Algebraic Localization of Wannier Functions Implies Chern Triviality in Non-periodic Insulators

For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to 0) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis elements satisfy \(\int |\varvec{x}|^2 |w(\varvec{x})|^2 \,\text {d}{\varvec{x}} < \infty \) ${\int |\mathit{x}|}^2{|w(\mathit{x})|}^2\text{d}\mathit{x}<\infty$ ). In this paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay ( \(\int |\varvec{x}|^{2+\epsilon } |w(\varvec{x})|^2 \,\text {d}{\varvec{x}} < \infty \) ${\int |\mathit{x}|}^{2+ϵ}{|w(\mathit{x})|}^2\text{d}\mathit{x}<\infty$ for any \(\epsilon > 0\) $ϵ>0$ ) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.