Time-dependent topological systems: A study of the Bott index

TL;DR

This paper demonstrates that the Bott index and Chern number remain constant over time for large, time-dependent quantum systems, including during ramps and periodic perturbations, extending their applicability.

Contribution

It establishes the time-invariance of the Bott index and Chern number in large, time-dependent systems, including under ramps and periodic perturbations.

Findings

Bott index remains constant over time in large systems.

Chern number is invariant during time evolution.

Results apply to systems with time-dependent Hamiltonians.

Abstract

The Bott index is an index that discerns among pairs of unitary matrices that can or cannot be approximated by a pair of commuting unitary matrices. It has been successfully employed to describe the approximate integer quantization of the transverse conductance of a system described by a short-range, bounded and spectrally gapped Hamiltonian on a finite two dimensional lattice on a torus and to describe the invariant of the Bernevig- Hughes-Zhang model even with disorder. This paper shows the constancy in time of the Bott index and the Chern number related to the time-evolved Fermi projection of a thermodynamically large system described by a short-range and time-dependent Hamiltonian that is initially gapped. The general situation of a ramp of a time-dependent perturbation is considered, a section is dedicated to time-periodic perturbations.