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

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 k0. The mode exists not due to symmetry principles, but results from non-trivial scattering properties of magnons with momentum k0 given by the anisotropy via cos k0 = ∆. 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 (PBR). We derive criteria under which the spectrum contains eigenstates with PBR, 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.