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For gapped periodic systems (insulators), it has been established that theinsulator is topologically trivial (i.e., its Chern number is equal to $0$) ifand only if its Fermi projector admits an orthogonal basis with finite secondmoment (i.e., all basis elements satisfy $\int |\boldsymbol{x}|^2|w(\boldsymbol{x})|^2 \,\textrm{d}{\boldsymbol{x}} < \infty$). In this paper,we extend one direction of this result to non-periodic gapped systems. Inparticular, we show that the existence of an orthogonal basis with slightlymore decay ($\int |\boldsymbol{x}|^{2+\epsilon} |w(\boldsymbol{x})|^2\,\textrm{d}{\boldsymbol{x}} < \infty$ for any $\epsilon > 0$) is a sufficientcondition to conclude that the Chern marker, the natural generalization of theChern number, vanishes.

Algebraic localization of Wannier functions implies Chern triviality in non-periodic insulators