the mass tensor and the dielectric tensor, and for simplicity we can set the latter to unity. In other words, we assume that Coulomb interaction is isotropic in space, hence its Fourier transform is $V(\mathbf{q}) = 1/q \equiv 1/\sqrt{q_x^2+q_y^2}$ (to model finite-width effects, we use the softened form of $V(\mathbf{q})$, following the Fang-Howard prescription). Projected to a single $n$th LL, the interaction part of the Hamiltonian becomes $H=\sum_{\{j_i\}}V_{j_1j_2j_3j_4}c_{j_1}^\dagger c_{j_2}^\dagger c_{j_3}c_{j_4}$, where $$\begin{aligned}
\label{matel}
\nonumber V_{j_1j_2j_3j_4} = \frac{1}{2\mathcal{S}} \sum_{\mathbf{q}}^{'} V(\mathbf{q}) \mathcal{L}_n \left( \frac{1}{2} q_g^2 \right)  \\
 e^{-\frac{1}{2} q_g^2 } e^{iq_x \left( X_{j_1}-X_{j_3}\right)} \delta_{t,j_1-j_4}^{'} \delta_{j_1+j_2,j_3+j_4}^{'},\end{aligned}$$ where $q_g^2 \equiv g^{ab}q_a q_b$ (e.g. in case of a diagonal mass tensor, $q_g^2=\alpha q_x^2 + q_y^2/\alpha$) and $\mathcal{L}_n$ is the Laguerre polynomial. The primed $\delta$-functions are to be taken $({\rm mod} \; N_\phi)$, and the sum over $\mathbf{q}$ extends over the reciprocal space (the prime on the sum indicates that the diverging $\mathbf{q}=0$ term is cancelled by the positive background charge).

Apart from the many-body translational symmetry, discrete symmetries can be used to further reduce the Hilbert space. Several types of Bravais lattices are possible, depending on the angle $\theta$ between the sides of the torus, $\mathbf{a}$ and $\mathbf{b}$, and the aspect ratio, $|\mathbf{a}|/|\mathbf{b}|$. Both of 