An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of

n

spins the ground states of a local Hamiltonian with energy gap

ε

are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0

DOI: 10.48550/arxiv.1403.0327

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An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of

n

spins the ground states of a local Hamiltonian with energy gap

ε

are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0

An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of

n

spins the ground states of a local Hamiltonian with energy gap

\epsilon

are constant-fold degenerate. Then, the Renyi entanglement entropy

R_\alpha(0<\alpha<1)

of any ground state across any cut is upper bounded by

\tilde O(\alpha^{-3}/\epsilon)

, and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension

2^{\tilde O(\epsilon^{-1/4}\log^{3/4}n)}

The Rényi area law conjecture is false below

α=1

in high dimensions

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.