Gaussian filters in quantum lattice systems: Applications to spectral flow, local perturbations, clustering, and the quantum Hall effect

We consider the locality and spectral properties of the smearing $${\tau }_f(A)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dtf(t){\tau }_t(A)$$ when applied to the dynamics ${\tau }_t$ of quantum spin systems. While recent applications of this map have used superpolynomially but not exponentially decaying functions $f$ to ensure exact spectral properties, we use here Gaussian filters. This improves the locality at the expense of errors on the spectral side. We propose a number of concrete applications, from quasi-adiabatic continuation to correlation decay, and exponential stability away from impurities. Finally, we discuss an application to the quantum Hall effect.