Abstract
Chaotic many-body dynamics typically generates volume-law entanglement from initially low-entangled states. We reveal an intricate, hierarchical entanglement structure in local quantum quenches, both in the canonical purification of locally quenched Gibbs states and in a companion pure-state circuit model. In either setting, the full state exhibits a Renyi-index-tuned transition: at long times, S_α>1 obeys an area law, while S_α≤ 1 is volume-law. More strikingly, the response linear in the quench strength is carried by only an O(1)-dimensional dominant Schmidt sector; the corresponding states exhibit their own area-to-volume-law transitions at critical indices α_c<1, implying polynomial-bond-dimension approximability in one dimension. We provide evidence that this hierarchy persists recursively: upon bipartitioning the dominant Schmidt states, their leading Schmidt sectors exhibit analogous structure. We derive the mechanism analytically in the circuit model, prove the S_α>1 area law for locally quenched Gibbs states, and support the hierarchy by exact diagonalization of random circuits and locally quenched Gibbs states of chaotic spin chains.