  The quadrupole insulator is a well-known instance of higher-order topological
insulators in two dimensions, which possesses midgap corner states in both the
energy spectrum and entanglement spectrum. Here, by constructing and exploring
a model Hamiltonian under a staggered $\mathbb{Z}_2$ gauge field that respects
momentum-glide reflection symmetries, we surprisingly find a quadrupole
insulator that lacks zero-energy corner modes in its energy spectrum, despite
possessing a nonzero quadrupole moment. Remarkably, the existence of midgap
corner modes is found in the entanglement spectrum. Since these midgap states
cannot be continuously eliminated, the quadrupole insulator cannot be
continuously transformed into a trivial topological insulator, thereby
confirming its topological nature. We show that the breakdown of the
correspondence between the energy spectrum and entanglement spectrum occurs due
to the closure of the edge energy gap when the Hamiltonian is flattened.
Finally, we present a model that demonstrates an insulator with corner modes in
the energy spectrum even in the absence of the quadrupole moment. In this
phase, the entanglement spectrum does not display any midgap states. The
results suggest that the bulk-edge correspondence of quadrupole insulators
generally manifests in the entanglement spectrum rather than the energy
spectrum.
