  We use quantum information measures to study the local quantum phase
transition that occurs for trapped spinless fermions in one-dimensional
lattices. We focus on the case of a harmonic confinement. The transition occurs
upon increasing the characteristic density and results in the formation of a
band-insulating domain in the center of the trap. We show that the ground-state
bipartite entanglement entropy can be used as an order parameter to
characterize this local quantum phase transition. We also study excited
eigenstates by calculating the average von Neumann and second Renyi eigenstate
entanglement entropies, and compare the results with the thermodynamic entropy
and the mutual information of thermal states at the same energy density. While
at low temperatures we observe a linear increase of the thermodynamic entropy
with temperature at all characteristic densities, the average eigenstate
entanglement entropies exhibit a strikingly different behavior as functions of
temperature below and above the transition. They are linear in temperature
below the transition but exhibit activated behavior above it. Hence, at
nonvanishing energy densities above the ground state, the average eigenstate
entanglement entropies carry fingerprints of the local quantum phase
transition.
