Electric field assisted amplification of magnetic fields in tilted Dirac cone systems

We show that the continuum limit of the tilted Dirac cone in materials such as $8Pmmn$-borophene and layered organic conductor $\alpha$-(BEDT-TTF)$_2$I$_3$ deformation of the Minkowski spacetime of Dirac materials. From its Killing vectors we construct an emergent tilted-Lorentz (t-Lorentz) symmetry group for such systems. With t-Lorentz transformations we are able to obtain the exact solution of the Landau bands for a crossed configuration of electric and magnetic fields. For any given tilt parameter $0\le\zeta<1$ if the ratio $\chi=v_FB_z/cE_y$ of the crossed magnetic and electric fields that satisfies $\chi\ge 1+\zeta$ one can always find appropriate t-boosts in both valleys labeled by $\pm$ in such a way the electric field can be t-boosted away, whereby the resulting pure effective magnetic field $B^\pm_z$ governs the Landau level spectrum around each valley. The effective magnetic field in one of the valleys is always larger than the applied perpendicular magnetic field. This amplification comes at the expense of of diminishing the effective field in the opposite valleyand can be detected in various quantum oscillation phenomena in tilted Dirac cone systems. Tuning the ratio of electric and magnetic fields to $\chi_{\rm min}=1+\zeta$ leads to valley selective collapse of Landau levels. Our geometric description of the tilt in Dirac systems reveals an important connection between the tilt and an incipient "rotating source" when the tilt parameter can be made to depend on spacetime in certain way.