  Within the emergent eigenstate solution to quantum dynamics [Phys. Rev. X 7,
021012 (2017)], one can construct a local operator (an emergent Hamiltonian) of
which the time-evolving state is an eigenstate. Here we show that such a
solution exists for the expansion dynamics of Tonks-Girardeau gases in optical
lattices after turning off power-law (e.g., harmonic or quartic) confining
potentials, which are geometric quenches that do not involve the boost
operator. For systems that are initially in the ground state and undergo
dynamical fermionization during the expansion, we show that they remain in the
ground state of the emergent local Hamiltonian at all times. On the other hand,
for systems at nonzero initial temperatures, the expansion dynamics can be
described constructing a Gibbs ensemble for the emergent local Hamiltonian (an
emergent Gibbs ensemble).
