The chiral edge states can be described in such simple terms, that you might doubt the fact that they cannot exist without a bulk. After all, couldn’t we just build a theory of a single chiral edge state, neglecting the bulk of the quantum Hall system and the existence of the other edge? Can’t a truly one-dimensional system just show the same behavior of the chiral edge state of the quantum Hall effect?

Let’s consider the equation \(E=\hbar v (k-k_F)\) which describes these chiral states. We can imagine that a constant electric field \(\mathcal{E}\) can be applied along the edge, pallel to the momentum \(k\). (In the Hall cylinder, this can be done by threading a time-dependent flux through the cylinder, as you have seen in the previous part of the lecture).

The momentum \(k\) changes according to the equation \(\hbar \dot{k} = -e\mathcal{E}\). After a time \(t\), the energy of a state with momentum \(k\) has changed to \(\hbar v (k - k_F - e\mathcal{E}t/\hbar)\). This increase corresponds to a time-dependent shift of the Fermi momentum, \(k_F\,\to\,k_F + e\mathcal{E}t/\hbar\). Recall that \(k_F = 2\pi N/L\) where \(N\) is the number of electrons, so the rate of change of \(k_F\) gives

\[\dot{N} = \mathcal{E}L/\Phi_0,\]

with \(\Phi_0=h/e\) a flux quantum! Since the number of electrons is changing, charge is not conserved. In particular, after a time such that \(\mathcal{E}L t = \Phi_0\), it seems that exactly one electron has popped out of nowhere at the edge.

At this point, you should understand what’s happening. This is just how the Laughlin pump manifests itself if you only look at one edge. The number of electrons at one edge can increase, because electrons are being depleted from the other edge (which is not included in our “theory”) and pumped through the bulk until they appear.

This property of the edge is referred to as the chiral anomaly. The chiral anomaly tells us that we cannot have a consistent theory for a chiral edge state without a bulk, which at the same time conserves electric charge. Chiral edge states, or anything else that exhibits a chiral anomaly, are an example of the bulk edge correspondence, since they can only appear at the edge of a two dimensional system and never in isolation.

We called the anomaly “harmless” since the non-conservation of charge at the edge has a very simple explanation when the rest of the system is included in the picture. If you ever encounter other “anomalous” theories, it might well be a sign that the system under consideration is only the edge of something else!