This post is an indirect follow-on from

here, and was spawned by a request that I discuss the "modern theory of polarization". I have to say, this has been very educational for me. Before I try to give a very simple explanation of the issues, those interested in some more technical meat should look

here, or

here, or

here, or at

this nice blog post.

Colloquially, an electric dipole is an overall neutral object with some separation between its positive and negative charge. A great example is a water molecule, which has a little bit of excess negative charge on the oxygen atom, and a little deficit of electrons on the hydrogen atoms.

Once we pick an origin for our coordinate system, we can define the electric dipole moment of some charge distribution as \(\mathbf{p} \equiv \int \mathbf{r}\rho(\mathbf{r}) d^{3}\mathbf{r}\), where \(\rho\) is the local charge density. Often we care about the *induced* dipole, the dipole moment that is produced when some object like a molecule has its charges rearrange due to an applied electric field. In that case, \(\mathbf{p}_{\mathrm{ind}} = \alpha \cdot \mathbf{E}\), where \(\alpha\) is the polarizability. (In general \(\alpha\) is a tensor, because \(\mathbf{p}\) and \(\mathbf{E}\) don't have to point in the same direction.)

If we stick a slab of some insulator between metal plates and apply a voltage across the plates to generate an electric field, we learn in first-year undergrad physics that the charges inside the insulator slightly redistribute themselves - the material *polarizes*. If we imagine dividing the material into little chunks, we can define the polarization \(\mathbf{P}\) as the electric dipole moment per unit volume. For a solid, we can pick some volume and define \(\mathbf{P} = \mathbf{p}/V\), where \(V\) is the volume over which the integral is done for calculating \(\mathbf{p}\).

We can go farther than that. If we say that the insulator is built up out of a bunch of little polarizable objects each with polarization \(\alpha\), then we can do a self-consistent calculation, where we let each polarizable object see both the externally applied electric field and the electric field from its neighboring dipoles. Then we can solve for \(\mathbf{P}\) and therefore the relative dielectric constant in terms of \(\alpha\). The result is called the

Clausius-Mossotti relation.

In crystalline solids, however, it turns out that there is a serious problem! As explained clearly

here, because the charge in a crystal is distributed periodically in space, the definition of \(\mathbf{P}\) given above is ambiguous because there are many ways to define the "unit cell" over which the integral is performed. This is a big deal.

The "modern theory of polarization" resolves this problem, and actually involves the electronic

Berry Phase. First, it's important to remember that polarization is

*really* defined experimentally by how much charge flows when that capacitor described above has the voltage applied across it. So, the problem we're really trying to solve is, find the integrated current that flows when an electric field is ramped up to some value across a periodic solid. We can find that by adding up all the contributions of the different electronic states that are labeled by wavevectors \(\mathbf{k}\). For each \(\mathbf{k}\) in a given band, there is a contribution that has to do with how the energy varies with \(\mathbf{k}\) (that's the part that looks roughly like a classical velocity), and there's a second piece that has to do with how the actual electronic wavefunctions vary with \(\mathbf{k}\), which is proportional to the Berry curvature. If you add up all the \(\mathbf{k}\) contributions over the filled electronic states in the insulator, the first terms all cancel out, but the second terms don't, and actually give you a well-defined amount of charge.

**Bottom line**: In an insulating crystal, the actual polarization that shows up in an applied electric field comes from how the electronic states vary with \(\mathbf{k}\) within the filled bands. This is a really surprising and deep result, and it was only realized in the 1990s. It's pretty neat that even "simple" things like crystalline insulators can still contain surprises (in this case, one that foreshadowed the whole topological insulator boom).