where $\omega_z$ is the harmonic frequency of the normal confinement. The mixing of the cyclotron frequency and the confinement frequency is parametrized by $\tan 2\bar{\theta}=2\omega_c\omega_\parallel/(\omega_c^2-\omega_z^2)$, where $\omega_\parallel=eB_\parallel/m$. Writing $\lambda_{i=1,2}=\omega_i/\omega_c$, the anisotropy parameter becomes $\alpha=\lambda_1\lambda_2/(\lambda_1\sin^2\bar{\theta}+\lambda_2\cos^2\bar{\theta})$. Parametrizing the in-plane magnetic field by $\mathbf{B}_\parallel=(B_\parallel\cos\phi,B_\parallel\sin\phi)$, the effective metric associated with the tilt is given by $$\begin{aligned}
\nonumber g=\left(\begin{array}{ccc}
\cosh2\theta+\sinh2\theta\cos2\phi &\sinh2\theta\sin2\phi\\
\sinh2\theta\sin2\phi & \cosh2\theta-\sinh2\theta\cos2\phi\end{array}\right), \\ \end{aligned}$$ where $\cosh 2\theta=\frac{1}{2}(\alpha+\frac{1}{\alpha})$. Therefore, the effect of tilting on the LLL single-particle levels can be captured by the variation of the mass tensor.

In order to study a finite, interacting system of $N_e$ electrons, it is convenient to choose a compact surface to represent the 2DES. As we emphasized in Sec.\[sec\_introduction\], the presence of mass anisotropy destroys rotational invariance, and one must use periodic boundary conditions [@yhl; @duncan_translations] i.e. put the 2DES on the surface of a torus. The unit cell can generally be chosen as a parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$ whose area $\mathcal{S}$ is quantized because of the magnetic translations algebra: $\mathcal{S}\equiv |\mathbf{a} \times \mathbf{b}| = 2\pi \ell_B^2 N_\phi$. The single-particle states compatible with periodic boundary conditions are given in the Landau gauge by $$\begin{aligned}
\label{landaugauge}
\nonumber \phi_{j,n}^\alpha (\mathbf{r}) = \frac{1}{\sqrt{\mathcal{N}}} \sum_k e^{i\left(X_j+ka\right)y -\frac{1}{2\alpha}\left(X_j + ka 