Correlations in incompressible quantum liquid states: constructions of electronic trial wavefunctions

Numerical studies indicate that incompressible quantum Hall states occur when the relation between the single particle angular momentum $l$ and the number $N$ of electrons in the partially filled Landau level is $2l = \nu^{-1}N-c_\nu$. Here, $\nu$ is the filling factor and $c_\nu$ is a ``finite size shift.'' The values of $c_\nu$ found numerically depend on correlations, and for $\nu=p/q\leq 1/2$ are given by $c_\nu = q+1-p$. This finite size shift points the way to constructing electronic trial wavefunctions. A trial wavefunction can always be written $\Psi = FC$, where $F = \prod_{i < j}z_{ij}$ and $C(z_{ij})$ is a symmetric correlation function caused by interactions. For the Moore-Read state, $C_{MR}(z_{ij})$ is a product of $F$ and the antisymmetric Pfaffian. $C_{MR}$ is not the only possible correlation function for this state. Another choice is the quadratic function $C_Q = S \left\{\prod_{i < j\in g_A} \prod_{k < l\in g_B}(z_{ij}z_{kl})^2\right\}$, where $S$ is a symmetrizing operator, and $g_A$ and $g_B$ each contain $N/2$ particles resulting from a partition of N into two sets. For the Jain states (e.g. $\nu=2/5$), different partitioning of $N$ particles into sets of unequal size gives appropriate correlation functions.