The Rényi area law conjecture is false below $α=1$ in high dimensions

Abstract

Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed $0<α<1$. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error $ε$ across a planar cut of cross-section $A$ scales as $\exp[\tildeΘ((A\logε^{-1})^{2/3})]$ when $\logε^{-1}\gtrsim A$. Thus $α=1$ is the sharp threshold for a universal Rényi area law within this adiabatically connected class.