General relativity (!) and band structure

Today we had a seminar at Rice by Qian Niu of the University of Texas, and it was a really nice, pedagogical look at this paper (arxiv version here).   Here's the basic idea.

As I wrote about here, in a crystalline solid the periodic lattice means that single-particle electronic states look like Bloch waves, labeled by some wavevector \(\mathbf{k}\), of the form \(u_{\mathbf{k}}(\mathbf{r}) \exp(i \mathbf{k}\cdot \mathbf{r})\) where \(u_{\mathbf{k}}\) is periodic in space like the lattice.  It is possible to write down semiclassical equations of motion of some wavepacket that starts centered around some spatial position \(\mathbf{r}\) and some (crystal) momentum \(\hbar \mathbf{k}\).  These equations tell you that the momentum of the wavepacket changes with time as due to the external forces (looking a lot like the Lorentz force law), and the position of the wavepacket has a group velocity, plus an additional "anomalous" velocity related to the Berry phase (which has to do with the variation of \(u_{\mathbf{k}}\) over the allowed  values of \(\mathbf{k}\)).

The paper asks the question, what are the semiclassical equations of motion for a wavepacket if the lattice is actually distorted a bit as a function of position in real space.  That is, imagine a strain gradient, or some lattice deformation.  In that case, the wavepacket can propagate through regions where the lattice is varying spatially on very long scales while still being basically periodic on shorter scales still long compared to the Fermi wavelength.

It turns out that the right way to tackle this is with the tools of differential geometry, the same tools used in general relativity.  In GR, when worrying how the coordinates of a particle change as it moves along, there is the ordinary velocity, and then there are other changes in the components of the velocity vector because the actual geometry of spacetime (the coordinate system) is varying with position.  You need to describe this with a "covariant derivative", and that involves Christoffel symbols.  In this way, gravity isn't a force - it's freely falling particles propagating as "straight" as they can, but the actual geometry of spacetime makes their trajectory look curved based on our choice of coordinates.

For the semiclassical motion problem in a distorted lattice, something similar happens.  You have to worry about how the wavepacket evolves both because of the local equations of motion, and because the wavepacket is propagating into a new region of the lattice where the \(u_{\mathbf{k}}\) functions are different because the actual lattice is different (and that also affects the Berry phase anomalous velocity piece).   Local rotations of the lattice can lead to an affective Coriolis force on the wavepacket; local strain gradients can lead to effective accelerations of the wavepacket.

(For more fun, you can have temporal periodicity as well.  That means you don't just have Bloch functions in 3d, you have Bloch-Floquet functions in 3+1d, and that's where I fell behind.)

Bottom line:  The math of general relativity is an elegant way to look at semiclassical carrier dynamics in real materials.  I knew that undergrad GR course would come in handy....