Topological states in quasicrystals

We find a connection between quasicrystals and topological matter, namely establishing that quasicrystals exhibit non-trivial topological phases attributed to dimensions higher than their own. Quasicrystals are materials which are neither ordered nor disordered, i.e. they exhibit long-range order only. Recently, the unrelated discovery of Topological Insulators defined a new type of materials classified by their topology. We show that the one-dimensional Harper and Fibonacci quasicrystals are topologically nontrivial. As a result they exhibit topologically-protected boundary states equivalent to the edge states of the two-dimensional Integer Quantum Hall Effect. Additionally, topological bulk phase transitions occur between topologically nonequivalent quasicrystals. We present experiments using photonic lattices, which harness the resultant boundary phenomena to adiabatically pump light across the quasicrystal, and study the bulk phase transitions using a smooth boundary between different quasicrystals.