Jan Wyzula, Milan Orlita, LNCMI-CNRS Grenoble and Mark O. Goerbig, LPS-CNRS Orsay
Ever since the advent of graphene and topological materials, relativistic physics has become an integral part of condensed-matter sciences. While emergent, it is important to stress that this type of relativity is pertinent beyond the dispersion of the low-energy excitations in different solids. Klein tunneling and the chiral anomaly represent well-known examples. One of the salient aspects of relativity is the particular dependence of energy on the frame of reference. For a particle of mass m moving at a speed u lower than the speed of light c, a Lorentz boost to the comoving frame of reference changes the particle‘s energy from E to E/γ = mc2, in terms of the Lorentz factor γ = 1/(√1−β2) and the rapidity β = u/c. A natural question that one may ask is whether one can observe this relativistic renormalization also in topological materials governed by the Dirac Hamiltonian, upon the replacement of c by a characteristic velocity v.
The effects of relativistic renormalization have been invoked theoretically in the past, for solid-state systems with (gapless or weakly gapped) tilted conical bands that are subjected to an externally applied magnetic field. This is because, in such a case, the motion of an electron becomes mathematically equivalent to the dynamics of a relativistic charge carrier in the crossed electric and magnetic fields. This motion is, therefore, governed by fully Lorentz-covariant Dirac and Maxwell equations. This covariant formulation, and thus the use of Lorentz transformations, allows us to calculate the energy spectrum in a reference frame where the (effective) electric field vanishes, meaning u = 0. This spectrum, or more directly, Landau-level spectrum becomes renormalized by the corresponding Lorentz factor calculated for the velocity parameter that describes the tilt.
Recent high-field experiments on niobium diarsenide (NbAs2)–– realized in a broad collaboration of researchers from Grenoble, Paris, Taipei, Brno, Zagreb, and Fribourg, and supported by theoretical modelling –– show that the effects of band renormalization can be traced experimentally. NbAs2 is a Dirac semimetal which hosts two nodal lines, the energy of which disperses with momentum, which are weakly gapped by spin-orbit interaction and which propagate approximately along the crystallographic a axis. The dispersion of electrons in the vicinity of these nodal lines has a form of a gapped conical band, with a tilt that depends on the choice of the particular plane in the reciprocal space. Applying the magnetic field in different directions with respect to the nodal line, we have observed, using the Landau-level spectroscopy technique, profound renormalization of the energy band gap driven by the corresponding Lorentz boost.
Figure: (a) Schematic band structure of a weakly gapped conical band, with the effective speed of light v, with and without tilting by an additional velocity parameter u. (b) The effective band gap deduced using Landau-level spectroscopy technique with the magnetic field applied perpendicular to different crystallographic planes of NbAs2. θD stands for the angle between the local nodal-line direction and the applied magnetic field.
Lorentz-Boost-Driven Magneto-Optics in a Dirac Nodal-Line Semimetal, J. Wyzula, X. Lu, D. Santos-Cottin, D. K. Mukherjee, I. Mohelský, F. Le Mardelé, J. Novák, M. Novak, R. Sankar, Y. Krupko, B. A. Piot, W.-L. Lee, A. Akrap, M. Potemski, M. O. Goerbig, and M. Orlita, Adv. Sci. 2022, 2105720.
https://ui.adsabs.harvard.edu/abs/2021arXiv211007266W/abstract
Contact: milan.orlita@lncmi.cnrs.fr
