The Effect of the Tilted Field in the Fractional Quantum Hall Systems: Numerical Studies for the Solid-liquid Transition

We construct a generalized Laughlin-liquid wave function and a variational electron solid wave function when the magnetic field is tilted. The energy of the liquid state is evaluated by Monte Carlo methods while the energy of the solid state is calculated by the optimization. Comparing these two energies for a given tilted angle $\theta$, it is seen that the critical filling factor $\nu_c$ of $\theta$ of the solid-liquid transition increases as the tilted angle. The implication to the experiment is that: i) the insulating phase may harder be melt as $\nu\to 1/5$ such that the width of the valley of the longitudinal resistance may become narrow as the filed is tilted; ii) it is expected that even in the vicinity of $\nu=1/3$ for the electron system in the presence of the tilted field, the insulating phase may be observed.