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

TL;DR

This paper investigates the energy levels of Dirac fermions in a spinning conical G"odel-type spacetime, revealing how curvature, rotation, and spin influence quantization and degeneracy, with detailed analytical and graphical analysis.

Contribution

It provides the first detailed derivation of relativistic and nonrelativistic energy spectra for Dirac fermions in this specific curved, rotating spacetime using the tetrads formalism.

Findings

Energy levels depend on quantum numbers and spacetime parameters.

Degeneracy is broken by the conical curvature parameter.

Energy levels can diverge as certain parameters approach critical values.

Abstract

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 $m_j$, 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 $\alpha$ and $\beta$ (describes the conical curvature and the angular momentum of the spinning cosmic string), and on the vorticity parameter $\Omega$ (describes the G\"odel-type spacetime). In particular, the quantization is a direct result of the existence of $\Omega$ (i.e. $\Omega$ acts as a kind of ``external field or potential''). We see that for $m_j>0$, the energy levels do not depend on $s$ and $u$; however, depend on $n$, $m_j$, $\alpha$, and $\beta$. In this case, $\alpha$ breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit $\frac{4\Omega\beta}{\alpha}\to 1$. Already for $m_j<0$, we see that the energy levels depends on $s$, $u$ and $n$; however, it no longer depends on $m_j$, $\alpha$ and $\beta$. 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 $\Omega$, $\alpha$, and $\beta$ for three different values of $n$ (ground state and the first two excited states).