Cluster-cluster correlations beyond the Laughlin state

Number of zeros seen by a particle around small clusters of other particles is encoded in the root partition, and partly characterizes the correlations in fractional quantum Hall trial wavefunctions. We explore a generalization wherein we consider the counting of zeros seen by a cluster of particles on another cluster. Numbers of such zeros between clusters in the Laughlin wavefunctions are fully determined by the root partition. However, such a counting is unclear for general Jain states where a polynomial expansion is difficult. Here we consider the simplest state beyond the Laughlin wavefunction, namely a state containing a single quasiparticle of the Laughlin state. We show numerically and analytically that in the trial wavefunction for the quasiparticle of the Laughlin state, counting of zeros seen by a cluster on another cluster depends on the relative dimensions of the two clusters. We further ask if the patterns in the counting of zeros extend, in at least an approximate sense, to wavefunctions beyond the trial states. Using numerical computations in systems up to $N=9$, we present results for the statistical distribution of zeros around particle clusters at the center of an FQH droplet in the ground state of a Hamiltonian that is perturbed away from the $V_1$ interaction (short-range repulsion). Evolution of this distribution with the strength of the perturbation shows that the counting of zeros is altered by even a weak perturbation away from the parent Hamiltonian, though the perturbations do not change the phase of the system.