Patterns of electromagnetic response in topological semimetals

Topological semimetals are gapless states of matter which have robust and unique electromagnetic responses and surface states. In this paper, we consider semimetals which have pointlike Fermi surfaces in various spatial dimensions D = 1,2,3 which naturally occur in the transition between a weak topological insulator and a trivial insulating phase. These semimetals include those of Dirac and Weyl types. We construct these phases by layering strong topological insulator phases in one dimension lower. This perspective helps us understand their effective response field theory that is generally characterized by a 1-form b which represents a source of Lorentz violation and can be read off from the location of the nodes in momentum space and the helicities/chiralities of the nodes. We derive effective response actions for the two-dimensional (2D) and 3D Dirac semimetals and extensively discuss the response of the Weyl semimetal. We also show how our work can be used to describe semimetals with Fermi surfaces with lower codimension as well as to describe the topological response of 3D topological crystalline insulators.