  We study the relationship between entanglement and spectral gap for local
Hamiltonians in one dimension. The area law for a one-dimensional system states
that for the ground state, the entanglement of any interval is upper-bounded by
a constant independent of the size of the interval. However, the possible
dependence of the upper bound on the spectral gap Delta is not known, as the
best known general upper bound is asymptotically much larger than the largest
possible entropy of any model system previously constructed for small Delta. To
help resolve this asymptotic behavior, we construct a family of one-dimensional
local systems for which some intervals have entanglement entropy which is
polynomial in 1/Delta, whereas previously studied systems, such as free fermion
systems or systems described by conformal field theory, had the entropy of all
intervals bounded by a constant times log(1/Delta).
