Invariant subspaces and elliptic spin-helix states in the integrable open spin- 12 XYZ chain

Abstract

We derive a criterion under which splitting of all eigenstates of an open $\mathrm{XYZ}$ Hamiltonian with boundary fields into two invariant subspaces, spanned by chiral shock states, occurs. The splitting is governed by an integer number, which has the geometrical meaning of the maximal number of kinks in the basis states. We describe the generic structure of the respective Bethe vectors. We obtain explicit expressions for Bethe vectors, in the absence of Bethe roots, and those generated by one Bethe root, and we investigate the single-particle subspace. We also describe in detail an elliptic analog of the spin-helix state, appearing in both the periodic and the open $\mathrm{XYZ}$ model, and we derive the eigenstate condition. The elliptic analog of the spin-helix state is characterized by a quasiperiodic modulation of the magnetization profile, governed by Jacobi elliptic functions.