Abstract

Sebas Eliens (University of Amsterdam)


Observables and topological symmetry breaking phase transition

Topological phases are receiving a great deal of attention from the condensed matter community and the first measurements of Majorana fermions in a quantum wire and non-abelian statistics in a fractional quantum Hall effect show the promise of the exciting new physics these phases have to offer. For planar systems, like a two dimensional electron gas showing the fractional quantum Hall effect, the topological order is characterized by a theory of Anyons, quasi particles with nontrivial exchange statistics that unlike bosons or fermoins can have spins of any fractional number, not just integer or half integer. A driving force in the research in this field is the proposed application of anyons to store and manipulate quantum information, a scheme known as topological quantum computation. An abstract but powerful way of thinking about anyonic charge is in terms of representations of a quantum group, which can often be shown to emerge as the full symmetry of the quantum theory. This approach has the benefit that it highlights the general features and algebraic structure - for example, it automatically takes into account the braid statistics of the anyons - but of course the relation to the microscopic origin becomes opaque. However, as a very interesting possibility it suggests a mechanism for phase transitions between topological phases hosting anyons by means of breaking the quantum group symmetry by the formation of a 'bosonic' condensate. This mechanism is known as topological symmetry breaking. In this talk, I will address the the fate of observables under a topological symmetry breaking phase transition. It turns out that one can do arbitrary diagrammatic calculations of observables in the broken phase, provided a finite number of coefficients are determined from consistency conditions on the condensate. In particular, this enables the direct calculation of the topological S matrix in the condensed phase, which is known to almost completely specify the topological order, and thus serves as a good (nonlocal) order parameter. It also relates to interferometry experiments with anyons. In lattice simulation of so called Discrete Gauge Theories, the condensation of bosonic charges can be induced by tuning the parameters of the model. When 'measuring' the S-matrix in these simulations, the same physical picture emerges. In particular, the coefficients appear to relate to a hidden symmetry that effects the observables of the theory only in nontrivial vacua.