# Microscopic interpretation of magnetization in a 2D electron gas

I'm studying the de Haas-Van Alphen (dHvA) effect in a 2D free electron gas, and I have a problem to interpret the microscopical meaning of the flip of magnetization during the dHvA oscillation. My teacher and textbook ignore the problem with nonchalance, so I ask you.

### Overview of the problem:

It's well known that energy spectrum of a free electron gas in a magnetic field $H$ is quantized into Landau Levels (LLs), each of them with energy: $$E_n = \hbar\omega_c$$ where $\omega_c=eH/mc$ is the cyclotron frequency. We could so expect that, since energy of levels increases with $H$, the ground state energy $E_0(H)$ increase with $H$. But this isn't the case, and we discover that $E_0(H)$ exhibit a more complex behavior. In fact, even if the LLs' energy increase with $H$, so degeneration of LLs (I will indicate with $N_L$) does: $$N_L=\frac{e}{hc}H\cdot S$$ where $S$ is the surface where my electron gas lives. Therefore, there will be a threshold value $H_t$ of the the applied magnetic field for which degeneration of LLs is equal to the total number of electrons $N_e$. If $H>H_t$, every electron is in the lowest LL. Let's now suppose to be over the threshold value $H_t$, and start to decrease the applied magnetic field. The ground state energy decrease and decrease, until $H=H_t$: when the value of magnetic field goes under this value, I can no more accomodate every electron in the lowest LL, so electrons start to occupy the first unoccupied level, who is $\hbar\omega_c$ higher in energy. Everytime I lower the magnetic field under a value $H_t/i$ (with $i$ integer), electrons start to occupy the lowest unoccupied LL, and $E_0(H)$ suddenly increases. These are the so called de Haas-Van Alphen oscillations in a 2D gas of electrons.

### Meaning of Magnetization:

What's my problem? From a thermodynamical point of view,magnetization $M$ is that quantity that gives me information about how much change energy of the system varying magnetic field. It's defined by: $$M=-\frac{1}{S}\frac{\partial E_0}{\partial H}$$ Since $E_0$ is oscillating and singular in $H=H_t$, I have no matter to understand why magnetization is alternatively positive and negative, and discontinue in $H=H_t$. What I don't understand, It's how can interpret microscopically the behavior of magnetization. In fact (from the courses of classical electromagnetism) I have the habit to interpret the magnetization as the sum of magnetic dipoles of elementary constituents of the system. In other words, I'd like to express magnetization as a sum like this:

$$\sum_\alpha \mu_{\alpha}$$ being $\{\alpha\}$ a set of quantum numbers labelling states of my electron gas, and $\mu_{\alpha}$ the expectation values on $|\alpha\rangle$ of an appropriate magnetic dipole operator. (In principle, it doesn't seem too complicated: I need only choosing a gauge, diagonalizing the Hamiltonian of electron gas in magnetic field, and evaluating element of matrix of the operatore $\mu$, that I can deduce from classical case)

Let's assume that such an explanation of magnetization is possible: how can I interpret, in this frame, flips of magnetization due to the dHvA effect? I mean, if I lower the magnetic field under $H_t$, even if to a slightly lower value, magnetization flips. It means that also a very small number of electrons migrating in the first unoccupied LL may brutally change the magnetization of the whole system! It's a quite stunning effect, if we see from this point of view. What does it imply, exactly? Maybe different Landau levels carry different magnetic moments? I don't think this is the answers: after all, every time $H=H_t/i$ and LLs are or completely filled or completely empty, magnetization is always the same (see the figure)And, finally: does all the states in the same LL carry the same magnetic dipole? And, if the answers to this last question is negative: so what happens when I start to occupy that level?