Laughlin wavefunction

Remarks

The function is antisymmetric: exchanging $z_i$ and $z_j$ multiplies $(z_i - z_j)^m$ by $(-1)^m = -1$ and permutes the other factors. It lies in the lowest Landau level, since its prefactor is a polynomial in the $z_j$. The modulus squared $\abs{\Psi_m}^2$ is the Boltzmann weight of a two-dimensional one-component plasma, which shows that the state is a uniform liquid of density $1/(2\pi m)$ in these units.

Example: for $m = 1$, $\prod_{i<j}(z_i - z_j)$ is, up to sign, the Vandermonde determinant $\det(z_i^{j-1})$, so $\Psi_1$ is the Slater determinant of the orbitals $z^k e^{-\abs{z}^2/4}$, $0 \le k \le N - 1$: the filled lowest Landau level, $\nu = 1$.

Non-example: for $m = 2$ the same formula is symmetric under exchange, so it is not an admissible state of electrons, which are fermions; it describes bosons at $\nu = 1/2$.