Figure 4. Dispersion curves obtained using full numerical simulations to show the effect of high pumping frequency.
(a) fα = 0, i.e., a system with only spatial modulation in stiffness, has a symmetric dispersion curve. The three prominent band-gaps: 1, 2, and 3 are marked. (b) Torsional wave (with fα = 2 kHz in this case) breaks the time-reversal symmetry, and thus results in asymmetric band-gaps for rightward (positive wavenumber) and leftward (negative wavenumber) propagating longitudinal wave. The frequency offsets for the band-gaps are indicated, and those are dictated by corresponding topological invariant, i.e., the summation of Chern numbers of bands below it, defined in quasi-static sense.