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

This comment corrects errors in a previous study by Sahan et al. regarding Dirac particles in a magnetized rotating curved background, deriving the complete energy spectra that correctly depend on both radial and angular quantum numbers rather than just one.

The Gist

Imagine you are trying to bake a very specific cake. The recipe (the original paper by Sahan et al.) claims to tell you exactly how much sugar and flour you need to get the perfect flavor (the energy of a particle). However, when you read the recipe, you notice something strange: it only lists the amount of flour (one number) but completely ignores the sugar (a second number).

In the world of quantum physics, particles like electrons (Dirac particles) spinning in a rotating, curved universe with a magnetic field are like that cake. Their "flavor" or energy is determined by two main ingredients:

1. How many times they wiggle up and down (the radial quantum number, $n$).
2. How fast they spin around the center (the angular quantum number, $m$).

The author of this new paper, R. R. S. Oliveira, looked at the original recipe and said, "Wait a minute. If you are baking in a spinning, curved kitchen, you must account for both the wiggles and the spin. If you ignore the spin, your cake won't taste right, and all the calculations about how the cake behaves (thermodynamics) will be wrong."

Here is a breakdown of what this paper does, using simple analogies:

1. The Missing Ingredient (The Problem)

The original paper tried to solve a complex math puzzle called the Dirac Equation. Think of this equation as the "laws of physics" for a tiny particle moving through a twisted, rotating space with a magnetic field.

The original authors solved the puzzle and got an answer for the particle's energy. But their answer only depended on one number (the wiggles). It was as if they solved a maze but forgot to count the turns. In physics, if you are in polar coordinates (like a clock face), the energy must depend on how far out you are ($n$) and what angle you are at ($m$). The original paper's answer was incomplete because it lost the "angle" information.

2. The Detective Work (The Correction)

Oliveira stepped in like a detective to find out where the "angle" went. He realized the original authors made a few small but critical mistakes in their math toolkit:

• Wrong Tools: They used a slightly incorrect set of "glasses" (mathematical matrices) to look at the problem.
• Sign Errors: They got a few plus and minus signs mixed up, like writing "add sugar" when the recipe actually said "subtract sugar."
• Missing Steps: They only solved the puzzle for half of the particle (the bottom part of the spin) and ignored the other half.

3. The New Recipe (The Solution)

Oliveira went back to the drawing board. He used the standard, correct "glasses" and fixed the signs. He then solved the math puzzle for both parts of the particle.

When he did this, the "missing ingredient" reappeared! The new energy formula he derived depends on two numbers:

• $n$ (The Wiggle): How many radial nodes (wiggles) the particle has.
• $m$ (The Spin): How much angular momentum (spin) the particle has.

4. Why It Matters (The Consequences)

Why should we care if a formula has one number or two?

• The Thermodynamics: The original paper used their incomplete energy formula to calculate things like heat capacity and entropy (how disordered the system is). It's like calculating the total weight of a suitcase by only weighing the clothes and forgetting the shoes. The result is wrong.
• The "Special Case": Oliveira found that the original paper's answer wasn't totally useless. It actually works as a special, limited case where the particle is spinning in a very specific, negative direction. But for the general case, the original answer was incomplete.

The Big Picture Analogy

Imagine a carousel (the rotating curved background) with horses (the particles).

• The original paper said: "The energy of a horse depends only on how high it is on the pole."
• Oliveira says: "That's wrong! The energy also depends on how fast the horse is spinning around the pole. If you ignore the spin, you can't predict how the horses will move or how much energy the whole carousel uses."

Conclusion

This paper is a "correction note." It doesn't tear down the original work entirely but fixes the math errors that caused a key variable (the angular momentum) to vanish. By restoring this variable, Oliveira provides the complete and correct energy map for these particles. This ensures that any future calculations about heat, magnetism, or phase transitions in this strange, rotating universe are built on a solid foundation.

In short: The original map was missing a street; this paper draws the street back in so you don't get lost.

Technical Summary

1. Problem Statement

The author addresses a critical inconsistency in the work of Sahan et al. [1], which investigated quantum and classical phase transitions of Dirac particles in a homogeneously magnetized, rotating, curved 2+1-dimensional spacetime (Gürses metric).

• The Discrepancy: Sahan et al. derived quantized energy spectra that depended only on the radial quantum number ($n$). However, standard literature on the Dirac equation in polar coordinates (both flat and curved spacetime) dictates that energy spectra must intrinsically depend on two quantum numbers: the radial quantum number ($n$) and the angular (momentum) quantum number ($m$).
• The Consequence: Because the energy spectra were incomplete (missing the $m$ dependence), the subsequent thermodynamic calculations (partition function, magnetization, heat capacity, entropy) derived by Sahan et al. were fundamentally incorrect.
• The Goal: To identify the source of the error, correct the Dirac equation formulation, derive the true second-order differential equation, and obtain the complete energy spectra dependent on both $n$ and $m$.

