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The Bott index is an index that discerns among pairs of unitary matrices thatcan or cannot be approximated by a pair of commuting unitary matrices. It hasbeen successfully employed to describe the approximate integer quantization ofthe transverse conductance of a system described by a short-range, bounded andspectrally gapped Hamiltonian on a finite two dimensional lattice on a torusand to describe the invariant of the Bernevig- Hughes-Zhang model even withdisorder. This paper shows the constancy in time of the Bott index and theChern number related to the time-evolved Fermi projection of athermodynamically large system described by a short-range and time-dependentHamiltonian that is initially gapped. The general situation of a ramp of atime-dependent perturbation is considered, a section is dedicated totime-periodic perturbations.

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