metric that maximizes the overlap with the family of Laughlin wavefunctions (Fig. \[fig\_intrinsic\]). With the usual definition of the momentum $|k|$, several roton minima appear. Different magneto-roton branches collapse onto the same curve if we plot them as a function of $\sqrt{g^{ab}k_ak_b}$. This is reasonable, because the magneto-roton mode is well approximated by single-mode approximation up to the roton minimum [@boyang], which is defined entirely in terms of the properties of the ground state. The anisotropy of the ground state structure factor (determined by the shape of elementary droplets) dictates the position of the roton minimum.

The analysis of this section in principle applies to other LLL states at fillings $\nu=p/(2p+1), p=2,3,\ldots$, though it is more involved because of the “multicomponent nature" of these states and typically a smaller excitation gap.

Higher Landau levels: quantum phase transitions driven by anisotropy {#sec_ll1}
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We have found that $\nu=1/3$ in the LLL is particularly robust with respect to anisotropy, and this is the case also with other prominent FQH states. In higher LLs, due to a number of nodes in the single-particle wavefunction, the region of the phase diagram where incompressible states occur becomes increasingly narrower, and compressible phases such as stripes 