+ x \right)^2} \\
H_n \left( \frac{X_j+ka+x}{\sqrt{\alpha}} \right),\end{aligned}$$ where $j=0,\ldots,N_\phi-1$, $X_j \equiv 2\pi j/b$, normalization factor is $\mathcal{N}=b\sqrt{\pi} \sqrt{\alpha}2^n n!$, and the sum over $k$ extends over all integers. We have set $\ell_B=1$. The wavefunction for the $n$th LL involves a Hermite polynomial $H_n$. For simplicity, we assumed the case of a rectangular torus and $g={\rm diag}\left[\alpha, 1/\alpha\right]$, but Eq.(\[landaugauge\]) can be generalized to an arbitrary shape/anisotropy using Jacobi theta functions.

Many-body states, like in the isotropic case, can be classified using a crystal quasimomentum $\mathbf{K}$ [@duncan_translations] defined in a Brillouin zone. With a suitable definition of the Brillouin zone, incompressible states always occur at $\mathbf{K}=0$, and are characterized by the gap in their excitation spectrum. The Hamiltonian for $N_e$ electrons is given by the sum of the kinetic term and the Coulomb interaction, $$\begin{aligned}
\label{fullham}
H^{\alpha,\phi} = \sum_i K_i^{\alpha,\phi} + \sum_{i<j} \frac{1}{|\mathbf{r}_i - \mathbf{r}_j|_\epsilon}.\end{aligned}$$ 2DES is embedded in a three-dimensional dielectric host material, which is characterized by its own dielectic tensor $\epsilon$, defining the metric for the distance $|\mathbf{r}_i-\mathbf{r}_j|_\epsilon$ between interacting electrons. This tensor in general can be different from the one that parametrizes the cyclotron orbits ($\alpha,\phi$). However, physical properties are determined only by the relative difference between 