Integer Quantum Hall Current

[Kirjava]

Hi PF,

Hoping somebody out there can help me to clear up what is probably a silly misunderstanding of the IQHE:

Since the quantized Hall current can be expressed as a property of occupied bulk bands (Chern number) why do we say that the current is carried by the edge states?

 

[Kirjava]

To clarify, is the Hall current shared between bulk and edge states in such a way as to preserve its precise quantization?

 

[Physics Monkey]

It won't surprise you to learn that this is a tricky question.

In short, yes, at the end of the day you can use the edge picture to safely compute the current in standard experiments even if some of the current is flowing in the bulk. Here is a simple (probably too simple) argument: consider a simply connected quantum Hall fluid and attach two metallic contacts to the edge. Now draw any curve you want that cuts the Hall fluid in half with one contact on each side. The total current through this curve, assuming a static flow (no charge build up), is the current from one contact to the other. This current is $I= \int_\gamma \hat{n}\cdot j$ where $\gamma$ is the curve, $\hat{n}$ is a normal to the curve, and $j$ is the current density. Now using $j = \sigma_{xy} \epsilon E$ with $\epsilon^{ab}$ the anti-symmetric tensor, we see that $\epsilon \hat{n} = \hat{t}$ with $\hat{t}$ the unit tangent. But then we just have $I = \sigma_{xy} \int \hat{t} \cdot E = \sigma_{xy} (V_1 - V_2)$ where $V_i$ are the voltages at the places where the curve hits the boundary of the Hall fluid. The fact that the Hall conductivity is a topological response suggests that it is indeed valid to assume it is constant everywhere provided the bulk gap remains finite.

However, its also important to understand the topology of current measurements. Imagine a Hall system that consists of an annulus of inner radius A and outer radius B.

1.
The standard current measurement performed on quantum Hall states amounts to putting your metallic leads all on B or all on A (physically A is usually zero so that we actually have a disk). In these cases you can measure a current and verify that it has no dissipation in the sense that the chemical potential (or voltage) is constant along the current.

For example, consider the following sequence of probes. A metallic contact at chemical potential $\mu_1$ is placed at $r=B,\theta=0$ and another metallic contact at chemical potential $\mu_2$ is placed at $r=B,\theta=\pi$. We also measure the voltage between $r=B,\theta=\pi/2$ and $r=B,\theta=3\pi/2$. If you now look carefully at these arrangements you'll see that the current sources/sinks (the metallic contacts) have a special topological relationship with the voltage probes. In this case you will measure a net current of $\sigma_{xy} (\mu_1 - \mu_2)$ flowing between the metallic contacts while the voltage measurement will see $\mu_1 - \mu_2$. Furthermore, this will remain true no matter where on the edge we choose to measure the voltage provided we maintain the ordering of the contacts and voltage probes i.e. contact probe contact probe.

This measurement is clearly at least partially probing the edge states.

2.
A way to more directly access the bulk conductivity is consider current from B to A instead of from B to B. If you put one contact on B and one on A in our situation above, you will typically measure no current even if $\mu_A \neq \mu_B$because the Hall state is an insulator. However, you can measure a current provided you pump flux through the center of the annulus. This will, be elementary electromagnetism, produce a circulating electric field that by virtue of the Hall conductivity lead to a radial flow of charge. Note that the contacts are still important otherwise one would build up a charge imbalance at A and B.

You can look up papers by Thouless and others for more information e.g. http://prb.aps.org/abstract/PRB/v35/i5/p2188_1 where he considers in some detail the structure of the edge. There are also other works using electro-optical methods, for example, to try to actually image where the current is flowing. I'll see if I can dig these up.