2D Dirac Oscillators Overview

• 2D Dirac Oscillators are relativistic quantum models featuring non-minimal coupling and spin-dependent spectra in planar systems.
• Their Hamiltonian structure, modified by external fields and singular potentials, yields infinite degeneracy and rich symmetry properties.
• They are pivotal in exploring quantum deformations, graphene-based nanostructures, and thermoelectric transport phenomena in Dirac materials.

The two-dimensional Dirac oscillator (2D DO) is a fundamental model in relativistic quantum mechanics characterized by a non-minimal linear coupling of the canonical momentum in the Dirac equation. It represents confined relativistic spin-½ fermions in planar geometries, with applications ranging from quantum field theory to condensed matter and thermoelectric transport in Dirac materials. The 2D DO notably features infinite degeneracy, spin-dependent spectra, sensitivity to electromagnetic backgrounds, and rich symmetry properties, making it a versatile testbed for generalizations, symmetry analysis, and practical device modeling.

1. Hamiltonian Structure and Spin Effects

The canonical 2D Dirac oscillator Hamiltonian is obtained by the non-minimal coupling prescription $p \to p - i m \omega \beta r$, where $m$ is the mass, $\omega$ is the oscillator frequency, $\beta$ is the Dirac matrix, and $r$ is the position vector. In the planar case (obtained via reduction from four to two components by exploiting translational symmetry along $z$), the spin connection $\Gamma$ in conical or cosmic string backgrounds introduces significant modifications. The total Hamiltonian includes:

$$H = H_0 + \frac{\phi\,s}{\alpha} \frac{\delta(r)}{r}$$

$$H_0 = -\frac{d^2}{dr^2} - \frac{1}{r} \frac{d}{dr} + \frac{j^2}{r^2} + M^2\omega^2 r^2$$

$$j = \frac{1}{\alpha} \left( m + \phi + \frac{1-\alpha}{2} s \right)$$

Here, $\phi$ is the magnetic flux, $s=\pm1$ indexes spin states, and the $\delta$-function term arises from the Zeeman interaction (Andrade et al., 2014). Its presence, neglected in earlier treatments, leads to singular point interactions at $r=0$ and allows for irregular solutions affecting both the energy spectrum and the structure of wave functions.

The complete energy quantization in the presence of the $\delta$-potential follows:

$$E = \pm \sqrt{M^2 + 2 M \omega \left[2 n \pm \frac{1}{\alpha} \left| m + \phi + \frac{1-\alpha}{2} s \right| - \frac{s}{\alpha}\left(m + \phi + \frac{1-\alpha}{2} s\right) \right]}$$

Boundary conditions are enforced via the self-adjoint extension method, making use of parameters sensitive to the singularity induced by the Zeeman term. In physical contexts such as cosmic string backgrounds, nontrivial spin–orbit and Zeeman couplings fundamentally modify spectral properties and must be accounted for in any quantum description.

2. Quantum Deformations and Generalized Algebras

The $\kappa$-deformed Dirac oscillator introduces quantum-gravity-inspired modifications through the $\kappa$-Poincaré–Hopf algebra. The deformation parameter $\epsilon$ (often related to the Planck mass) generates first-order corrections in the Dirac Hamiltonian (Andrade et al., 2014):

$$H \psi = [H_0 - (\epsilon/2)(H_0^2 - \pi_i \pi_i - m \beta H_0)] \psi = E \psi$$

Wave functions and energy levels become explicitly $\epsilon$-dependent:

$$E = \pm \sqrt{m^2 + 2 m \omega (2n + |l| - s l)} - m \epsilon \omega (2n + |l| - s l)$$

Charge conjugation symmetry is broken by these corrections, resulting in nondegenerate particle/antiparticle spectra. The deformation also alters the distance between adjacent levels, preserving infinite degeneracy but reducing the level spacing with increasing $\epsilon$. This framework is essential for explorations of relativistic quantum systems at Planck-scale energies and for understanding possible quantum gravity corrections in Dirac materials.

3. Oscillator in External Fields and Potentials

The energy spectrum and eigenfunctions of the 2D Dirac oscillator are heavily influenced by external fields and potentials. In magnetic fields and with antidot (repulsive) potentials (Akcay et al., 2016), the radial equation takes a Laguerre polynomial form. The energy spectrum is given by:

$$E^2 - m^2 c^4 = (\hbar c)^2 \{ (2n + 1 + \tilde{y})[2(E + m c^2)/(\hbar c)^2] + (m + \alpha) \}$$

$\alpha$ incorporates contributions from the AB flux and antidot potential, while $n, m$ parameterize radial and magnetic quantum numbers. The probability density is depleted near the origin for nonzero antidot strength, a direct manifestation of the repulsive character of the potential, and Landau level degeneracy is lifted for negative $m$ or in the presence of AB flux.

Comparison with the Schrödinger equation shows that while the nonrelativistic limit reproduces familiar Landau/Fock–Darwin levels, the Dirac oscillator framework provides additional relativistic corrections and a richer dependence on quantum numbers and potential strength.

4. Generalizations: Isotonic, PT-Symmetric, and Darboux-Transformed Oscillators

The isotonic oscillator generalization introduces a nonlinear superpotential:

$$W_x = a x + \frac{b x}{x^2 + y^2}, \quad W_y = a y + \frac{b y}{x^2 + y^2}$$

with SUSY QM factorization yielding exact solutions in terms of confluent hypergeometric functions. The corresponding energy spectrum remains equispaced and the system can be mapped onto anti-Jaynes–Cummings-like Hamiltonians, with spin operators coupled to supercharges (Ghosh et al., 22 Feb 2025).