2. Methodology and Corrections

Oliveira performs a rigorous re-derivation of the problem, identifying and correcting several specific errors in Sahan et al.'s formulation:

• Dimensional Analysis: The original Dirac equation used by Sahan et al. was dimensionally inconsistent. Oliveira restores the correct dimensions by properly including $\hbar$ and $c$ (or using natural units consistently) and ensuring the electromagnetic coupling term is $A_\mu/\hbar$.
• Gamma Matrix Representation:
  • Sahan et al. used a specific representation of gamma matrices that led to sign errors in the spin connection.
  • Oliveira adopts the standard definition for (2+1)-dimensions where $\beta = \sigma_3$ and $\vec{\alpha} = \vec{\sigma}$, leading to curved sigma matrices $\tilde{\sigma}^a = (\sigma_3, i\sigma_2, -i\sigma_1)$. This ensures the Clifford algebra is satisfied correctly.
• Metric and Tetrad Corrections:
  • Metric Sign Error: The inverse metric tensor $g^{\mu\nu}$ in the original paper contained sign errors in the off-diagonal components ($g^{t\phi}$ and $g^{\phi t}$). Oliveira corrects these based on the Gürses metric definition.
  • Spin Connection: Due to the incorrect gamma matrices and metric signs, the spin connection components ($\Gamma_\mu$) calculated by Sahan et al. were flawed. Oliveira recalculates the spinorial connection $\Gamma_\mu$ using the correct commutators of the Pauli matrices.
• Spinor Formulation:
  • Sahan et al. only derived the differential equation for the lower component of the Dirac spinor.
  • Oliveira derives the coupled first-order equations for both the upper ($\chi_+$) and lower ($\chi_-$) components.
  • He decouples these to form a single second-order differential equation valid for both components, introducing a parameter $s = \pm 1$ to distinguish between the upper and lower spinor components.

3. Key Results

A. The Corrected Differential Equation

Oliveira derives the correct radial Dirac equation, which leads to the following second-order differential equation for the spinor components $\chi_s(r)$:
$$\left[ \frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr} - \frac{(m - s/2)^2}{r^2} - \frac{(E k + m_e \omega_c)^2}{4}r^2 + E^2 - \left(m_e - \frac{k}{4}\right)^2 - (E k + m_e \omega_c)\left(m + \frac{s}{2}\right) \right] \chi_s(r) = 0$$
Where:

• $m$ is the angular quantum number ($m \neq 0$).
• $s = +1$ for the upper component, $s = -1$ for the lower component.
• $\omega_c = eB/m_e$ is the corrected cyclotron frequency (without the factor of 2 used in the original paper).

B. The Complete Energy Spectra

By solving the equation using confluent hypergeometric functions and applying the quantization condition ($a = -n$), Oliveira obtains the complete energy spectra:
$$E_{\pm} = kN \pm \sqrt{k^2 N^2 + 2m_e \omega_c N + \left(m_e - \frac{k}{4}\right)^2}$$
Where the effective quantum number $N$ is defined as:
$$N = n + \frac{1}{2} + \left| m - \frac{s}{2} \right| + \frac{m + s/2}{2}$$

• Dependence: The energy $E$ now explicitly depends on the radial quantum number ($n$), the angular quantum number ($m$), and the spinor component ($s$).
• Recovery of Previous Results: Oliveira notes that the spectra from Sahan et al. [1] are recovered only as a specific, incomplete limit where $m < 0$ (negative angular momentum) and $s = -1$ (lower component), and even then, only if a sign error in the term $-m_e k/2$ is ignored.

4. Significance and Implications

• Correction of Thermodynamics: The primary significance is that the thermodynamic properties (partition function, magnetization, entropy) calculated by Sahan et al. were based on an incomplete spectrum. Since the partition function requires a sum over all quantum states (including all $m$), the previous results are invalid. The corrected spectra must be used to re-evaluate the thermodynamic behavior of Dirac particles in this background.
• Theoretical Consistency: The work restores the fundamental requirement that Dirac particles in polar coordinates must exhibit dependence on the angular momentum quantum number. It highlights the sensitivity of curved spacetime Dirac equations to the precise definition of gamma matrices and spin connections.
• Methodological Rigor: The paper serves as a technical guide for correctly handling the Dirac equation in (2+1)-dimensional rotating curved spacetimes, correcting common pitfalls regarding metric signs, spin connections, and cyclotron frequency definitions.

Conclusion

R. R. S. Oliveira demonstrates that the energy spectra derived by Sahan et al. were incomplete due to algebraic and definitional errors. By correcting the Dirac equation and spin connection, he provides the physically correct energy spectrum that depends on both radial and angular quantum numbers. This correction is essential for any future study involving the thermodynamics or phase transitions of Dirac fermions in magnetized rotating curved backgrounds.