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

TL;DR

This paper explores the use of Gaussian filters in quantum lattice systems to improve locality properties, with applications to spectral flow, correlation decay, stability, and the quantum Hall effect.

Contribution

It introduces Gaussian filters for quantum dynamics, balancing locality and spectral accuracy, and demonstrates their applications in various quantum phenomena.

Findings

Gaussian filters enhance locality in quantum dynamics.

Application to quantum Hall effect demonstrates practical utility.

Improved stability and correlation decay results in quantum spin systems.

Abstract

We consider the locality and spectral properties of the smearing \[ \tau_f(A) = \int_{-\infty}^\infty dt \, f(t) \, \tau_t(A) \] 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.