PT-symmetric extensions, obtained via $p \rightarrow p - i q r$, produce stationary, localized wave packets characterized by Gaussian envelopes and Bessel function oscillatory behavior, constructed as eigenstates of $J_z$ (the third component of total angular momentum) (Faruque et al., 2019). Such solutions are particularly pertinent to applications in graphene quantum dots and confined Dirac–Weyl systems.

The Darboux transformation methodology enables construction of new families of exactly solvable 2D Dirac equations with position-dependent mass, matrix potentials, and generalized oscillator couplings. These approaches yield explicit zero-energy states and allow direct modeling of Dirac materials under various inhomogeneous field profiles (Schulze-Halberg et al., 2021).

5. Symmetry Properties and Electromagnetic Interpretation

The 2D Dirac oscillator notably manifests an electromagnetic duality: it describes a 1/2-spin relativistic fermion under a uniform, perpendicular magnetic field. The oscillator frequency is directly proportional to the field strength ($B = m \omega / q$), and the coupling arises naturally via a covariant four-potential $A^\mu = (0, p y, -p x)$, producing $B = -2p^2$ with $E = 0$ (Moyano et al., 2020).

While global $U(1)$ gauge symmetry is preserved, the Dirac oscillator explicitly breaks local chiral symmetry ($U(1)_R \times U(1)_L$) due to the linear interaction term. This symmetry breaking is independent of dimensionality and is significant for understanding chiral phase transitions and anomaly structures in relativistic matter.

6. Thermoelectric and Nanostructure Applications

Dirac oscillators serve as effective models for impurity-like perturbations induced by localized strain fields in graphene, such as nanobubbles or ripples, which generate pseudomagnetic fields and modify transport properties. The semiclassical Boltzmann transport formulation, employing the Dirac oscillator scattering potential and phase-shift analysis, yields analytical expressions for conductivity, Seebeck coefficient, and thermal conductivity (Cañas et al., 4 Sep 2025):

$$\sigma_{\alpha\beta}(T) = 2\frac{q^2}{h} \delta_{\alpha\beta} \left\{\frac{\mathcal{E}_F}{\hbar} \tau_{tr}(\mathcal{E}_F) + \frac{\pi^2}{6} \frac{(k_B T)^2}{\hbar}[2 \tau'_{tr}(\mathcal{E}_F) + \mathcal{E}_F \tau''_{tr}(\mathcal{E}_F)] \right\}$$

The temperature dependence of nanobubble density modulates scattering relaxation time and thus controls the transport coefficients, opening avenues for strain engineering of graphene thermoelectric devices. While the Seebeck coefficient and figure of merit $ZT$ are ratios and thus more resilient to changes in scatterer density, direct tuning of strain provides a mechanism to optimize device performance.

7. Special Functions and Analytic Solutions

Exact analytic solutions of the 2D Dirac oscillator—whether in the presence of smooth waveguide potentials, cosmic string backgrounds, or generalized couplings—are expressible through special functions. For instance, Heun confluent functions govern the wavefunctions in Pöschl–Teller potential profiles (Hartmann et al., 2017), spheroidal wave functions emerge for supercritical states, and associated Laguerre polynomials appear in systems with magnetic and antidot potentials (Akcay et al., 2016), as well as in rotating cosmic string backgrounds (Oliveira, 2019). These analytic forms bridge relativistic and non-relativistic limits, provide control over zero-energy bound state thresholds, and quantify device-relevant conduction phenomena.

Summary Table: Core Features of 2D Dirac Oscillator Models

Feature	Mathematical Representation / Key Phenomenon	Notable References
Non-minimal linear coupling	$p \to p - i m \omega \beta r$	(Andrade et al., 2014, Andrade et al., 2014)
Electromagnetic interpretation	Uniform $B$ field; $A^\mu = (0, p y, -p x)$, $B = -2p^2$	(Moyano et al., 2020)
Spin and Zeeman effects	$\delta(r)/r$ term; modified spectrum via self-adjoint extension	(Andrade et al., 2014)
Quantum deformations ($\kappa$-algebra, GUP)	Spectral corrections $\propto \epsilon$ or deformed algebra	(Andrade et al., 2014, Stetsko, 2018)
Higher symmetry and superpotential generalizations	Isotonic, anti-Jaynes–Cummings mapping	(Ghosh et al., 22 Feb 2025)
Analytic solutions (special functions)	Heun confluent, Laguerre, spheroidal wave functions	(Hartmann et al., 2017, Akcay et al., 2016, Oliveira, 2019)
Thermoelectric transport in graphene	Relaxation time via DO scattering, strain engineering	(Cañas et al., 4 Sep 2025)

Concluding Remarks

The breadth of recent developments demonstrates that the 2D Dirac oscillator is not simply a quantum mechanical curiosity but an integrative tool connecting high-energy physics, condensed matter, quantum optics, and nanotechnology. Its frameworks—whether incorporating cosmic backgrounds, quantum deformations, external fields, nontrivial potentials, or symmetry analysis—provide analytic and computational access to realistic physical phenomena in low-dimensional systems. The interplay of spin, topology, and external interactions ensures continued relevance in both theoretical investigations and advanced device engineering.