Comment on "Quantum phase transitions of Dirac particles in a magnetized rotating curved background: Interplay of geometry, magnetization, and thermodynamics"

TL;DR

This comment corrects and completes the energy spectra for Dirac particles in a curved spacetime, clarifying their dependence on two quantum numbers and fixing minor errors in a previous study.

Contribution

It provides the complete energy spectra depending on two quantum numbers and corrects errors in the prior work by Sahan et al. in the context of Dirac particles in curved spacetime.

Findings

Derived the true second-order differential equation for the problem.

Obtained the complete energy spectra depending on quantum numbers n and m.

Corrected minor errors in the previous study.

Abstract

In this comment, we obtain the complete energy spectra for the paper by Sahan et al. [1], that is, the energy spectra dependent on two quantum numbers, namely, the radial quantum number (given by $n\geq 0$) and the angular quantum number (given by $m\neq 0$). In particular, what motivated us to carry out such a study was the fact that the quantized energy spectra for Dirac particles in a curved or flat spacetime in polar coordinates explicitly depend on two quantum numbers. From this, the following question arose: Why do the energy spectra in the paper by Sahan et al. [1] depends on only one quantum number and not two, given that they worked with the Dirac equation in polar coordinates? So, using several important papers in the literature on the Dirac equation in curved spacetimes, as well as the most commonly used definition for Dirac gamma matrices in (2+1)-dimensions, we corrected some minor errors in the paper by Sahan et al. [1]. Consequently, we obtain the true second-order differential equation for their problem, as well as the complete energy spectra, which explicitly depend on both $n$ and $m$. Finally, we note that for $m<0$ (negative angular momentum) with $s=-1$ (lower component of the Dirac spinor), we obtain (except for one term with the incorrect sign) the particular energy spectra of Sahan et al. [1].