Boris Shklovskii: Superfluid-insulator transition with strong disorder

(Delayed Blog posting from Week 2, Thursday 9th Aug).

Boris Shklovskii spoke about the critical density of particles, n, needed to enter
a superfluid state from a localized insulator. He considered the phase diagram
as a function of n and the Cooper pair size xi. For small xi, this becomes a boson
localization problem, while for large xi, we instead have to consider the metal-insulator transition of fermions (it is assumed that once the fermions become metallic, the BCS instability will convert the metal into a superconductor). Boris argued that bosons localize more easily than fermions: for bosons one has to compare the variation in the effective potential energy with the kinetic energy (of order 1/L^2, where L is the size of a wavefunction) of a single state, while for fermions the potential energy has to be compared with the Fermi energy. Consequently the critical density for small xi was parametrically larger than the critical density for large xi - this was one of the central results.


A crucial ingredient in the argument was the computation of the effective potential energy experienced by a single boson or fermion. This was determined in a classical (Hartree) manner, by adding the interaction energy to the random potential. In particular, Boris considered a model of a strongly compensated semiconductor, with many impurities of both positive and negative charges. Then a self-consistent non-linear screening argument was used, similar to that described in the classic book by Efros+Shklovskii.