Area law in one dimension: Degenerate ground states and Renyi entanglement entropy

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