independent of the magnetic field. Nevertheless, we shall see that the temperature makes the observation of the Zeeman splitting in the magnetization be strongly dependent of $B$.

MO at nonzero temperature
=========================

The effects of nonzero temperature can be taken into account in equation (\[M sawtooth\]) by introducing the temperature reduction factor $R_{T}$, which for these systems has the form [@Sharapov2004]

$$R_{T}=\frac{4\pi^{2}\mu pkT/\alpha^{2}B}{\sinh\left(4\pi^{2}\mu pkT/\alpha^{2}B\right)},\label{Rt}$$

where $\alpha=\upsilon_{F}\sqrt{2e\hbar}$. Thus the MO at nonzero temperature becomes

$$M_{T}=\sum_{s=\pm1}A_{s}\sum_{p=1}^{\infty}\frac{R_{T}}{\pi p}\sin\left[2\pi p\omega\left(\frac{1}{B}+s\Delta\right)\right].\label{M temperature}$$

In general, the series given by equation (\[M temperature\]) cannot be evaluated, but it can be simplified under some approximations. The usual one is to take the limit $B\rightarrow0$, in which case $1/\sinh\left(4\pi^{2}\mu pkT/\alpha^{2}B\right)\sim\exp\left(-4\pi^{2}\mu pkT/\alpha^{2}B\right)$. Nevertheless, this limit implies low magnetic field, which in turn makes it difficult to observe the MO. Instead we will impose the low temperature limit, but such that the MO are still observable.

To obtain an approximation for $M_{T}$ at low temperatures, it is convenient to start from the expression of the grand potential $\Omega$ at $T\neq0$ and $\mu>0$. It can be shown that in the absence of impurities, one can use the non-relativistic grand potential to obtain the magnetization [@Tabert2014]. Hence we start with

$$\Omega_{T}=-kT\int_{-\infty}^{\infty}\rho_{0}(\varepsilon)\ln\left[1+e^{\beta\left(\mu-\varepsilon\right)}\right]d\varepsilon,$$

where $\beta=1/kT$ 