Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
Vladislav Popkov, Xin Zhang, Andreas Klümper
Abstract
We demonstrate the existence of a special chiral ``phantom'' mode with some analogy to a Goldstone mode in the anisotropic quantum $XXZ$ Heisenberg spin chain. The phantom excitations contribute zero energy to the eigenstate, but a finite fixed quantum of momentum ${k}_{0}$. The mode exists not due to symmetry principles, but results from nontrivial scattering properties of magnons with momentum ${k}_{0}$ given by the anisotropy via $cos{k}_{0}=\mathrm{\ensuremath{\Delta}}$. Different occupations of the phantom mode lead to energetical degeneracies between different magnetization sectors in the periodic case. This mode originates from special string-type solutions of the Bethe ansatz equations with unbounded rapidities, the phantom Bethe roots (PBRs). We derive criteria under which the spectrum contains eigenstates with PBRs, both in open and periodically closed integrable systems, for spin-$1/2$ and higher spins, and discuss the respective chiral eigenstates. The simplest of such eigenstates, the spin helix state, which is a periodically modulated state of chiral nature, is built up from the phantom excitations exclusively. Implications of our results for experiments are discussed.