Noncommutative Landau problem in graphene: a gauge-invariant analysis with the Seiberg–Witten map

We investigate the relativistic quantum dynamics of a massless electron in graphene in a two-dimensional noncommutative (NC) plane under a constant background magnetic field. To address the issue of gauge invariance, we employ an effective massless NC Dirac field theory, incorporating the Seiberg–Witten (SW) map alongside the Moyal star (\(\star\)) product. Using this framework, we derive a manifestly gauge-invariant Hamiltonian for a massless Dirac particle, which serves as the basis for studying the relativistic Landau problem in graphene in NC space. Specifically, we analyze the motion of a relativistic electron in monolayer graphene within this background field and compute the energy spectrum of the NC Landau system. The NC-modified energy levels are then used to explore the system’s thermodynamic response. Notably, in the low-temperature limit, spatial noncommutativity leads to a spontaneous magnetization—a distinct signature of NC geometry in relativistic condensed matter systems like graphene.