Geometric phase invariance in spatiotemporal modulated elastic system

We study the topological characteristics of elastic waves in a one-dimensional mass and spring elastic superlattice that exhibits non-reciprocal elastic wave propagation due to extrinsic application of a single sinusoidal spatiotemporal modulation of its spring stiffness. We employ a computational procedure to generate the band structure, traveling modes' amplitudes and phases, and subsequently the geometric phases to characterize the global vibrational behavior of the system and its topological character. The elastic superlattice demonstrates the notion of non-conventional band structure, where hybridization gaps arise as bands appear and cross for certain wavenumber values. We estimate these hybridization points using multiple time scale perturbation theory for low modulation velocity. At the hybridization points, both the theoretical and numerical analyses display a discontinuity in the traveling modes’ amplitudes. Consequently, we find a multiple of π, sometimes zero, geometric phase value in the spatiotemporal modulated elastic system, if the unit cell has inversion symmetry as imposed by the values of the spring constant at the initial time. The temporal modulation, which creates hybridization gaps, changes the nature of the geometric phase from a closed loop geometric phase when the modulation is only spatial, to an open loop geometric phase. This open loop geometric phase is invariant to the temporal modulation.