in the $y$-direction and now $m_y$ is an integer. We can estimate the degeneracy here as well. The magnetic flux $\Phi_B$ between the centres of two successive $m_y$ eigenstates $\Psi_{n,m_y+1}$ and $\Psi_{n,m_y}$ is $$\Phi_B=Ba(x_{k+1}-x_k)=Ba\left(\frac{2\pi (m_y+1)}{a}l^2_B-\frac{2\pi m_y}{a}l^2_B\right)=B\frac{2\pi\hbar}{eB}=\frac{h}{e}=\Phi_0,
   \label{degeneracy_landau}$$ so, as in the Symmetric gauge, each Landau gauge eigenstate contains roughly a single quantum of flux, $\Phi_0$. Thus, the expected degeneracy $G$ in a finite system of area $A$ is $G = BA/(h/e) = \Phi/\Phi_0$, as before. The infinite degeneracy in both cases allows the wave functions in the two gauges to differ by more than the simple “textbook” phase, as discussed by Swenson[@swenson89] and Wakamatsu et al.[@wakamatsu18]

In summary, the two gauges discussed both represent the same uniform magnetic field $\mathbf{B}=B\boldsymbol{\hat{z}}$. They result in the same energy spectrum, and yet they produce two very different-looking eigenstates (see Fig. \[fig: fs symm vs landau\]). There are gauge-invariant similarities as well; both gauges produce infinitely degenerate Landau levels in free space and both gauges have degeneracies that are expected to be approximately $G \equiv BA/\Phi_0$ in a finite space with area $A$. Eq. (\[psi\_transform\]) suggests that the two eigenstates should differ by a simple phase factor. That does not happen here, because of 