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 $int |boldsymbol{x}|^2 |w(boldsymbol{x})|^2 ,textrm{d}{boldsymbol{x}} < infty$). In this paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay ($int |boldsymbol{x}|^{2+epsilon} |w(boldsymbol{x})|^2 ,textrm{d}{boldsymbol{x}} < infty$ for any $epsilon > 0$) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.