Fractional Pumping and Quasi-energy Winding in Space-time Crystals

Periodically driven systems exhibit exotic topological phenomenons with no analogue at equilibrium have been discovered[1,2,3]. In some cases, the space and time translation symmetry operations can be intertwined giving rise to a so-called space-time crystal[5]. In this case, the conventional Floquet formulation can not be applied. We construct a space-time translation symmetric 1+1D system and demonstrate that this system exhibits a fractional polarization change as well as a fractional quasi-energy winding during a cyclic process. The polarization change indicates a fractional adiabatic pumped charge, while, the quasi-energy corresponds to an electric flux penetrating the space-time unit-cell. When combined, we show that the two fractions are required to be quantized to an integer value, i.e., the underlying Chern number. By utilizing a Rice-Mele model, which exhibits the Thouless pumping effect[6,7], we demonstrate our general formulation numerically.


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[6] S. Nakajima, et. al, Nat. Phys. 12, 296 (2016).
[7] M. Lohse, et. al, Nat. Phys. 12, 350 (2016).