Quantization and enhancement of non-linear responses in topological matter

In this talk I will discuss three examples of nonlinear responses of topological matter, which can be quantized and/or remarkably large. The first and simplest is a quantized circular dichroism in Chern insulators: it can be probed by heating and it is related to an absorbed power sum rule. It does not rely on translational invariance, and its therefore applicable to fractional quantum Hall states. The second concerns the circular photogalvanic effect, the part of the photocurrent which changes sign when the light's polarization flips. In Weyl semimetals it is quantized in units of e^3/h^2 times the Weyl monopole charge, with no material-dependent parameters. It is predicted to be obsevable in mirror free Weyl semimetals (e.g. SrSi2) and three-dimensional Rashba materials (e.g. doped Te) and its magnitude is relatively large, which enables the direct detection of the monopole charge with current techniques. Lastly, I will discuss the recently observed giant and anisotropic second harmonic generation found in TaAs. In particular I will provide evidence that a class of ferroelectric materials have an upper bound for non-linear effects, and that TaAs is close to saturating it for a particular frequency range.