Electronic structure and a couple of fun links

Real life has been very busy recently.  Posting will hopefully pick up soon.  

One brief item.  Earlier this week, Rice hosted Gabi Kotliar for a distinguished lecture, and he gave a very nice, pedagogical talk about different approaches to electronic structure calculations.  When we teach undergraduate chemistry on the one hand and solid state physics on the other, we largely neglect electron-electron interactions (except for very particular issues, like Hund's Rules).  Trying to solve the many-electron problem fully is extremely difficult.  Often, approximating by solving the single-electron problem (e.g. finding the allowed single-electron states for a spatially periodic potential as in a crystal) and then "filling up"* those states gives decent results.   As we see in introductory courses, one can try different types of single-electron states.  We can start with atomic-like orbitals localized to each site, and end up doing tight binding / LCAO / Hückel (when applied to molecules).  Alternately, we can do the nearly-free electron approach and think about Bloch waves.  Density functional theory, discussed here, is more sophisticated but can struggle with situations when electron-electron interactions are strong.

One of Prof. Kotliar's big contributions is something called dynamical mean field theory, an approach to strongly interacting problems.  In a "mean field" theory, the idea is to reduce a many-particle interacting problem to an effective single-particle problem, where that single particle feels an interaction based on the averaged response of the other particles.  Arguably the most famous example is in models of magnetism.  We know how to write the energy of a spin \(\mathbf{s}_{i}\) in terms of its interactions \(J\) with other spins \(\mathbf{s}_{j}\) as \(\sum_{j} J \mathbf{s}_{i}\cdot \mathbf{s}_{j}\).  If there are \(z\) such neighbors that interact with spin \(i\), then we can try instead writing that energy as \(zJ \mathbf{s}_{i} \cdot \langle \mathbf{s}_{i}\rangle\), where the angle brackets signify the average.  From there, we can get a self-consistent equation for \(\langle \mathbf{s}_{i}\rangle\).  

Dynamical mean field theory is rather similar in spirit; there are non-perturbative ways to solve some strong-interaction "quantum impurity" problems.  DMFT is like a way of approximating a whole lattice of strongly interacting sites as a self-consistent quantum impurity problem for one site.  The solutions are not for wave functions but for the spectral function.  We still can't solve every strongly interacting problem, but Prof. Kotliar makes a good case that we have made real progress in how to think about many systems, and when the atomic details matter.

*Here, "filling up" means writing the many-electron wave function as a totally antisymmetric linear combination of single-electron states, including the spin states.

PS - two fun links:

• Don't forget to fill out your grant reports, or you can hold up the works for an entire university!
• PIs, don't forget to keep running on the funding hamster wheel!