  We find a new physical regime in the trapped Bose-Hubbard Hamiltonian using
time-evolving block decimation. Between Mott-insulating and superfluid phases,
the latter induced by trap compression, a spatially self-organized state
appears in which non-local entropy signals entanglement between spatially
distant superfluid shells. We suggest a linear rather than harmonic potential
as an ideal way to observe such a self-organized system. We also explore both
quantum information and thermal entropies in the superfluid regime, finding
that while the former follows the density closely the latter can be strongly
manipulated with the mean field.
