advantage of having adopted the symmetric gauge with this geometry is that the resulting Hamiltonian is diagonal in one of the quantum numbers, $\ell$. This represents a tremendous computational gain. We mention it here because we will [*not*]{} have this possibility when we study the square geometry in the next section. There we will have to diagonalize $N^2 \times N^2$ matrices, where $N$ is a large number (here typically something like $50$ or $200$. Also, for circular geometry we can monitor the quantum number $\ell$ precisely, which means we can track the radial quantum number $n_r$ as well. In the Symmetric gauge the Landau level quantum number $n$ is given simply by $n = n_r + (\ell + |\ell|)/2$. We will not have this possibility with the square geometry.

[Results and Discussion]{}
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To be clear, in practice in this section we diagonalize $N \times N$ matrices, and only show results that have converged as $N$ increases (recall that $N$ should be infinite, but is chosen to be finite and represents the number of radial basis states (i.e. number of $n$) for a given $\ell$ in Eq. (\[circ\_basis\]) to be used in the matrix diagonalization). We will do this for 