Gauge theory and topological quantum matter
Bad Honnef Physics SchoolBericht
16 Sep - 21 Sep 2018
Physikzentrum Bad Honnef
Prof. Martin Zirnbauer, U Köln
Among the principles underlying the physics of charged matter, the gauge-invariant coupling to gauge fields stands out by its significance and power. Taking a geometric perspective, the Summer School starts with a thorough pedagogical introduction to the gauge principle in single-particle quantum mechanics. Then, in the wider context of many-body systems, we discuss the fact that non-relativistic condensed matter has, in addition to the well-known U(1) electromagnetic gauge symmetry, a local gauge invariance with respect to SU(2) spin transformations. Along the way, we review such concepts and phenomena as Berry curvature, Dirac quantization condition, spin structure, and the Aharonov-Casher effect.
The series of lectures then turns to the response of charged matter to external gauge fields (under the constraint of gauge invariance, or the lack thereof in the case of an anomaly) in the scaling limit. For gapped systems of the sort of a topological insulator, the effective action for the bulk response may have a gauge anomaly which is canceled by the corresponding anomaly of the boundary theory. This so-called bulk-boundary correspondence serves to classify different phases of topological quantum matter. The final goal of the School is to lead students to the frontier of current research, by developing the notion of symmetry-protected topological phases and their effective description by topological field theories.