Orbital magnetism of a two-dimensional noncommutative confined system

We study a system of spinless electrons moving in a two-dimensional noncommutative space subject to a perpendicular magnetic field B-bar and confined by a harmonic potential type 1/2mw0r2. We look for the orbital magnetism of the electrons in different regimes of temperature T, magnetic field B-bar and noncommutative parameter θ. We prove that the degeneracy of Landau levels can be lifted by the θ-term appearing in the electron energy spectrum at weak magnetic field. Using the Berezin-Lieb inequalities for thermodynamical potential, it is shown that in the high-temperature limit, the system exhibits a magnetic θ-dependent behaviour, which is missing in the commutative case. Moreover, a correction to susceptibility at low T is observed. Using the Fermi-Dirac trace formulas, a generalization of the thermodynamical potential, the average number of electrons and the magnetization is obtained. There is a critical point where the thermodynamical potential becomes infinite in both of the two methods above. So at this point we deal with the partition function by adopting another approach. The standard results in the commutative case for this model can be recovered by switching off the θ-parameter. (author)