Topological charge pumping in quasiperiodic systems characterized by Bott index

TL;DR

This paper introduces a real-space formalism using the Bott index to characterize topological charge pumping in one-dimensional quasiperiodic systems, overcoming the limitations of momentum-space approaches.

Contribution

The authors extend the Bott index method to quasiperiodic systems for topological charge pumping analysis, enabling systematic topological characterization regardless of system details.

Findings

Quasiperiodic systems exhibit multi-level topological charge pumping.

The Fibonacci-Rice-Mele model demonstrates fractal-induced multi-level pumping.

Real space renormalization explains the multi-level behavior.

Abstract

We study topological charge pumping in one-dimensional quasiperiodic systems. Since these systems lack periodicity, we cannot use the conventional approach based on the topological Chern number defined in the momentum space. Here, we develop a general formalism based on a real space picture using the so-called Bott index. We extend the Bott index that was previously used to characterize quantum Hall effects in quasiperiodic systems, and apply it to topological charge pumping in quasiperiodic systems. The Bott index allows us to systematically compute topological indices of charge pumping, regardless of the detail of quasiperiodic models. We apply this formalism to the Fibonacci-Rice-Mele model which we made from Fibonacci lattice, a well-known quasiperiodic system, and Rice-Mele model. We find that these quasiperiodic systems show topological charge pumping with a multi-level behavior due to the fractal nature of the Fibonacci lattice. Such multi-level pumping behaviors can be understood by a real space renormalization group analysis.