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The quadrupole insulator is a well-known instance of higher-order topologicalinsulators in two dimensions, which possesses midgap corner states in both theenergy spectrum and entanglement spectrum. Here, by constructing and exploringa model Hamiltonian under a staggered $\mathbb{Z}_2$ gauge field that respectsmomentum-glide reflection symmetries, we surprisingly find a quadrupoleinsulator that lacks zero-energy corner modes in its energy spectrum, despitepossessing a nonzero quadrupole moment. Remarkably, the existence of midgapcorner modes is found in the entanglement spectrum. Since these midgap statescannot be continuously eliminated, the quadrupole insulator cannot becontinuously transformed into a trivial topological insulator, therebyconfirming its topological nature. We show that the breakdown of thecorrespondence between the energy spectrum and entanglement spectrum occurs dueto the closure of the edge energy gap when the Hamiltonian is flattened.Finally, we present a model that demonstrates an insulator with corner modes inthe energy spectrum even in the absence of the quadrupole moment. In thisphase, the entanglement spectrum does not display any midgap states. Theresults suggest that the bulk-edge correspondence of quadrupole insulatorsgenerally manifests in the entanglement spectrum rather than the energyspectrum.

Quadrupole Insulator without Corner States in the Energy Spectrum