  Investigating translationally invariant qudit spin chains with a low local
dimension, we ask what is the best possible tradeoff between the scaling of the
entanglement entropy of a large block and the inverse-polynomial scaling of the
spectral gap. Restricting ourselves to Hamiltonians with a "rewriting"
interaction, we find the pair-flip model, a family of spin chains with nearest
neighbor, translationally invariant, frustration-free interactions, with a very
entangled ground state and an inverse-polynomial spectral gap. For a ground
state in a particular invariant subspace, the entanglement entropy across a
middle cut scales as $\log n$ for qubits (it is equivalent to the XXX model),
while for qutrits and higher, it scales as $\sqrt{n}$. Moreover, we conjecture
that this particular ground state can be made unique by adding a small
translationally-invariant perturbation that favors neighboring letter pairs,
adding a small amount of frustration, while retaining the entropy scaling.
