Fermions in $(1+2)$-dimensions modified by nonminimal coupling and its applications to condensed matter physics

TL;DR

This work analyzes planar ($1+2$-dimensional) fermions with a nonminimal coupling to the electromagnetic field, deriving a nonrelativistic limit and applying it to condensed-matter relevant problems. The authors show that the coupling constant $g$ shifts Landau levels and alters Hall conductivity plateaus, while also modifying Stark shifts and the polarizability of a 2D harmonic oscillator in an electric field. Key results include $\mathcal{E}_{nm}=[n-m\Theta(-m)]\hbar\omega_c+gB$ and $\sigma_H(\epsilon_F,0)/\sigma_0 = -\left\lfloor \frac{m_{\psi}\epsilon_F}{eB}+m\Theta(-m)-g\right\rfloor$, indicating a tunable transport spectrum via $g$. The findings suggest that nonminimal couplings can engineer transport and optical responses in planar quantum systems and motivate further studies in anisotropies, renormalization, and topological materials. The work links Lorentz-violating effective theories to experimentally accessible condensed-matter phenomena, offering a bridge between high-energy-inspired models and 2D materials.

Abstract

Fermions in two-dimensional space, commonly called $(1+2)$-dimensional fermions, exhibit intriguing and distinctive characteristics that distinguish them from their higher-dimensional counterparts. This paper offers a comprehensive theoretical examination of planar fermionic systems, presenting novel findings by incorporating nonminimal coupling. Our analysis includes the computation of the non-relativistic limit up to second-order corrections in the Dirac equation. We also explore the Schrödinger equation under the influence of a harmonic potential and an electric field. Furthermore, we investigate how the coupling parameter affects physical properties relevant to condensed matter systems. Our results demonstrate that this parameter significantly impacts electronic properties and Hall conductivity. The interplay between an external electric field and the coupling parameter also influences energy levels and the system's polarizability. These findings underscore the novel effects of including nonminimal coupling in wave equations, offering new insights into the physics of coupled systems.

Figures (8)

• Figure 1: DOS given by Eq. \ref{['eq:DOS']} as a function of energy for two different values of $g$ (fixed $B = 1.0$).
• Figure 2: Hall conductivity at zero temperature as a function of the magnetic field for various values of $g$ and $m$. Fixed: $m_{\psi}\epsilon_F/e = 170$.
• Figure 3: Hall conductivity at zero temperature as a function of the magnetic field for different Fermi energies $\epsilon_F$ with $g = 0.1$ and fixed $m = 0$.
• Figure 4: The total energy as a function of the applied electric field for varying coupling parameter $g$. We use fixed $n=1$ and $\omega=1$.
• Figure 5: The total energy as a function of the quantum number $n$ for varying coupling parameter $g$. We use fixed $E_x=2.5$ and $\omega=1$.