Fig. 3: Semiclassical analysis of spectra and addressing the shape of isotropic massive Dirac band.

a Transition energies \(\omega\) scaling with momenta given by the semiclassical Lifshitz-Onsager relation at different magneto-optical geometries: Faraday geometry with \({{\bf{B}}}{{{\parallel }}}[001]\) (i); Voigt geometries with \({{\bf{B||}}}[010]\) (ii), \({{\bf{B||}}}[100]\) (iii) and \({{\bf{B||}}}[110]\) (iv). With a nontrivial Berry phase π, all extracted transition energies collapse to the same joint massive Dirac band profile (blue curve in each panel), indicating an isotropic massive Dirac band in LaAlSi. b, c A direct comparison between the cases of \(\gamma =0\) and \(\gamma =1/2\). The obtained massive Dirac band (blue curve) \(E=\sqrt{{\varDelta }^{2}+{\hslash }^{2}{v}_{{{\rm{F}}}}^{2}{k}^{2}}\) exhibits a finite gap (Dirac mass) at zero-momentum and a nearly linear energy dispersion in high energy approaching the zero-mass limit (black dashed curves).