Floquet-Bloch topology of nonreciprocal sound transport

A traveling-wave stiffness modulation in an active metamaterial picks out a particular spacetime direction for sound waves, thereby serving as a mechanism for reciprocity breaking in acoustic transport which has been exploited in a variety of platforms [see review by Nassar et al, Nature Reviews Materials 2020]. One way to quantify the breaking of reciprocity in such systems is to measure the asymmetry in the dispersion relation upon comparing momenta of equal magnitude but opposite directions; the frequencies of waves with equal and opposite momenta are guaranteed to be equal in passive structures, but can differ in active systems with reciprocity-breaking modulation. This asymmetry, termed band tilting, has been shown to be quantized in the limit of adiabatic stiffness modulation [Nassar et al, Physical Review B 97, 014305 (2018)] using perturbative methods. Using exact Floquet-Bloch theory, we show that band tilting is in fact quantized at all modulation speeds by virtue of the topological winding of Floquet quasifrequencies on a (d+1)-dimensional torus for d-dimensional systems. We discuss consequences of this quantization for systems with flat bands.