  Motivated by the recent experimental observation of negative absolute
temperature states in systems of ultracold atomic gases in optical lattices
[Braun et al., Science 339, 52 (2013)], we investigate theoretically the
formation of these states. More specifically, we consider the relaxation after
a sudden inversion of the external parabolic confining potential in the
one-dimensional inhomogeneous Bose-Hubbard model. First, we focus on the
integrable hard-core boson limit which allows us to treat large systems and
arbitrarily long times, providing convincing numerical evidence for relaxation
to a generalized Gibbs ensemble at negative temperature T<0, a notion we define
in this context. Second, going beyond one dimension, we demonstrate that the
emergence of negative temperature states can be understood in a dual way in
terms of positive temperatures, which relies on a dynamic symmetry of the
Hubbard model. We complement the study by exact diagonalization simulations at
finite values of the on-site interaction.
