Analytic evaluation of Kane fermion magneto-optics in two and three dimensions

We calculate and present an analytic form of the magneto-optical conductivity for the gapped low-energy Kane model in two and three dimensions separately. The two-dimensional case maps onto the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}₃$ model at a particular value of $\ensuremath{\alpha}=1/\sqrt{3}$. In two dimensions, two chiral sectors exist, between which there are no optically activated transitions. In three dimensions, the extra dimension of dispersion mixes the two sectors so that intra- and intersector transitions can occur. The latter type of transition can be separated out via circular polarizations of light, and it shows a distinct signature in the transverse conductivity.