Electrons at the Nanoscale
The last few decades have seen the discovery of materials with exotic properties few would have imagined. Superconductors exhibit zero resistivity up to 135 Kelvin. Multi-ferroic materials allow a magnetic field to write electric domains and an electric field to write magnetic domains. Colossal magnetoresistance materials change their electrical conductivity by orders of magnitude upon application of a magnetic field. Heavy fermion materials host electrons which behave thousands of times heavier than their actual mass, whereas the electrons in graphene behave as if they were massless. Existing materials display fractional charges in two dimensions, and the recently discovered topological insulators may extend these fractional charges into three dimensions.
These diverse behaviors all trace to electron interactions. In conventional metals, electrons barely interact with each other: Fermi liquid theory describes their tendency to screen local charge, so that electrons can be treated as isolated particles in a homogenous background. In contrast, the common feature to all of these exotic materials is that their electrons do interact, and we lack broadly successful theoretical language to describe these 'correlated electron materials'.