a FQH system with band mass anisotropy. In Sec. \[sec\_laughlin\] we discuss the intrinsic metric of the Laughlin state. We define a family of the Laughlin wavefunctions characterized by the varying shape of their elementary droplets. Intrinsic metric is determined variationally by optimizing the overlap between this family of wavefunctions and the exact Coulomb ground state. In Sec.\[sec\_ll1\] we perform exact diagonalization of finite systems at several filling factors to explore quantum phase transitions that occur as a function of the anisotropy. Our conclusions are presented in Sec.\[sec\_conclusion\].

Model {#sec_model}
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Consider an electron moving in the plane with the perpendicular magnetic field $B\hat{z}=\nabla\times \mathbf{A}(\mathbf{r})$. The Hamiltonian can be written in the following, manifestly covariant form: $$\begin{aligned}
K^{\alpha,\phi} =\frac{1}{2m} g^{ab} \pi_{a} \pi_{b}.\end{aligned}$$ Here $\pi_a= p_a - \frac{e}{c} A_a(\mathbf{r}), a=x,y$ represents the dynamical momentum, and $g$ is the mass tensor parametrized by the anisotropy $\alpha$ and the angle of the principal axis $\phi$. The mass tensor is unimodular $\det g=1$. In the isotropic case when $g$ is the unit matrix, we can obtain the single-particle energies (Landau levels) by choosing, for example, a symmetric gauge $A_x=By/2, A_y=-Bx/2$. In this case, the dynamical momenta become $\pi_x=-i\hbar \frac{\partial}{\partial x}+\frac{\hbar}{2\ell_B^2}y$ and $\pi_y=-i\hbar \frac{\partial}{\partial y} 