  We study the entanglement spectrum of noninteracting band insulators, which
can be computed from the two-point correlation function, when restricted to one
part of the system. In particular, we analyze a type of partitioning of the
system that maintains its full translational symmetry, by tracing over a subset
of local degrees of freedom, such as sublattice sites or spin orientations. The
corresponding single-particle entanglement spectrum is the band structure of an
entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a
nontrivial topological phase is a gapless entanglement spectrum with an
"entanglement Fermi surface." Furthermore, we derive a relation between the
entanglement spectrum and the quantum geometry of Bloch states contained in the
Fubini-Study metric. The results are illustrated with lattice models of Chern
insulators and Z_2 topological insulators.
