The thermal properties of a two-dimensional Dirac oscillator under an external magnetic field

In the language of creation and annihilation operators, we study a relativistic spin-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{1} {2}$$\end{document} fermion subject to a Dirac oscillator (DO) coupling and a constant magnetic field in both commutative and non-commutative (NC) spaces. All dynamical physical variables, in a two-dimensional complex formalism, are expressed in terms of the creation and annihilation operators via az, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar a_z$$\end{document} and az, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar a_z$$\end{document} in the commutative space, and dz, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar d_z$$\end{document} and dz, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar d_z$$\end{document} in the NC space. The eigensolutions of our problem have been determinated, and the exact connection with both Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) models has been established. In addition, we revealed the existence of the quantum phase transition in both commutative and NC spaces. The thermal properties of the Dirac oscillator under a magnetic field, calculated from the partition function, have been investigated, and the effect of the non-commutative parameters on these properties has been tested.