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Topological phases, quantized transport and edge states in open lattice models

Michael Fleischhauer

Since the discovery of the quantum Hall effect, topological states of matter have attracted the attention of scientists in many fields of physics. It was long believed that all phases of matter can be classified by spontaneously broken symmetries described by local order parameters. Topological phases do not fit into this scheme but have to be characterized by non-local integer invariants. The existence of these invariants leads to a number of fascinating properties such as protected edge states or edge modes and exotic elementary excitations such as anyons.
By now there is a rather good understanding of topological order for non-interacting systems. An exhaustive classification of gapped topological states of non-interacting fermions (the “ten-fold way”) can be achieved solely on the basis of general symmetry properties of the Hamiltonian upon time reversal, particle-hole exchange and sub-lattice transformation. In contrast, our understanding of topology in interacting many-body systems is very limited and the extension to open system is entirely in its infancy. The goal of the present theoretical project is to shed new light onto topology in open systems both with and without interactions. In closed systems topological invariants such as the Zak phase in one spatial dimension or the Chern number in two dimensions are related to the single-particle bandstructure or their generalizations to the many-body wavefunction of a fixed number of particles. A generalization to open systems, where the particle number is no longer conserved, is not yet established. In this project we aim at developing an understanding of the notion of topology in open systems. To this end we will search for topological phases in open systems and expect to identify invariants or other suitable measures of topology. We will study analogues of topological signatures known from unitary systems, such as quantized charge pumps, edge states or edge currents for open systems and, using those as a basis, we will identify appropriate invariants. Most importantly we will investigate whether topological protection in closed systems can be generalized to perturbations such as particle losses. The ultimate goal is to combine open system control and topological protection to create interesting non-trivial quantum states or quantum degrees of freedom, and to achieve a higher degree of robustness and protection than known until now. Finally we will develop suitable experimental realisations of open topological systems


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