Searching for topological phases of matter and their electromagnetic signatures
This talk starts by reviewing known examples of how topological materials generate new kinds of electrodynamic couplings and effects. One category of topological phases can be understood in terms of nearly free electrons, but still manage to show interesting behavior such as the integer quantum Hall effect. We start with known cases and build up to an understanding of how the recently discovered topological Weyl semimetals can show unique electromagnetic responses; in nonlinear optics there should be a new approximately quantized effect, which may have been seen experimentally. This nonlinear effect has a natural quantum e^3/h^2 and appears in chiral Weyl semimetals over a finite range of frequencies. We then turn to fractional topological phases and their possible appearance in frustrated magnets ("spin liquids"). Emerging computational techniques allow a nearly unbiased comparison of spin liquid candidates in the original model studied by Anderson, the triangular lattice Hubbard model, and analytical methods help understand intuitively why a particular kind of topological spin liquid is an attractive candidate. We close with some comments on the remarkable recent progress in using quantum computers and emulators to create topological states.