Abstract

Condensed matter physicists have studied topological phases of matter for over 40
years due to, among other things, their potential applications for quantum
computing and backscattering-free electronic transport. All these promising
applications hinge on the existence of mysteriously protected conducting states
residing on the material's boundary while the material's bulk is insulating.
The jewel in the crown of theoretical developments is the discovery of the
corresponding "periodic table" listing all possible topological quantum matter.

The mechanism behind the mysterious protection of conducting states is believed
to be that of separated twins: in every wire, left- and right-moving electrons
are 'twins' that always come together. In two-dimensional materials, one can
separate the twins by placing them on opposite edges, hence, rendering
backscattering impossible. This scenario is realized in the famous Quantum Hall
Insulator. Still, the twins keep their connection via delocalized 'bridge'
states residing in the material's bulk, the mechanism known as the 'spectral flow
principle'. The only known exception to this principle appears in
the one-dimensional systems where it is possible to have disconnected twins. In this
special case, the twins are called Majorana fermions --- fractional
zero-energy electrons that reside on the wire ends. Since Majorana fermions are
spectrally disconnected from the bulk and spatially separated from each other,
they are ideal candidates for storing quantum information.

In this talk I show that many three-dimensional phases in the periodic table have
much richer boundary phenomenology than previously believed, where the two
extreme cases correspond to the two scenarios mentioned above: separated twins
communicating via the delocalized bridge states and disconnected twins that
appear only at zero energy. The hope is that these findings will open the door to
even more exciting applications of topological materials.