One thing that physics and mechanical engineering students learn early on is that there are often analogies between charge flow and heat flow, and this is reflected in the mathematical models we use to describe charge and heat transport. We use Ohm's law, \(\mathbf{j}=\tilde{\sigma}\cdot \mathbf{E}\), which defines an electrical conductivity tensor \(\tilde{\sigma}\) that relates charge current density \(\mathbf{j}\) to electric fields \(\mathbf{E}=-\nabla \phi\), where \(\phi(\mathbf{r})\) is the electric potential. Similarly, we can use Fourier's law for thermal conduction, \(\mathbf{j}_{Q} = - \tilde{\kappa}\cdot \nabla T\), where \(\mathbf{j}_{Q}\) is a heat current density, \(T(\mathbf{r})\) is the temperature distribution, and \(\tilde{\kappa}\) is the thermal conductivity.

We know from experience that the electrical conductivity really has to be a tensor, meaning that the current and the electric field don't have to point along each other. The most famous example of this, the Hall effect, goes back a long way, discovered by Edwin Hall in 1879. The phenomenon is easy to describe. Put a conductor in a magnetic field (directed along \(z\)), and drive a (charge) current \(I_{x}\) along it (along \(x\)), as shown, typically by applying a voltage along the \(x\) direction, \(V_{xx}\). Hall found that there is then a *transverse* voltage that develops, \(V_{xy}\) that is proportional to the current. The physical picture for this is something that we teach to first-year undergrads: The charge carriers in the conductor obey the Lorentz force law and curve in the presence of a magnetic field. There can't be a net current in the \(y\) direction because of the edges of the sample, so a transverse (\(y\)-directed) electric field has to build up.

There can also be a thermal Hall effect, when driving heat conduction in one direction (say \(x\)) leads to an additional temperature gradient in a transverse (\(y\)) direction. The least interesting version of this (the Maggi–Righi–Leduc effect) is in fact a consequence of the regular Hall effect: the same charge carriers in a conductor can carry thermal energy as well as charge, so thermal energy just gets dragged sideways.

Surprisingly, *insulators* can also show a thermal Hall effect. That's rather unintuitive, since whatever is carrying thermal energy in the insulator is not some charged object obeying the Lorentz force law. Interestingly, there are several distinct mechanisms that can lead to thermal Hall response. With phonons carrying the thermal energy, you can have magnetic field affecting the scattering of phonons, and you can also have intrinsic curving of phonon propagation due to Berry phase effects. In magnetic insulators, thermal energy can also be carried by magnons, and there again you can have Berry phase effects giving you a magnon Hall effect. There can also be a thermal Hall signal from topological magnon modes that run around the edges of the material. In special magnetic insulators (Kitaev systems), there are thought to be special Majorana edge modes that can give quantized thermal Hall response, though non-quantized response argues that topological magnon modes are relevant in those systems. The bottom line: thermal Hall effects are real and it can be very challenging to distinguish between candidate mechanisms.

(Note: Blogger now compresses the figures, so click on the image to see a higher res version.)