Dirac fermions in a spinning conical G\"odel-type spacetime

In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical G\"odel-type spacetime in $(2+1)$-dimensions, where we work with the curved Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels of the fermion and antifermion, where such levels are quantized in terms of the radial and total magnetic quantum numbers n and mⱼ, and explicitly depends on the spin parameter s (describes the ``spin''), spinorial parameter u (describes the two components of the spinor), curvature and rotation parameters α and β (describes the conical curvature and the angular momentum of the spinning cosmic string), and on the vorticity parameter Ω (describes the G\"odel-type spacetime). In particular, the quantization is a direct result of the existence of Ω (i.e. Ω acts as a kind of ``external field or potential''). We see that for mⱼ>0, the energy levels do not depend on s and u; however, depend on n, mⱼ, α, and β. In this case, α breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit 4Ωβ/α→ 1. Already for mⱼ<0, we see that the energy levels depends on s, u and n; however, it no longer depends on mⱼ, α and β. In this case, it is as if the fermion ``lives only in a flat G\"odel-type spacetime''. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of Ω, α, and β for three different values of n (ground state and the first two excited states).