Electrons in Flatland: Recent Developments and New Vistas

A two-dimensional electron system subjected to a large perpendicular
magnetic field is a veritable goldmine for the condensed matter
physicist. The original discovery of the integer quantized Hall effect
(IQHE) in semiconductor heterostructures was followed by the even more
unexpected discovery of its fractional counterpart (FQHE) soon
thereafter. During the thirty years that have elapsed since those
Nobel Prize winning discoveries, this system has continued to provide
an increasing number of exotic phenomena that have literally
revolutionized our understanding of condensed matter. Examples include
fractionalization of the electronic charge, composite particles,
Abelian and non-Abelian quantum states, and topological spin
excitations, in addition to charge-density-wave phases, including
electron crystals, bubble and striped phases.

After an introduction to the rich and fascinating world of
two-dimensional electrons, I will describe two recent efforts in our
continual search for new phenomena. First, I will consider electrons
in the two-dimensional material graphene, where the band dispersion is
of the relativistic, Dirac form, instead of the parabolic
non-relativistic spectrum in Gallium Arsenide. Then, I will discuss
the effect of mass anisotropy, which exists in systems based on
many-valley semiconductors such as AlAs, Si and Ge, or in isotropic
GaAs when the magnetic field is tilted away from the plane normal;
this has resulted in a new understanding of the FQHE. Using exact
numerical many-body calculations for both types of systems, I will
discuss how these systems add to the richness and diversity of quantum
Hall phases, including (i) the possibility of optimizing properties,
and (ii) uncovering universal chiral Luttinger liquid behavior, which
has remained elusive for semiconductor based electron gases in the
quantum Hall regime.