This paper begins with a summary of a powerful formalism for the study of electronic states in condensed matter physics called “gauge theory of states/phases of matter.” The chiral anomaly, which plays quite a prominent role in that formalism, is recalled. I then sketch an application of the chiral anomaly in 1 + 1 dimensions to quantum wires. Subsequently, some elements of the quantum Hall effect in two-dimensional (2D) gapped (“incompressible”) electron liquids are reviewed. In particular, I discuss the role of anomalous chiral edge currents and of the anomaly inflow in 2D gapped electron liquids with explicitly or spontaneously broken time reversal, i.e., in Hall and Chern insulators. The topological Chern–Simons action yielding transport equations valid in the bulk of such systems and the associated anomalous edge action are derived. The results of a general classification of “Abelian” Hall insulators are outlined. After some remarks on induced Chern–Simons actions, I sketch results on certain 2D chiral photonic wave guides. I then continue with an analysis of chiral edge spin-currents and bulk response equations in time-reversal invariant 2D topological insulators of electron gases with spin–orbit interactions. The “chiral magnetic effect” in 3D systems and axion-electrodynamics are reviewed next. This prepares the ground for an outline of a general theory of 3D topological insulators, including “axionic insulators.” Some remarks on Weyl semi-metals, which exhibit the chiral magnetic effect, and on Mott transitions in 3D systems with dynamical axion-like degrees of freedom conclude this review.
See also Ref. 82 for some mathematical aspects of the theory of edge currents in non-interacting electron liquids.
I offer my apologies to all those colleagues whose contributions may not be properly highlighted in my paper, due to my lack of knowledge of their work or oversights, or because they are too recent to be included in this paper; (I am not familiar with very recent results on topological states of matter). Results concerning topologically protected states of systems of non-interacting quantum particles are neither reviewed, nor referred to in the following; but see Refs. 39 and 83. They will surely be discussed extensively in other contributions to this volume.
A fairly extensive list of references to relevant work will be given later in the text.
Caused, e.g., by an external magnetic field or by the vorticity of a velocity field that generates the motion of an ionic background
These formulae can be traced back to Tomonaga’s work on 1D electron liquids.
In classical physics, the role of is played by .
Instead, we could consider a very large Corbino disk and study charge transport from the inner to the outer boundary circle when the magnetic flux threaded through its hole changes.
States exhibiting quasi-particles with non-abelian braid statistics are discussed in my work36 with Pedrini, Schweigert and Walcher.
The same conclusion is reached by noticing that all Wilson loop expectations have perimeter decay and then invoking “ ’tHooft duality.”
A general reference where the role of electron spin and spin currents and the spin Hall effect are analyzed is Ref. 86.
Often dubbed “Berry connection.”
At level 1, for non-interacting electrons.
In this section we use units such that ℏ = 1.
I am indebted to H.-G. Zirnstein for instructive discussions of this point.
I am indebted to Greg Moore for a very instructive discussion of this effect.
This follows from the celebrated Nielsen-Ninomiya theorem; see Ref. 88 and references given there.
To read, to mention but one example (see Ref. 89), that it is only during the past decade that the chiral anomaly has been used in the study of transport phenomena in condensed matter physics makes me wonder whether communication channels within the scientific community are congested (to put it politely). Of course, there are also examples of this congestion in the mathematical physics community. – In all modesty I claim some credit for my collaborators and myself for having developed a useful formalism, emphasizing connections to particle physics and quantum field theory including gauge theory and the chiral anomaly, for the study of “topological states of matter,” at least thirty years ago; see Refs. 1–10 and 77.
A technique that owes some of its early momentum to a paper90 containing the first mathematically rigorous proof of existence of the Berezinskii-Kosterlitz-Thouless transition.