Charged Dirac fermions with anomalous magnetic moment in the presence of the chiral magnetic effect and of a noncommutative phase space

TL;DR

The authors derive and solve the noncommutative Dirac equation for charged fermions with an AMM in the presence of the CME, using a cylindrical-coordinate formulation and a unitary transformation to simplify gamma matrices. Through Frobenius analysis of the radial equations, they obtain a closed-form relativistic spectrum that generalizes relativistic Landau levels and depends on the CME parameters, AMM, magnetic field, and NC parameters $\theta$ and $\eta$ via the NC-cyclotron frequency $\omega_c^{NC}$ and two NC angular frequencies $\omega_\theta$ and $\omega_\eta$. The resulting energy levels are given by a compact expression with an effective quantum number $N_{eff}$ that unifies multiple branches, and special limits reproduce known results in Chaudhuri, Connell, Fukushima, Bubnov, and others. Graphical analysis reveals distinct trends for electrons and positrons, including NC-induced level splitting and asymmetric responses to chemical potentials, illustrating the significant impact of NC geometry on CME-enabled relativistic spectra. Overall, the work broadens the landscape of relativistic quantum systems under CME by incorporating AMM and NC phase space effects, with potential implications for high-field QED, QCD-inspired models, and condensed-matter analogs.

Abstract

In this paper, we analyze the relativistic energy spectrum (or relativistic Landau levels) for charged Dirac fermions with anomalous magnetic moment (AMM) in the presence of the chiral magnetic effect (CME) and of a noncommutative (NC) phase space, where we work with the $(3+1)$-dimensional Dirac equation in cylindrical coordinates. Using a similarity transformation, we obtain four coupled first-order differential equations. Subsequently, obtain four non-homogeneous second-order differential equations. To solve these equations exactly and analytically, we use a change of variable, the asymptotic behavior, and the Frobenius method. Consequently, we obtain the relativistic spectrum for the electron/positron, where we note that this spectrum is quantized in terms of the radial quantum number $n$ and the angular quantum number $m_j$, and explicitly depends on the position and momentum NC parameters $θ$ and $η$ (describes the NC phase space), cyclotron frequency $ω_c$ (an angular frequency that depends on the electric charge $e$, mass $m$, and external magnetic field $B$, i.e., $ω_c=eB/m$), anomalous magnetic energy $E_m$ (an energy generated through the interaction of the AMM with the external magnetic field), $z$-momentum $k_z$ (linear momentum along the $z$-axis), and on the fermion and chiral chemical potential $μ$ and $μ_5$ (describes the CME). However, through $θ$, $η$, and $m$, we define two types of ''NC angular frequencies'', given by $ω_θ=4/mθ$ and $ω_η=η/m$ (our spectrum depends on three angular frequencies). Comparing our spectrum with other papers, we verified that it generalizes several particular cases found in the literature. Besides, we also graphically analyze the behavior of the spectrum as a function of $B$, $μ$, $μ_5$, $k_z$, $θ$, and $η$ for three different values of $n$ and $m_j$.

Figures (6)

• Figure 1: Behavior of $E_{n}(B)$ vs. $B$ for $n=0,1,2$ with $m_j=\pm 1/2$ (a) and the behavior of $E_{m_j}(B)$ vs. $B$ for $m_j=\pm 1/2,\pm 3/2,\pm 5/2$ with $n=0$ (b).
• Figure 2: Behavior of $E_{n}(\mu)$ vs. $\mu$ for $n=0,1,2$ with $m_j=\pm 1/2$ (a) and the behavior of $E_{m_j}(\mu)$ vs. $\mu$ for $m_j=\pm 1/2,\pm 3/2,\pm 5/2$ with $n=0$ (b).
• Figure 3: Behavior of $E_{n}(\mu_5)$ vs. $\mu_5$ for $n=0,1,2$ with $m_j=\pm 1/2$ (a) and the behavior of $E_{m_j}(\mu_5)$ vs. $\mu_5$ for $m_j=\pm 1/2,\pm 3/2,\pm 5/2$ with $n=0$ (b).
• Figure 4: Behavior of $E_{n}(k_z)$ vs. $k_z$ for $n=0,1,2$ with $m_j=\pm 1/2$ (a) and the behavior of $E_{m_j}(k_z)$ vs. $k_z$ for $m_j=\pm 1/2,\pm 3/2,\pm 5/2$ with $n=0$ (b).
• Figure 5: Behavior of $E_{n}(\theta)$ vs. $\theta$ for $n=0,1,2$ with $m_j=\pm 1/2$ (a) and the behavior of $E_{m_j}(\theta)$ vs. $\theta$ for $m_j=\pm 1/2,\pm 3/2,\pm 5/2$ with $n=0$ (b).