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

@misc{kuwahara2026rényi,
  title = {{The Rényi area law conjecture is false below $α=1$ in high dimensions}},
  author = {Tomotaka Kuwahara},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15199},
  eprint = {2610.15199},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15199},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The Rényi area law conjecture is false below α=1 in high dimensions
AU  - Tomotaka Kuwahara
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15199
UR  - https://arxiv.org/abs/2610.15199
AB  - 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[Θ̃((Alogε^−1)^2/3)] when logε^−1≳ A. Thus α=1 is the sharp threshold for a universal Rényi area law within this adiabatically connected class.
ER  -