# Tag Archives: Geometry

## Modern Theory of Polarization

It is quite curious that the simple concept of polarization in a solid was not understood until the early to mid-90s. The solution to the problem actually came from the computational physics community because of their inability to calculate accurately the polarization in solids. Prior to the solution, there were papers seriously discussing whether polarization was a bulk phenomenon or whether it was a property of a crystal surface. The solution to the problem is commonly associated with these papers by Resta (pdf!) and King-Smith and Vanderbilt (pdf!) .

There are a few startling realizations that arose from the modern theory of polarization:

1. Polarization is not a well-defined quantity in that it is multi-valued.
2. Only polarization difference has any physical meaning.
3. Experiments only measure differences in polarization.
4. Polarization is deeply rooted in the concept of the Berry phase.
5. (not startling) Polarization is a bulk phenomenon.

There is a fantastic pedagogical introduction to the subject by Spaldin entitled A beginner’s guide to the modern theory of polarization.

As just a little preview of the Spaldin paper, let me outline the issues as she does. The problem with defining polarization as the dipole moment per unit cell can easily be understood using the picture of the one-dimensional chain below:

Using the box on the left, one would calculate the dipole moment per unit length as:

$p =\frac{e}{a}\sum_i q_id_i = \frac{e}{a}(\frac{a}{4}\times -1 + \frac{3a}{4}\times 1) = \frac{e}{2}$

whereas the box on the right gives:

$p =\frac{e}{a}\sum_i q_id_i = \frac{e}{a}(\frac{a}{4}\times 1 + \frac{3a}{4}\times -1) = \frac{-e}{2}$.

That these don’t match has to do precisely with the fact that polarization in a solid is multi-valued, as alluded to earlier. Now consider the following one-dimensional lattice with a distortion:

The polarization now for the distorted lattice (second row in the image above) for the left and right boxes are respectively:

Left Box: $p = \frac{e}{2} +\frac{ed}{a}$

Right Box: $p = \frac{-e}{2} +\frac{ed}{a}$

We can therefore see that for both boxes, the change in polarization is $\delta p = \frac{ed}{a}$, which is a single-valued and experimentally well-defined quantity.

While this illustration was classical, when one includes the wavefunction of the electrons, one is forced to consider the Berry phase of the Bloch electrons. While I have known about this result for some time now, I still find it quite surprising that the simple concept of polarization in a solid has any relationship to the Berry phase at all. I strongly recommend Spaldin’s eminently readable article as an excellent introduction to the subject.

## It’s Not Just a (Berry) Phase

Just a few years after Berry’s breakthrough paper (pdf!) on the phase now named after him, it was recognized by Zak (pdf!) that this effect could play a role in a solid state setting. Zak realized that in the Bloch Hamiltonian, the crystal momentum, k, could be treated as a “parameter” similar to how other “parameters” had been treated in Berry’s original work. More concretely:

$H(\textbf{k}) = e^{-i\textbf{k}\cdot r}He^{i\textbf{k}\cdot r}$

While phenomena that are now considered prototypical Berry phase effects were understood before the Berry paper, it was Berry who unified many concepts under a general framework. For instance, the Aharonov-Bohm effect, the Dirac monopole problem and the Integer Quantum Hall Effect were all understood separately, but the Berry phase concept presented a huge leap forward in consolidating these seemingly disparate phenomena.

It should be mentioned that the three cases mentioned above are all instances where the Berry phase is topological and results in a robust quantization through a topological invariant (known as the Chern number). However, the Berry phase does not have to be topological in nature and can have “geometrical” consequences. Many of these “geometrical” consequences are discussed in a great review article by Xiao, Chang and Niu.

One particular “geometrical” consequence stands out: the semiclassical equations of motion for electrons in a solid, as detailed in textbooks such as Ashcroft and Mermin, are incomplete! The application of an electric field in crystals with either broken time-reversal symmetry or broken inversion symmetry may, under certain circumstances, exhibit a transverse velocity associated with a non-zero Berry curvature (i.e. an application of an electric field in the x-direction will result in electrons travelling in the y-direction!). This experimental paper by NP Ong’s group demonstrates this effect (pdf!), and this paper by Karplus and Luttinger (paywall!) theoretically explains the transverse velocity in ferromagnets, again before Berry’s seminal paper.

I’ll also mention briefly that the seemingly mundane phenomenon of electrical polarization in solids is another case that cannot be explained without Berry-esque concepts, but I’ll leave that topic for another day.

It truly is stunning how many effects can be accounted for under the umbrella of Berry/Zak ideas. I’ll even restrain myself from mentioning topological insulators and Chern insulators (paywall) / the quantum anomalous hall effect (paywall).