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

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, $γ$, and show that it induces entanglement transitions of the ground state. At very strong inhomogeneity, the rainbow state survives for $0 \leq γ\leq 1$, 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 $γ$. In all regimes we have found that the entanglement properties are invariant under the transformation $γ\longleftrightarrow 1 - γ$, whose fixed point $γ= \frac{1}{2}$ corresponds to the usual rainbow model. Finally, we study the robustness of non trivial topological phases in the presence of the defect.

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