Quasiperiodicity and 2D topology in 1D charge-ordered materials

The mathematical description of 1D quasicrystals has recently been linked to that of 2D quantum Hall states. The topological classification of 1D quasicrystals and the corresponding interpretation of their observed charge transport have been widely discussed. We demonstrate the equivalence of both 1D quasicrystals and 2D quantum Hall states to a mean-field treatment of charge order. Using the fractal nature of the spectrum of charge-ordered states we consider incommensurate order as a limit of commensurate. The topological properties of both are identical, arising from a 2D parameter space of phase and wave vector, and fit into class A of the Tenfold Way. The topological nature of all the systems can be tested by measuring a quantized particle transport upon dragging the charge order.