Why do Weyl points come in pairs in solid state system?

Question

Quoted from "Beyond Band Insulators: Topology of Semi-metals and Interacting Phases" by Turner and Vishwanath:

In 3D lattice models, Weyl points always come in pairs of opposite helicity; this is the fermion doubling theorem.

Explanation from that paper (in my words):

The integral of the Berry flux of any sufficient small surface around a Weyl point is $\pm 2\pi$, depending on its chirality (helicity?). However, the total flux in Brillouin zone has to be zero. Therefore Weyl points must comes into pairs if they exist.

So why is the integral of the Berry flux around the Brillouin zone surface zero?

Answer 1

A simple argument: suppose our Brillouin zone is a cubic. If we have a Chern flux penetrating the upper surface, then the same flux with the same magnitude and direction will penetrate the lower surface. Same is true for the front and back, left and right surface. Therefore the total flux penetrating the whole BZ is zero.

This is caused by the periodicity of the Brillouin zone, the upper and lower surface of the Brillouin zone are the same, so they should have same Berry curvature, and hence the same Berry flux.

Answer 2

There are many ways of looking at this, summarized by a no-go theorem by Nielsen and Ninomiya. The doubling is caused by the introduction of lattice. Be it solid state or lattice gauge theory when a d-dimensional Lagrangian is discretized $2^d$ number of Weyl particles arise, they can be seen as left and right (chiral) moving particles. Among these the extra particles (also called Wilson fermions) have mass that is inversely proportional to lattice spacing, so under zero spacing, i.e. continuum, they get decoupled and the theory converge consistently to the continuum.

Now why do these Wilson fermions show up? One reason is that, the naive replacement of the continuum first order derivative by a discrete difference operator connecting two neighboring lattice sites. This substitution makes the various corners of the BZ energetically equivalent or a $2^d$-fold degeneracy in the spectrum shows up in the momentum space.

Another way to look at this is from anomaly point of view. There is nice discussion in Fujikawa's book on anomalies (Chap. 9). The continuum Dirac theory is anomaly free. The lattice formulation can be viewed as one way of regularizing it, but the lattice regulator is anomalous. However, the catch is it turns out that half of the $2^d$ modes have positive anomaly and the other half have negative anomaly. So by coming in pair they make sure the theory is anomaly free.