a root of unity, thus restricting the phenomenon to the critical region $|\De|<1$. For the open chain, PBR solutions appear for any value of anisotropy, both in the critical and in the non-critical region, provided that the boundary fields satisfy a criterion which we derive in this paper. There exist PBR solutions with all Bethe roots being phantom, and PBR solutions that consist of phantom roots as well as regular (finite) roots. Implications of our results for an experiment are discussed.