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

We investigate the relativistic quantum dynamics of amassless 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 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.