Gauge invariance and anomalies in condensed matter physics

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.

See Refs. 9, 21, and 22; they were later applied to studies in cold-atom physics – see Ref. 66 and references given there.

An effect discovered in a preliminary form in Ref. 64 and studied systematically in Ref. 8.

Functional- or path integral methods were originally introduced by Dirac in Ref. 84. An operator formalism to derive expressions for effective actions has been sketched elsewhere; see, e.g., Refs. 2 and 6, and references given there.

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 $eE⃗$ is played by $MV⃗̇+M2∇⃗|V⃗|2$⁠.

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.”

See, e.g., Ref. 85 for information about integral lattices and their invariants.

This has been done in unpublished work on QED₃, by J. Magnen, the late R. Sénéor and myself, in 1976. Explicit expressions were first published by Deser et al. in Ref. 49; see also Ref. 50.

A general reference where the role of electron spin and spin currents and the spin Hall effect are analyzed is Ref. 86.

See Refs. 4 and 6 for a more detailed discussion.

Note that if $W⃗$ is expressed in terms of $B⃗$ and $E⃗$⁠, as in (61), (63), and (64), one recovers expression (67). The effective action in (68) first appeared in a paper with Studer⁴ in 1993 !

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.

A conjecture proposed by S.-C. Zhang and coworkers,⁸⁷ inspired by the work in Ref. 9.

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.