equations (\[freq\]) and (\[phase\]).](fig3.eps)

$$\Delta M_{s}=A_{+}\left(1-\frac{2}{\pi}\arctan\left[\cot\left(2\pi\omega\Delta\right)\right]\right),\label{Delta Ms}$$

where $A_{+}\equiv\left|A_{1}+A_{-1}\right|/2=\mu/\phi$. A better measurement of the effect is to take the relation between $\Delta M_{s}$ and the amplitude of the peaks, which can be approximated as $A_{+}$ (because $A_1\simeq A_{-1}$). On the other hand, we can also analyze the splitting peak width, which is given by the phase difference $2\Delta=4\mu_{\mathrm{B}}/\mu$. This width also affect the visualization of the spin splitting in $M$, for at low values of Fermi energy, $2\Delta$ becomes insignificant. Then we can compare this width with the oscillation period, given by $1/\omega=\alpha^{2}/\mu^{2}.$ In figure \[fig3\] it is plotted $\Delta M_{s}/A_{+}$ and $2\Delta\omega$ as a function of the Fermi energy. We see that for $\mu\lesssim100$ meV we have $\Delta M_{s}/A_{+}\sim10^{-2}$, while $2\Delta\omega$ is also very small, which makes the spin splitting effect practically unobservable. But when we get to higher Fermi energies, about $\mu\gtrsim250$ meV, we have $\Delta M_{s}/A_{+}\sim0.1$ so the effect starts to become more evident. Consequently, in order to account for the Zeeman effect in $M$ we will take Fermi energies above 200 meV. Notice also that $\Delta M_{s}/A_{+}$ does not depend on $B$, so the effect of the Zeeman splitting in the magnetization at zero temperature is 