Space-time phononic crystals with anomalous topological edge states

It is well known that an interface created by two topologically distinct structures could host nontrivial edge states that are immune to defects. In this paper, we introduce a one-dimensional space-time phononic crystal and study the associated anomalous topological edge states when the phononic crystal's density is modulated both in space and time. While preserving the key topological feature of the system, the time modulation also duplicates the edge state mode across the spectrum, both inside and outside the band gap. It is shown that, in contrast to conventional topological edge states which are excited by frequencies in the Bragg regime, the time-modulation-induced frequency conversion can be leveraged to access topological edge states at a deep subwavelength scale where the entire phononic crystal size can be as small as 1/5.1 of the wavelength. This feature could open another route for designing miniature devices that are based on topological physics.