Quasiperiodicity and 2D Topology in 1D Charge Ordered Materials

It has recently been argued that individual 1D quasicrystals can be ascribed 2D topological quantum numbers and a corresponding set of topologically protected edge modes. Here, we demonstrate the equivalence of such 1D quasicrystals to a mean-field treatment of incommensurate charge order in 1D materials. Using the fractal nature of the spectrum of commensurate charge-ordered states we consider incommensurate order as a limiting case of commensurate orders. We show that their topological properties arise from a 2D parameter space spanned by both phase and wave vector, bringing the observation of 2D edge modes in line with the standard classification of topological order.The equivalence also provides a set of real-world quasiperiodic materials which can be readily experimentally examined. We propose an experimental test of both the quasicrystalline and topological character of these systems in the form of a quantized adiabatic particle transport upon dragging the charge-ordered state.