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arXiv · 2605.09249

Bound-State Spectra of a Lifshitz-Type Dirac Equation in (2+1) Dimensions

Lucas K. R. Queiroz·Van Sérgio Alves·Nilberto Bezerra·Luis Fernández·Francisco Peña
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Abstract

We investigate a Dirac-type equation in $(2+1)$ dimensions modified by Lifshitz spatial derivatives with dynamical exponent $z=2$, focusing on the spectral properties of bound-states under radial confinement. Analytical solutions are obtained for constant backgrounds, hard-wall confinement, and harmonic potentials, while logarithmic confinement is treated numerically via the Numerov method and complemented by a semiclassical WKB analysis. The resulting spectra exhibit characteristic scaling laws governed by the Lifshitz parameter $b$, including $E - M \propto b/R_0^2$ for hard-wall confinement, $E - M \propto \sqrt{2b}\,\omega$ for harmonic trapping, and $E - M \sim \alpha \ln\sqrt{b}$ in the semiclassical regime of logarithmic confinement, where $M$ is a mass scale. These results provide a consistent characterization of how higher-order spatial derivatives modify the energy spectra in two-dimensional Dirac systems and may be relevant for effective descriptions of materials with quadratic low-energy dispersion, such as bilayer graphene and related anisotropic 2D systems.

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Lucas K. R. Queiroz, Van Sérgio Alves, Nilberto Bezerra, Luis Fernández, Francisco Peña. 2026-05-10. Bound-State Spectra of a Lifshitz-Type Dirac Equation in (2+1) Dimensions. https://arxiv.org/abs/2605.09249

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