Expansion in spherical harmonics in cubic symmetry

Question

suppose I have an electrostatic potential which I expand in spherical harmonics via

$$\sum_{l,m} A^l_m r^n P_l^{|m|}(\cos \theta) e^{im\varphi}$$

and I know that the field has cubic symmetry. Is there something I can say about the coefficients $A^n_m$? I ask because this would be the case for the crystal field in a cubic crystal, where it is known that the $e_g$ and the $t_{2g}$ orbitals are splitted. I am trying to show that the matrix-element is the same for all $e_g$, and the same for all $t_{2g}$. I tried it using the Wigner-Eckart theorem, but that seems not to be enough; I also need some symmetry relations for the $A^n_m$...

Answer 1

Yes, of course, cubic symmetry is a hugely constraining condition on the coefficients $A_{nlm}$ - I suppose that you meant that the coefficients depend on $l$ as well.

The cubic symmetry is generated by a few generators such as some rotations by 90 degrees. For example, if the state is symmetric with respect to the 90-degree rotation around the $z$ axis, it means that $\exp(\pi i J_z/2)$ has to keep the state invariant. It implies that $m$ i.e. the eigenvalue of $J_z$ has to be a multiple of four. All coefficients $A_{nlm}$ for which $m$ is not a multiple of four have to vanish.

If the state is also symmetric with respect to the $z\to(-z)$ reflection (parity), it means that $l$ has to be even: all coefficients with $l$ odd vanish.

Finally, if the state is invariant under the rotation by 90 degrees around the $x$ axis, there is a similar condition for the coefficients that is slightly more complicated than that $m$ has to be a multiple of four (the condition will constrain some linear combinations of the coefficients), but it is essentially the same thing. If you impose this extra condition, you will find out that only 1/48 of the coefficients may be nonzero because 48 is the order of the group. (Replace 48 by 24 if you don't include the parity-odd transformations.)

Answer 2

This exact problem came up in cosmology years ago, when people first started to consider the possibility of "topologically small" universes. These are models in which the universe is spatially flat, but has the topology of a 3-torus rather than of $R^3$. In other words, the Universe has a finite cell in the shape of a rectangular solid, with periodic boundary conditions. If the size of the cell is smaller than the horizon, then the cosmic microwave background temperature field has the cubic symmetry you describe (in the simplest case of a cubic fundamental cell). Since the connection between CMB observations and theory is generally done in a spherical harmonic expansion, the mathematical problem is identical.

The upshot is that there are a bunch of selection rules of the sort Lubos describes, but no nice simple relations of the sort I think you're looking for.

There was a flurry of activity in the CMB community around these models in the 1990s. If you want to see what people came up with, you could try Steven, Scott, and Silk (1993), de Oliveira Costa and Smoot (1995), or a bunch of other papers that cite these two.