Phantom Bethe roots in the integrable open spin $1/2$ $XXZ$ chain

TL;DR

This paper explores special solutions to the Bethe Ansatz equations in open XXZ spin chains involving phantom roots that do not affect energy, revealing new invariant subspaces and chiral states with unique properties.

Contribution

It introduces the concept of phantom Bethe roots, analyzes their impact on eigenstates, and derives explicit Bethe vectors and correlation functions for these novel solutions.

Findings

Eigenstates split into two invariant subspaces with chiral properties

Bethe vectors can be explicitly constructed for states with phantom roots

Spin current and magnetization profiles exhibit quasi-periodic modulation

Abstract

We investigate special solutions to the Bethe Ansatz equations (BAE) for open integrable $XXZ$ Heisenberg spin chains containing phantom (infinite) Bethe roots. The phantom Bethe roots do not contribute to the energy of the Bethe state, so the energy is determined exclusively by the remaining regular excitations. We rederive the phantom Bethe roots criterion and focus on BAE solutions for mixtures of phantom roots and regular (finite) Bethe roots. We prove that in the presence of phantom Bethe roots, all eigenstates are split between two invariant subspaces, spanned by chiral shock states. Bethe eigenstates are described by two complementary sets of Bethe Ansatz equations for regular roots, one for each invariant subspace. The respective "semi-phantom" Bethe vectors are states of chiral nature, with chirality properties getting less pronounced when more regular Bethe roots are added. For the easy plane case "semi-phantom" Bethe states carry nonzero magnetic current, and are characterized by quasi-periodic modulation of the magnetization profile, the most prominent example being the spin helix states (SHS). We illustrate our results investigating "semi-phantom" Bethe states generated by one regular Bethe root (the other Bethe roots being phantom), with simple structure of the invariant subspace, in all details. We obtain the explicit expressions for Bethe vectors, and calculate the simplest correlation functions, including the spin-current for all the states in the single particle multiplet.