Introduction

Ashvin Vishwanath from the University of California, Berkeley will introduce Weyl semimetals and other examples of gapless, yet topological, systems.

Topological invariants of Fermi surfaces

The idea that leads us to the topology of gapless systems is extremely simple. It is:

If we consider momentum as an external conserved parameter, we can study topological closings of the gap in momentum space.

Let's consider the simplest type of topological invariant, one we learned about at the very beginning of this course. Remember the simplest topological invariant of a 0D Hamiltonian, the number of filled energy levels? What if we take two points in momentum space, $\mathbf{k}_1$ and $\mathbf{k}_2$, and consider a Hamiltonian such that the number of filled states changes by $n$ between these two points? We can conclude that there are at least $n$ Fermi surfaces that lie on every path between $\mathbf{k}_1$ and $\mathbf{k}_2$ in momentum space.

Now we just need to take this idea and apply it to more interesting systems and topological invariants!

What types of topological invariants are relevant? Aside from special circumstances, we cannot make use of time-reversal or particle-hole symmetries: in momentum space these only have an immediate effect in isolated $\mathbf{k}$-points, where every momentum component is either $0$ or $\pi$. So there are no paths in momentum space for which either of the symmetries is effective in each point.

So we are left with only two symmetry classes: A and AIII (no symmetry at all or sublattice/chiral symmetry), and with only two invariants: if there is a sublattice symmetry, a winding number can be defined, and without it there's a Chern number.

Graphene and protected Dirac cones

We've already analysed the 0D Chern number that stabilizes the usual Fermi surfaces. Let's go one dimension higher, and study winding numbers in systems with sublattice symmetry around 1D loops.

For a winding number to be nonzero, we need to consider 1D loops in momentum space. As a reminder, with sublattice symmetry the Hamiltonian can always be brought to the form

$$ H = \begin{pmatrix} 0 & h(\mathbf{k}) \\ h^\dagger(\mathbf{k}) & 0 \end{pmatrix} $$

The topological invariant is a nonzero winding of $\det h(\mathbf{k})$ when $\mathbf{k}$ goes around some contour. Since $h(\mathbf{k})$ is continuous, this means that its determinant also has to vanish somewhere inside this contour.

To study a particular example where this appears, let's return to graphene, which we studied as a simple limit of Haldane model. For graphene we have the Hamiltonian

$$h(k_x, k_y) = t_1 e^{i \mathbf{k} \cdot \mathbf{a_1}} + t_2 e^{i \mathbf{k} \cdot \mathbf{a_2}} + t_3 e^{i \mathbf{k} \cdot \mathbf{a_3}},$$

where $t_1, t_2, t_3$ are the three hoppings connecting a site in one of the two graphene sublattices, and $a_1, a_2, a_3$ are the lattice vectors connecting one unit cell to its neighbors.

To consider something specific, let's take $t_2 = t_3 = t$ and vary $t_1$. This is what the band structure and $\det h$ look like: