Quantum Phase Trasition Of Commutative And Noncommutative Planar Dirac Ascillator

In this paper we study the Quantum phase transition of the commutative and noncommutative planar Dirac oscillator interacting with a perpendicular uniform magnetic field.Our work shows that there are intrinsic relations between the Dirac oscillator and the Anti-Jaynes-Cummings (AJC)and Jaynes-Cummings (JC) models in Quantum optics on the commutative planar. What’s more, we find that there are competitions between AJC and JC interactions when the intensity of magnetic field changes. And there are abrupt transitions between JC and AJC interactions when the intensities of the magnetic fields cross the critical points. Interestingly, at the critical point, all the phenomena are identical to the Quantum phase transition. These quantum phase transitions, accompanied with the flipping of the chiralites of phonons, are named as Chiralitiy quantum phase transitions.Then, we pay our attentions on the models of the noncommutative planar Dirac oscillators interacting with the perpendicular magnetic fields. We study the conventional、exotic andZhang manners of coupling the magnetic fields respectively. We find that, similar with the commutative case, there are intrinsic relations between Dirac oscillators on the noncommutative plane and the quantum optics models which contain both AJC and JC interactions simultaneously. And there are completions between these two interactions when the magnitude of the magnetic field changes.In addition, when the magnitudes of the magnetic field cross critical points, the chirality quantum phase transitions will happen. However, we find that for the conventional and Zhang’s manner, there is only one critical point. But the critical points of conventional and Zhang’s coupling are different. While for the exotic manner, there are two critical points. We also investigate the non-relativistic limit of these models and find that the properties of chirality quantum phase transitions survive the relativistic reductions. Our studies may offer a new approach to determine the manners of coupling the magnetic fields in noncommutative quantum mechanics. In addition, we hope that the conclusions of our research work can contribute to the understanding of the new finding low dimensional material (Graphene).