Seeing Topological Order

Abstract

Different phases of matter can be distinguished by their symmetries. This information is captured by order parameters that summarize the essential properties of the phase. Order parameters are usually defined in terms of local operators that can be measured in the laboratory. Topological insulators are materials with symmetries that depend on the topology of the energy eigenstates of the system. These materials are of interest because they can give rise to robust spin transport effects with potential applications ranging from sensitive detectors to quantum computation. However, direct measurement of topological order has been up to now impossible due to its non-local character. In this work we provide a general methodology to perform a direct measurement of topological order in cold atom systems. As an application we propose the realisation of a characteristic topological model, introduced by Haldane, using optical lattices loaded with fermionic atoms in two internal states. We demonstrate that time-of-flight measurements directly reveal the topological order of the system in the form of momentum space skyrmions.