# 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.

### 4 responses to “Modern Theory of Polarization”

1. Hi Anshul,
This is one of my favorite subjects in condensed matter physics, because it really tests our understanding of several concepts at once!

In my mind, an important realization (that took me a long time to get) is that all this formalism is necessary only because we model solid materials as periodic systems. This isn’t actually correct; as you know, your materials have finite extent. So if we had an enormous computer and could just simulate the mm-sized chunk of material directly, we could calculate polarization in the normal way and there would be no ambiguities and no Berry phases.

There’s another paper, which is very mathematical, but interesting once you get it here: http://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.1666 . They explain the problem from a very different angle that I like. When we use periodic boundary conditions to simulate a bulk solid, we actually cut out some of our Hilbert space compared to the open boundary situation. Part of that ‘lost’ Hilbert space includes the result of operating X on a periodic wave function! So if we’d like to get at any expectation values of X, then we have to approach it in a different way. The Berry phases come about from doing this for one-body wave functions.

Cheers,

Like

• Anshul

Hi Lucas,

Yes, you are absolutely correct. It seems to me like when calculating the Berry phase in solids, even more generally, like in the Quantum Hall Effect, there is the tacit assumption of periodic boundary conditions. This of course has to be relaxed to obtain the edge states.

Thanks for the paper, I will take a look at it when I get the chance — a brief glance through looks like it may not be too far over my head. 8^)

Like

2. Piotr

Hi,
the introduction by Spaldin seems to be good, however, I am a little confused about this. How can you choose unit cell to obtain resultant polarization equal to 3/2 or 5/2??
Also, if you choose unit cell to begin at one of the ions, the polarization will be equal to zero in the non-polar chain, unlike Spaldin claims that it is …-5/2, -3/2, -1/2, 1/2, 3/2, 5/2…

Like