Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 5 / a / Solution

A magnetic field quantizes the transverse cyclotron motion into Landau levels. For a parabolic band, this is a harmonic oscillator with $E_n=\hslash {\omega }_c(n+1/2)$, $n=0,1,\dots$, where ${\omega }_c=eB/m_c$ and the half arises from its zero-point energy. Equivalently, semiclassical orbit quantization gives the stated flux sequence in increments of $h/e$.

To relate real and momentum space, take $\mathbf{B}=B\hat{\mathbf{z}}$ and electron charge $-e$. The semiclassical Lorentz force isConsequently $dx=\hslash d{k}_y/(eB)$ and $dy=-\hslash d{k}_x/(eB)$. The real-space orbit is a rotated copy of the transverse wavevector orbit, scaled by $\hslash /(eB)$. Their areas satisfy $S_r=[\hslash /(eB){]}^2{S}_k$. Since $B{S}_r=(n+1/2)h/e$,This is Onsager quantization for the ordinary half-integer phase. For a circular parabolic orbit, $S_k=\pi k^2$ reproduces the Landau energy spacing. The flux increment $h/e$ here concerns a single electron; it is distinct from the superconducting pair flux unit.