  High harmonic generation (HHG) has developed in recent years as a promising
tool for ultrafast materials spectroscopy. At the forefront of these
advancements, several works proposed to use HHG as an all-optical probe for
topology of quantum matter by identifying its signatures in the emission
spectra. However, it remains unclear if such spectral signatures are indeed a
robust and general approach for probing topology. To address this point, we
perform here a fully ab-initio study of HHG from prototypical two-dimensional
topological insulators in the Kane-Mele quantum spin-Hall and anomalous-Hall
phases. We analyze the spectra and previously proposed topological signatures
by comparing HHG from the topological and trivial phases. We demonstrate and
provide detailed microscopic explanations of why in these systems none of the
thus far proposed observables uniquely and universally probe material topology.
Specifically, we find that the: (i) HHG helicity, (ii) anomalous HHG
ellipticity, (iii) HHG elliptical dichroism and (iv) temporal delays in HHG
emission, are all unreliable signatures of topological phases. Our results
suggest that extreme care must be taken when interpreting HHG spectra for
topological signatures, and that contributions from the crystal symmetries and
chemical nature might be dominant over those from topology. They hint that a
truly universal topological signature in nonlinear optics is unlikely, and
raise important questions regarding possible utilization and detection of
topology in systems out-of-equilibrium.
