Variational Monte Carlo calculation of the spin gap in theν=1quantum Hall liquid

A variational method of calculating the spin 1 excitation energies of the $\nu =1\mathrm{}$ quantum Hall liquid is presented in detail. We are able to include Landau-level mixing and finite-thickness effects. The many-body wave functions of the states near $\nu =1\mathrm{}$ filling are taken to be a product of a lowest-Landau-level Laughlin-type wave function and a Jastrow factor. A significant reduction in the calculated spin gap is found due both to Landau-level mixing $(\sim 40\%$ at $r_s=1)$ and to the effect of the finite thickness of the liquid (a further $\sim 30\%$ at $r_s=1)$. We find our calculations to be consistent with recent photoluminescence experiments in $\delta$-doped samples.