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.

The paper asks whether locality and a uniform spectral gap enforce entanglement area laws for every fixed Rényi index $0<\alpha<1$ in higher dimensions, and determines the worst-case accuracy dependence of approximate ground-state projector (AGSP) Schmidt rank. It constructs finite-range lattice Hamiltonians with local dimension at most six, bounded interaction strength, unique ground states and a size-independent gap. For every fixed $0<\alpha<1$ and spatial dimension $\nu\ge2$, suitable families have $S_\alpha=\Omega(n)$ while $S_1=O(\mathcal A)$. In two dimensions the rectangular aspect ratio depends on $\alpha$; in dimensions at least three, one hypercubic family works for every fixed $\alpha<1$. These states admit uniformly gapped local paths from product states. The construction uses parallel history-state rows, local activation constraints and one attractive root well. A fixed product projection extracts $Mm$ Bell pairs with probability at least $\exp[-C(M+\sqrt M\,m^{3/2})]$. This small-weight sector yields both Rényi volume laws and approximation-rank lower bounds. At fixed gap and $\log(1/\epsilon)\gtrsim\mathcal A$, the worst-case AGSP rank satisfies $\log D=\widetilde\Theta[(\mathcal A\log(1/\epsilon))^{2/3}]$. The matching upper bound combines regional energy projection, Chebyshev filtering and a cut-count decomposition. Optimality concerns approximate, not exactly ground-state-preserving, AGSPs, with lower-bound system sizes selected for the target accuracy. Gap dependence and matching lower bounds at all accuracies remain open. The results do not establish efficient global tensor-network construction or perturbative stability of the Rényi volume law, and do not resolve the general higher-dimensional von Neumann area-law question.