Invariant subspaces and explicit Bethe vectors in the integrable open spin $1/2$ $\XYZ$ chain

TL;DR

This paper investigates the eigenstate structure of the open XYZ spin-1/2 chain with boundary fields, deriving conditions for invariant subspace splitting, explicit Bethe vectors, and analyzing an elliptic spin-helix state with quasi-periodic magnetization.

Contribution

It provides a criterion for eigenstate splitting into invariant subspaces, explicit Bethe vectors without roots, and an elliptic spin-helix state characterization.

Findings

Eigenstate splitting governed by an integer related to kinks.

Explicit Bethe vectors for specific cases.

Elliptic spin-helix state with quasi-periodic magnetization.

Abstract

We derive a criterion under which splitting of all eigenstates of an open $\XYZ$ 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 investigate the \multiplet. We also describe in detail an elliptic analogue of the spin-helix state, appearing in both the periodic and the open $\XYZ$ model, and derive the eigenstate condition. The elliptic analogue of the spin-helix state is characterized by a quasi-periodic modulation of the magnetization profile, governed by Jacobi elliptic functions.