Topological States and Adiabatic Pumping in Quasicrystals

We find a connection between quasicrystals and topological matter, namely that quasicrystals exhibit non-trivial topological phases attributed to dimensions higher than their own [1]. Quasicrystals are materials which are neither ordered nor disordered, i.e. they exhibit only long-range order [2]. This long-range order is usually expressed as a projection from a higher dimensional ordered system. Recently, the unrelated discovery of Topological Insulators [3] defined a new type of materials classified by their topology. We show theoretically and experimentally using photonic lattices, that one-dimensional quasicrystals exhibit topologically-protected boundary states equivalent to the edge states of the two-dimensional Integer Quantum Hall Effect. We harness this property to adiabatically pump light across the quasicrystal, and generalize our results to higher dimensional systems. Hence, quasicrystals offer a new platform for the study of topological phases while their topology may better explain their surface properties.\\[4pt] [1] Y.~E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, arXiv:1109.5983 (2011).\\[0pt] [2] C. Janot, \textit{Quasicrystals} (Clarendon, Oxford, 1994), 2nd ed.\\[0pt] [3] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. \textbf{82}, 3045 (2010).