Study of a two-dimensional electron gas in a magnetic field. I. Weak field

A many-body theory of a two-dimensional electron gas in a magnetic field is presented. The field dependence of the Fermi momentum shows a two-dimensional peculiarity. It consists of the spin-dependent paramagnetic part and the orbital diamagnetic part. The former increases more strongly with $r_s$ than the latter. The paramagnetic and diamagnetic susceptibilities are found as functions of $r_s$. The field-dependent ground-state energy is evaluated. Its paramagnetic part decreases steadily with $r_s$, while the diamagnetic part shows a maximum. No such maximum has been found for the three-dimensional case in the same approximation which is valid for high density and low magnetic fields. From the susceptibilities, an effective $g$ factor and an effective mass are derived. They are given approximately by $\frac{g^{*}}{g}=1+0.87\mathrm{}\times {10}^6{n}^{-\frac{1}{2}}$ or more accurately by $\frac{g^{*2\mathrm{}}}{g^2}=1+1.74\mathrm{}\times {10}^6{n}^{-\frac{1}{2}}-0.708\mathrm{}\times {10}^{12}{n}^{-1\mathrm{}}$ and by $\frac{m^{*}}{m}=1+3.21\mathrm{}\times {10}^4{n}^{-\frac{1}{2}}$, where $n$ is the number density.