Quantization of band tilting in modulated phononic crystals

A general theory of the tilting of dispersion bands in phononic crystals whose properties are being slowly and periodically modulated in space and time is established. The ratio of tilt to modulation speed is calculated, for the first time, in terms of Berry's phase and curvature and is proven to be a robust integer-valued Chern number. Derivations are based on a version of the adiabatic theorem for elastic waves demonstrated thanks to WKB asymptotics. Findings are exemplified in the case of a 3-periodic discrete spring-mass lattice. Tilted dispersion diagrams plotted using fully numerical simulations and semianalytical calculations based on a numerically gauge invariant expression of Berry's phase show perfect agreement. One-way blocking of waves due to the tilt, and ultimately to the breaking of reciprocity, is illustrated numerically and shown to be highly significant across a limited number of unit cells, suggesting the feasibility of experimental demonstrations. Finally, a version of the bulk-edge correspondence principle relating the tilt of bulk bands to the number of one-way gapless edge states is demonstrated.