The formation of a composite object (the polaron) when an electron interacts with the lattice deformations (phonons) is a problem studied for decades in condensed matter physics. Even so, there is no exact solution for the simplest model -- the Holstein Hamiltonian. Most of the analytical approaches were based on asymptotic limits (weak-strong coupling), while numerical solutions are computationally intensive due to the infinite size of the Hilbert space (Monte Carlo, exact diagonalization, etc).

A new analytical method -- the Momentum Average approximation (MA), has been developed in the group of Prof. Mona Berciu. This method is computationally fast and exact in the asymptotic limits and accurate for all electron-phonon coupling strengths. While all the sum rules are accurate, the spectral function can be readily calculated within this approximation.

Using the Momentum Average approximation, I considered problems that were previously even hard to solve, such as the existence of multiple phonon branches, ZO phonons in graphene, phonons in the presence of spin-orbit coupling and the effect of surfaces states on the polaron.

For
details on MA see:

Phys. Rev. Lett. 97, 036402 (2006)

2. The Green's function of the Holstein polaron, Glen L. Goodvin, Mona Berciu, and George A. Sawatzky

Phys. Rev. B 74, 245104 (2006)

3. Systematic improvement of the Momentum Average approximation for the Green's function of a Holstein polaron, Mona Berciu and Glen L. Goodvin

Phys. Rev. B 76, 165109 (2007)

###### Multiple phonons

Europhysics Lett. 80, 67001 (2007)
We accurately show the interplay between two phonon branches in the
formation of the polaronic state. The Momentum Average approximation is
very useful in this situation, the phonon space increases such that
other numerical methods become prohibitive.

###### Polarons in graphene

Phys. Rev. Lett. 100, 256405 (2008)We show that in rippled graphene, modeled by out-of-plane optical phonons, Dirac quasiparticles are well defined even for large electron-phonon coupling.