Piercing the rainbow: entanglement on an inhomogeneous spin chain with a defect

TL;DR

This paper investigates how a tunable defect affects the entanglement structure of the rainbow state in an inhomogeneous spin chain, revealing entanglement transitions, volume law behavior, and robustness of topological phases.

Contribution

It introduces a tunable defect in the rainbow state model, analyzing its impact on entanglement transitions and topological phase stability in an inhomogeneous spin chain.

Findings

Rainbow state persists for 0 ≤ γ ≤ 1 under strong inhomogeneity.

Entanglement entropy follows a volume law with an effective central charge.

Entanglement properties are symmetric under γ ↔ 1 - γ, with γ=1/2 as a fixed point.

Abstract

The {\em rainbow state} denotes a set of valence bond states organized concentrically around the center of a spin 1/2 chain. It is the ground state of an inhomogeneous XX Hamiltonian and presents maximal violation of the area law of entanglement entropy. Here, we add a tunable exchange coupling constant at the center, $\gamma$, and show that it induces entanglement transitions of the ground state. At very strong inhomogeneity, the rainbow state survives for $0 \leq \gamma \leq 1$, while outside that region the ground state is a product of dimers. In the weak inhomogeneity regime the entanglement entropy satisfies a volume law, derived from CFT in curved spacetime, with an effective central charge that depends on the inhomogeneity parameter and $\gamma$. In all regimes we have found that the entanglement properties are invariant under the transformation $\gamma \longleftrightarrow 1 - \gamma$, whose fixed point $\gamma = \frac{1}{2}$ corresponds to the usual rainbow model. Finally, we study the robustness of non trivial topological phases in the presence of the defect.