Spectral localizer in trivial metals as topological insulator zero-modes

Metals and topological insulators have in common that they cannot be described by exponentially localized wave-functions. Here we establish a relationship between these two seemingly unrelated observations. The connection is explicit in the low-lying states of the spectral localizer of trivial metals, an operator that measures the obstruction to finding localized eigenstates. The low-lying spectrum of the spectral localizer of metals is determined by the zero-mode solutions of Dirac fermions with a varying mass parameter. We use this observation, valid in any dimension, to determine the difference between the localizer spectrum of trivial and topological metals, and conjecture the spectrum of the localizer for fractional quantum Hall edges. Because the localizer is a local real-space operator, it may be used as a tool to differentiate between non-crystalline topological and trivial metals, and to characterize strongly correlated systems, for which local topological markers are scarce.