Anomalous successes and a surprising failure: The Dirac equation and topological materials*

The Dirac equation was formulated nearly a century ago, with the goal of combining quantum mechanics and special relativity. The elegance and simplicity of its construction infused the entire project of quantum physics, through the idea that a minimal combination of relativistic and quantum principles should define the basic building blocks of physical theories. This philosophy produced the most-tested, successful theories of physics, such as quantum electrodynamics (QED). In condensed matter physics, the Dirac equation has become a source of fascination in more recent times. Versions of the Dirac equation describe chiral edge states of quantum Hall droplets, "ultrarelativistic" carriers in graphene, Weyl semimetals, and moire materials, and surface states of 3D topological phases.

In this talk, after reviewing some of these successes, I will discuss a very surprising recent development wherein the emergent Dirac equation that is supposed to describe surface states of most "strong" 3D topological phases fails to determine even the most basic properties of the surface. One such question is whether surface particles can conduct energy in the presence of impurities. This development implies that 2+1-D quantum field theories meant to describe surface states of topological matter are fundamentally incomplete. This is analogous to requiring an understanding of physics at the Planck scale in order to predict properties of ordinary 3+1-D quantum matter. The "missing" information turns out to be a type of symmetry-preserving, hidden surface Berry curvature that resides outside the minimal Dirac description. We show that surface Berry curvature can interupt bulk-boundary spectral flow and Anderson localize most surface states. Our work reveals that three-dimensional phases in the periodic table have a much richer boundary phenomenology than previously believed, where the extreme cases correspond to (1) spectrum-wide quantum-critical surface delocalization, with a bulk "bridge" connecting boundary states at different surfaces, vs. (2) bulk-disconnected surface states that avoid localization only at zero energy.