is the one that minimizes the variational energy $E_g$, $$\label{correnergy}
E_g = \frac{\langle \Psi_L(g) | H(g_m, g_C)  | \Psi_L (g) \rangle}{\langle \Psi_L(g) | \Psi_L (g) \rangle}.$$

To find the intrinsic metric in the microscopic calculation, we use a slightly different criterion that $g$ should maximize the overlap with the exact ground state of $H(g_m,g_C)$. In the rotationally-invariant case, the ground state of the Coulomb interaction at $\nu=1/3$ is known to have a remarkably high overlap with the Laughlin wavefunction. The overlap is defined as a scalar product between two normalized vectors, and in this particular case it is typically greater than 97%. Therefore, we expect the intrisic metric chosen to maximize the overlap also to minimize the correlation energy (\[correnergy\]). To obtain the anisotropic Laughlin states, we perform exact diagonalization of the “$V_1$ Hamiltonian" on the torus. This Hamiltonian gives $\Psi_L^{\nu=1/3}$ as a unique and densest zero-energy ground state [@haldane_prange]. Note that any translationally-invariant interaction can be expanded in terms of the Laguerre polynomials, $V(\mathbf{q})=\sum_m V_m \mathcal{L}_m(\mathbf{q}^2)$, where the coefficients $V_m$ are the Haldane pseudopotentials [@haldane_prange]. Truncating this expansion at the first term, $V_1 \mathcal{L}_1(\mathbf{q}^2)$, singles out the strongest (“hard-core") component, which defines the Laughlin state at filling $\nu=1/3$. As 