Abstract
Starting with a description of the motivation underlying the analysis
presented in this paper and a brief survey of the chiral anomaly, I proceed to
review some basic elements of the theory of the quantum Hall effect in 2D
incompressible electron gases in an external magnetic field, ("Hall
insulators"). I discuss the origin and role of anomalous chiral edge currents
and of anomaly inflow in 2D insulators with explicitly or spontaneously broken
time reversal, i.e., in Hall insulators and "Chern insulators". The topological
Chern-Simons action yielding the large-scale response equations for the 2D bulk
of such states of matter is displayed. A classification of Hall insulators
featuring quasi-particles with abelian braid statistics is sketched.
Subsequently, the chiral edge spin currents encountered in some time-reversal
invariant 2D topological insulators with spin-orbit interactions and the bulk
response equations of such materials are described. A short digression into the
theory of 3D topological insulators, including äxionic insulators", follows
next. To conclude, some open problems are described and a problem in cosmology
related to axionic insulators is mentioned. As far as the quantum Hall effect
and the spin currents in time-reversal invariant 2D topological insulators are
concerned this review is based on extensive work my collaborators and I carried
out in the early 1990's.
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