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

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.

arXiv comment18 pages, 3 figures

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Cited by in corpus (15)

• Transverse spin dynamics in the anisotropic Heisenberg model realized with ultracold atoms
• Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
• Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ spin-$\frac{1}{2}$ chain
• Quantum spin helices more stable than the ground state: onset of helical protection
• Boundary driven $XYZ$ chain: Exact inhomogeneous triangular matrix product ansatz
• Universality in relaxation of spin helices under the $XXZ$- spin chain dynamics
• Chiral basis for qubits and spin-helix decay
• Unveiling chiral states in the XXZ chain: Finite-size scaling probing symmetry-enriched $c=1$ conformal field theories
• Invariant subspaces and explicit Bethe vectors in the integrable open spin $1/2$ $\XYZ$ chain
• All product eigenstates in Heisenberg models from a graphical construction
• A Pedestrian's Way to Baxter's Bethe Ansatz for the Periodic XYZ Chain
• Rational Q-systems at Root of Unity I. Closed Chains
• Exact NESS of XXZ circuits boundary driven with arbitrary resets or fields
• Generalized Josephson effect with arbitrary periodicity in quantum magnets
• Universal scaling of higher-order cumulants in quantum isotropic spin chains