The hottest off-shoot of topological insulators these days is the study of nodal electronic structures.  By this I mean the situation in which two different bands touch along some locus in momentum space.  This started a few years ago with the re-discovery of Weyl points – I wrote a commentary some time ago with the poetic title, “Weyl electrons kiss”, using kissing as a metaphor for band touching.  The actual contact for Weyl fermions is linear in energy versus momentum, so a bit angular compared to lips.  The kissing image seems more apt for quadratic band touching, which also occurs and is interesting in its own right.

The contact locus between bands can be extended, and form for example a loop in momentum space.  This is often called a “nodal loop”.  An exotic phenomena that happens in such cases is that there can be a branch of surface bound states which exists only for surface momenta which project inside the nodal loop.  If one plots the energy of the surface state versus the two dimensional momentum of the surface, it forms a “drumhead” that stretches across the projection of the nodal loop into the 2d surface Brillouin zone.  I think this drumhead surface state was first pointed out by Volovik, though I do not recall the reference at the moment.

To a good approximation such a drumhead state is “flat”, i.e. its energy is nearly constant. A flat band has no kinetic energy, so one might think it should be susceptible to interactions.  With Jianpeng Liu, a postdoc at KITP, we studied this and indeed found various instabilities due to electron-electron forces.   Here is a phase diagram from our paper, for a simple limit of our simple model:

You can see in the inset the charge density wave pattern that forms in our simple model – it looks a bit like the drumhead surface state is “ringing”, though this is really oscillation of the charge density as a function of distance into the material.

This sort of tendency to form ordered states means that nodal loop semimetals should be an interesting place to study two-dimensional electronic surface phase transitions.    A neat feature of this situation is that the nodal loop means there are gapless states in the bulk, and that the surface bound states become bulk-like as they approach the momenta of the loop.  These attributes mean that surface quantum criticality in this situation is different from the criticality of a purely 2d system, and is instead inextricably tied to the 3d nature of the system.

You can read our paper if you want to know the details.