Chern Invariants of Topological Continua; a Self-Consistent Nonlocal Hydrodynamic Model

The bulk-edge correspondence is a fundamental principle of topological wave physics, which states that the difference in gap Chern numbers between the interfaced materials is equal to the net number of topological edge modes. Although this principle works well for periodic photonic topological insulators, difficulties arise in the case of continuous systems with no intrinsic periodicity, due to the absence of a finite Brillouin zone, which may lead to an ill-behaved response for diverging wavenumbers. This problem has been solved previously by introducing an ad hoc material model including a spatial cutoff wavenumber. However, this method introduces other difficulties, such as an unphysical response at large wavenumbers, and the need to interpolate the interfaced materials permittivity functions. In this work, we show that the inclusion of realistic spatial dispersion (e.g., hydrodynamic nonlocality) formally restores the bulk-edge correspondence by avoiding an unphysical response at large wavenumbers, forming a complete, self-consistent model. This model clarifies the subtle and rich topological wave physics of continuous media.