function (a) represent the onset of compressibility and CDW ordering.[]{data-label="fig_s0qonethird"}](s0q_onethird.eps){width="0.8\linewidth"}

Although idealized numerical calculations with pure (projected) Coulomb interaction work exceedingly well in $n=0$ LL, more realistic models are required to describe phases in $n=1$ LL. In particular, the inclusion of finite width effects [@peterson] and varying a few strongest Haldane pseudopotentials in necessary to determine the phase diagram. We find that varying the $V_1$ pseudopotential leads to the following outcomes: (i) generically, for $\delta V_1<0$, the system is pushed deeper into a compressible phase; (ii) for $\delta V_1>0$, finite-size calculations on systems up to $N_e=9$ electrons permit the existence of two regimes: for $0<\delta V_1^a < \delta V_1 < \delta V_1^b$, the ground state is in the Laughlin universality class, but the lowest excitation is not the magneto-roton; for $\delta V_1 > \delta V_1^b$, the ground state *and* the excitation spectrum is the same as in $n=0$ LL. For smaller systems, $\delta V_1^b$ is estimated to be around $0.1e^2/\epsilon \ell_B$, while $\delta V_1^a$ is around $0.04e^2/\epsilon \ell_B$. Larger systems suggest that these two points might merge in the thermodynamic limit, when only a small modification of the interaction might be needed for the Laughlin physics to appear at $\nu=1/3$ 