Orbital Magnetism of Two-Dimension Noncommutative Confined System

We study a system of spinless electrons moving in a two dimensional noncommutative space subject to a perpendicular magnetic field $\vec B$ and confined by a harmonic potential type ${1\over 2}mw_{0}r^2$. We look for the orbital magnetism of the electrons in different regimes of temperature $T$, magnetic field $\vec B$ and noncommutative parameter $\te$. We prove that the degeneracy of Landau levels can be lifted by the $\te$-term appearing in the electron energy spectrum at weak magnetic field. Using the {\it Berezin-Lieb} inequalities for thermodynamical potential, it is shown that in the high temperature limit, the system exibits a magnetic $\te$-dependent behaviour, which is missing in the commutative case. Moreover, a correction to susceptibility at low $T$ is observed. Using the {\it 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 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 $\te$-parameter.