Dirac Materials and Non-Uniform Magnetic Fields Maintain Landau Level Spectra.

The behaviour of electrons within graphene, a single-layer sheet of carbon atoms, continues to reveal nuanced quantum mechanical properties. Recent research focuses on the formation of Landau levels, discrete energy levels that arise when charged particles, such as electrons, are subjected to a magnetic field. These levels are typically observed with uniform magnetic fields, but a new investigation explores whether non-uniform fields can also produce this characteristic spectrum. Aritra Ghosh, from the School of Basic Sciences at the Indian Institute of Technology Bhubaneswar, and colleagues demonstrate that specific non-uniform magnetic fields, constructed through isospectral deformations of uniform fields, indeed support the formation of Landau levels. Their work, detailed in the article “Landau levels of a Dirac electron in graphene from non-uniform magnetic fields”, provides explicit mathematical expressions for these fields, furthering understanding of electron behaviour in this important material.

Dirac materials, characterised by their linear energy dispersion relation near specific points in momentum space known as Dirac points, exhibit unusual electronic properties and are subject to intense investigation. Researchers now demonstrate that Landau level formation, a quantum mechanical phenomenon typically associated with uniform magnetic fields, can also occur under specific, non-uniform field configurations. This challenges the conventional understanding that field uniformity is a prerequisite for observing these quantized energy states.

Landau levels arise when a charged particle, such as an electron, moves within a magnetic field. The field quantises the electron’s motion into discrete energy levels, forming these Landau levels. The current work employs isospectral deformations, a mathematical technique that alters a quantum system while preserving its energy spectrum, to derive explicit expressions for the non-uniform magnetic fields capable of sustaining this level structure. This approach establishes a direct correspondence between uniform and non-uniform field configurations, allowing for a comparative analysis of their effects on quantum systems.

The study rigorously addresses the Schrödinger equation, the fundamental equation of quantum mechanics, utilising isospectral deformation techniques to identify magnetic field profiles that produce identical Landau levels to those generated by a constant field. This relies on mathematical transformations that ensure the preservation of the energy spectrum throughout the deformation process, offering a nuanced understanding of magnetic field influence. The implications are particularly relevant to condensed matter physics, where Landau levels are crucial for understanding the electronic properties of materials exhibiting behaviours near Dirac points.

Researchers systematically generate non-Hermitian Hamiltonians, mathematical operators describing the total energy of a system, from their Hermitian counterparts through the application of Darboux transformations. These transformations effectively reshape the potential energy landscape of the system, offering a novel approach to modelling complex behaviours and expanding the scope of possible system descriptions. Importantly, these transformations do not invalidate established physical predictions, but rather broaden the range of accessible system descriptions, frequently exhibiting pseudo-Hermiticity, a property ensuring real energy eigenvalues despite the non-Hermitian nature of the operator.

Applying Darboux transformations to the Landau level problem reveals that these non-uniform magnetic fields produce energy spectra consistent with Landau levels, mirroring the behaviour observed under uniform fields and enabling direct comparison with experimental results. This research highlights connections between these non-Hermitian systems and diverse areas of physics, including condensed matter physics and optics, demonstrating the interdisciplinary relevance and broad applicability of the methodology.

The study offers a refined understanding of how magnetic fields influence quantum systems and provides a framework for exploring novel quantum states in engineered materials. Future research will focus on exploring the implications of these findings for the development of novel electronic devices and materials.