Anderson Localization: A Floquet operator Krylov space perspective

Abstract

The problem of Anderson localization, as well as the single particle localization-delocalizaton transition of the Aubry-Andr\'e model, is studied employing operator Krylov space methods. It is shown that even when the dynamics is generated by a Hamiltonian, studying the dynamics at stroboscopic rather than continuous times has its advantages. In particular, mapping the dynamics to an effective Floquet problem results in an operator Krylov space description where quantities such as the spectral function can be computed with fewer computational resources, while a moment method exists that allows for the extraction of Krylov parameters directly from the discrete time autocorrelation function. For stroboscopic dynamics, the operator Krylov space corresponds to the dynamics of an edge operator of an inhomogeneous Floquet transverse field Ising model, with the parameters of this effective model generated recursively. The Krylov parameters show disorder-averaged renormalization with their distribution narrowing as the recursion step increases. It is shown that a more physical spectral function is obtained from the Krylov parameters obtained from the disorder-averaged autocorrelation function, rather than the disorder-averaged Krylov parameters. The delocalized (localized) phase is shown to correspond to the appearance (absence) of a Porter-Thomas distribution, a ballistically propagating (localized) wavefront in operator Krylov space, and a smooth (discrete) Berstein-Szeg\"o power-spectrum. The localization-delocalization transition is also demonstrated in operator Krylov space. A Porter-Thomas distribution is also observed at the critical point. The long-time dynamics and the inverse participation ratio at the critical point is shown to exhibit behavior consistent with a multi-fractal scaling with system size.