Quantum geometry - some intuition

There has been a great growing interest in quantum geometry in recent years.  Last week, I heard an excellent talk by Raquel Queiroz about this that gave me a more physically intuitive interpretation  of this topic.  The more formal write-up is in this preprint from this past April, which I'd missed at the time.

Caution:  Math incoming.  I will try to give a more physical picture at the end.  I know that this won't be very readable to non-experts.    

As I've written before,  (e.g. here and a bit here), the electronic states in crystalline solids are often written as Bloch waves of the form \(u_{n\mathbf{k}}(\mathbf{r})\exp(i \mathbf{k}\cdot \mathbf{r})\), where \(u_{n\mathbf{k}}(\mathbf{r})\) is periodic in the spatial period of the crystal lattice.  For many years, the \(\mathbf{k}\) dependence of \(u_{n\mathbf{k}}(\mathbf{r})\) was comparatively neglected, but now it is broadly appreciated that this is the root of all kinds of interesting physics, including the anomalous Hall effect and its quantum version.  

We can compute how much \(u_{n\mathbf{k}}(\mathbf{r})\) changes with \(\mathbf{k}\).  The Berry connection is related to the phase angle racked up by moving around in \(\mathbf{k}\), and it's given by \( \mathbf{A}(\mathbf{k}) = i \langle u_{n\mathbf{k}}| \nabla_{\mathbf{k}}| u_{n\mathbf{k}} \rangle \).  One can define \(\mathbf{\Omega} \equiv \nabla \times \mathbf{A}(\mathbf{k})\) as the Berry curvature, and the "anomalous velocity" is given by \(-\dot{\mathbf{k}}\times \mathbf{\Omega}\).  

If we worry about possible changes in the magnitude as well, and \( |\langle u_{n\mathbf{k}}| u_{n\mathbf{k+dk}} \rangle |^{2} = 1 - g^{n}_{\mu \nu}dk_{\mu}dk_{\nu}\) plus higher order terms.  The quantity \(g^{n}_{\mu \nu}\) is the quantum metric, and it can be written in terms of dipole operators:  \(g^{n}_{\mu \nu}= \sum_{m\ne n}\langle u_{n,\mathbf{k}}|\hat{r}_{\mu}|u_{m \mathbf{k}}\rangle \langle u_{m,\mathbf{k}}|\hat{r}_{\nu}|u_{n \mathbf{k}}\rangle\).  The quantum metric quantifies the "distance between" the Bloch states as one moves around in \(\mathbf{k}\).  

That last bit is what I really learned from the talk.  Basically, if you try to consider electrons localized to a particular lattice site in real space, this can require figuring in states in multiple bands, and the matrix elements involve dipole operators.  The quantum geometric tensor \(g_{\mu \nu}\) quantifies the dipole fluctuations in the electronic density.  You can define a lengthscale \(\ell_{g}\equiv \sqrt{Tr g}\), and this can tell you about the spatial scale of polarization fluctuations relative to, e.g., the lattice spacing.  Metals will have essentially divergent fluctuation lengthscales, while insulators have nicely bound charges (that give peaks in the optical conductivity at finite frequency).   The quantum geometry then influences all kinds of experimentally measurable quantities (see here).  

Neat stuff.  Someday I'd like to return to this with a nice cartoon/animation/presentation for non-experts.  The idea that there is so much richness within even relatively "boring" materials still amazes me.