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

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 lattice on a finite two-dimensional 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 system described by a short-range, bounded, 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.

Figure 1

Left panel: with $U_1$ and $U_2$ unitary the spectrum of $U_1{U}_2{U}_1^{†}{U}_2^{†}-1$ is a set of points on the black circle of radius equal to 1. $det\left(U_1{U}_2{U}_1^{†}{U}_2^{†}\right)=1$. The only point at a distance equal to 2 from the origin is on the real negative axis. The thick red line indicates the branch cut of the logarithm. Right panel: the spectrum of $V_1{V}_2{V}_1^{†}{V}_2^{†}$ as given in Eq. (18) is confined in a small region close to (1,0). The unitary time evolution of the system is such that $P$ and $Q$ are replaced by $U(t,t_0)P\left(t_0\right)U^{†}(t,t_0)$ and $U(t,t_0)Q\left(t_0\right)U^{†}(t,t_0)$ as a consequence the eigenvalues of $V_1\left(t\right)V_2\left(t\right)V_1^{†}\left(t\right)V_2^{†}\left(t\right)$ can move in time but nevertheless they stay confined within the unit disk in an area close by (1,0) schematically drawn as an ellipse. The thick red line indicates the branch cut of the logarithm.