Anomalies and Kinetic Theory

Abstract

In this chapter, we will make an overview of the quantum anomalies, as quantities that are no longer conserved when passing from the classical to the quantum realm. We will focus on the chiral anomaly. The discussion will be made in terms of the semiclassical kinetic theory, where the classical Boltzmann transport equation is supplemented by the equations of motion that explicitly contain the Berry connection. In this regard, we will make explicit the connection between the chiral anomaly and the non-trivial topological structure of Weyl semimetals. We will make the discussion beyond the different relaxation time approaches that are commonly used in the literature. This approach introduces some mathematical complexities but also reveals some less known features of transport in Weyl semimetals. Finally, we will discuss other quantum anomalies that have been of interest recently in Condensed Matter Physics.

Appendix

In this appendix, I will explain some technicalities about the way to solve inhomogeneous Fredholm integral equations of second kind. General approaches, like the resolvent formalism, are based on iterative series, but when the integral kernel is separable (as the ones we find when we solve the Boltzmann equation in series of $\alpha$), the problem can be reduced to an algebraic one.

Let us consider the following integral equation, where g(x) is the function we want to find, and satisfies the following equation:

$$\begin{aligned} g(x)=f(x)+\int ^{b}_{a}\mathrm{d}x^{\prime }K(x,x^{\prime })g(x^{\prime }). \end{aligned}$$

(7.59)

The kernel $K(x,x^{\prime })$ is separable when we can write it as a (finite) sum of products of functions of x and $x^{\prime }$:

$$\begin{aligned} K(x,x^{\prime })=\sum _{n}P_{n}(x)Q_{n}(x^{\prime }). \end{aligned}$$

(7.60)

Putting this expression back into our equation, we get

$$\begin{aligned} g(x)=f(x)+\sum _{n}P_{n}(x)\int ^{b}_{a}\mathrm{d}x^{\prime }Q_{n}(x^{\prime })g(x^{\prime }), \end{aligned}$$

(7.61)

and write $b_{n}\equiv \int ^{b}_{a}\mathrm{d}x^{\prime }Q_{n}(x^{\prime })g(x^{\prime })$, so

$$\begin{aligned} g(x)=f(x)+\sum _{n}P_{n}(x)b_n. \end{aligned}$$

(7.62)

If we now multiply both sides by $Q_{m}(x)$ and integrate, we have

$$\begin{aligned} b_m=\int ^{b}_{a}\mathrm{d}x Q_{m}(x)f(x)+\sum _{n}\int ^{b}_{a}\mathrm{d}x Q_{m}(x)P_{n}(x)b_n. \end{aligned}$$

(7.63)

If we define the parameters $c_{m}=\int ^{b}_{a}\mathrm{d}x Q_{m}(x)f(x)$ and $a_{mn}=\int ^{b}_{a}\mathrm{d}x Q_{m}(x)P_{n}(x)$, we get an algebraic set of equations

$$\begin{aligned} b_m=c_{m}+\sum _{n}a_{mn}b_n, \end{aligned}$$

(7.64)

or

$$\begin{aligned} \sum _{n}(\delta _{mn}-a_{mn})b_n=c_{m}. \end{aligned}$$

(7.65)

Once we obtain the coefficients $b_{m}$, we can go back to (7.62) and plug them into the expression for g(x). It is clear that if the matrix $a_{mn}$ has at least one eigenvalue equal to one, the algebraic equation (and therefore the integral equation) has no solutions, provided that the corresponding element of the vector $\mathbf {c}$ is nonzero. This result, when properly stated, is known as the Fredholm alternative [38].