Chiral Integrable Boundary States of ABJM Spin Chain from Reflection Equations

We develop a general framework for constructing $2n$-site chiral integrable matrix product states in Aharony-Bergman-Jafferis-Maldacena spin chain, based on reflection equations and the fusion procedure. For four-site chiral integrable product states, we propose their exact overlap formulas with Bethe states. We also investigate the chiral integrable subspaces numerically.

💡 Research Summary

The paper presents a comprehensive framework for constructing chiral integrable matrix product states (MPS) in the ABJM spin chain, leveraging reflection equations (RE) and the fusion procedure. After a concise introduction to the relevance of integrable MPS in both high‑energy and statistical physics, the authors focus on the SU(4) alternating spin chain that describes the scalar sector of ABJM theory at two‑loop order. They review the Hamiltonian, the four R‑matrices (fundamental and anti‑fundamental), the monodromy and transfer matrices, and the nested Bethe ansatz that yields three sets of rapidities (u, w, v) satisfying standard S‑matrix scattering relations.

A central concept is the distinction between chiral and achiral integrable boundary states. Chiral states enforce that Bethe roots of the same type pair with each other, while achiral states allow mixing of different types. The authors adopt the chiral condition expressed through the transfer matrices: τ(u)|B⟩ = Π τ(u) Π|B⟩, equivalently τ(u)|B⟩ = τ(−u−2)|B⟩, where Π reverses the order of sites.

The core technical development starts in Section 3, where two families of K‑matrices are introduced: soliton‑preserving (SP) and soliton‑non‑preserving (SNP). The SP K‑matrices (K_a and \bar K_a) satisfy the standard reflection equation, while the SNP K‑matrices (˜K_a and ˜K_{\bar a}) obey a twisted version involving the barred R‑matrices. By mixing an SP K‑matrix with an SNP K‑matrix, the authors derive a mixed reflection equation (3.5) that guarantees the resulting two‑site state |ϕ⟩⊗|ϕ⟩… satisfies the chiral integrability condition. The two‑site building block is defined as |ϕ(u)⟩ =