Noncommutative 2+12+1 dimensional Dirac oscillator and quantum phase transition

We study a charged Dirac oscillator on a noncommutative plane coupling to a uniform perpendicular magnetic field. We find that there is an exact map from this model onto a quantum optics model which contains both Anti-Jaynes–Cummings (AJC) and Jaynes–Cummings (JC) interactions simultaneously. And these two interactions compete each other when the dimensionless parameter κκ changes. Furthermore, this model behaves as a quantum phase transition when κκ crosses the critical point. However, different from the non-relativistic charged particles coupling to a uniform perpendicular with a harmonic oscillator potential on the noncommutative plane, we find that the critical point of this model is shifted from κ=0κ=0. And it also deviates from the critical point of its commutative counterpart because of spatial noncommutativity. Therefore, it may afford a method to detect the spatial noncommutativity experimentally. Finally, we investigate several characteristics of quantum phase transition.