Generalized Integrable Boundary States in XXZ and XYZ Spin Chains

TL;DR

The study generalizes integrable boundary states in XXZ and XYZ spin chains to both periodic and twisted boundaries and to odd system sizes, deriving factorized boundary states from the KT relation. It extends the standard $T$-$Q$ framework to off-diagonal and elliptic (XYZ) settings, establishing explicit selection rules that constrain Bethe roots for nonzero overlaps with boundary states. The work shows that only specific boundary-state sectors survive (e.g., $| ext{Ψ}_{+,e} angle$ and $| ext{Ψ}_{-,o} angle$ in many cases) and provides detailed conditions under which other sectors exist or do not exist, both for XXZ and XYZ. These results illuminate non-thermalizing dynamics and overlaps in integrable quenches, with potential relevance for cold-atom experiments and broader integrable-QFT/CFT contexts, and lay groundwork for computing exact overlaps via generalized $T$-$Q$ relations.

Abstract

We investigate integrable boundary states in the anisotropic Heisenberg chain under periodic or twisted boundary conditions, for both even and odd system lengths. Our work demonstrates that the concept of integrable boundary states can be readily generalized. For the XXZ spin chain, we present a set of factorized integrable boundary states using the KT-relation, and these states are also applicable to the XYZ chain. It is shown that a specific set of eigenstates of the transfer matrix can be selected by each boundary state, resulting in an explicit selection rule for the Bethe roots.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• proof
• proof