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

Abstract

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 $\hat{I}\mu$ are constant-fold degenerate. Then, the Renyi entanglement entropy $R_{\hat{I}}\pm (0<\hat{I}\pm <1)$ of any ground state across any cut is upper bounded by $\tilde{O}(\hat{I}{\pm }^{-3}/\hat{I}\mu )$, and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension $2^{\tilde{O}(\hat{I}{\mu }^{-1/4}{\log }^{3/4}n)}$.

close to the final version; to be published