Holomorphic state within a Landau level: a new discrete formulation in periodic (torus) geometry

It is widely believed that the holomorphic character of model states of particles projected into a Landau level (e.g., the Laughlin state) results from their being ``lowest Landau level Schr\"{o}dinger wavefunctions''. In fact this ``common wisdom'' is incorrect, as should have been obvious after the observation of a Laughlin-type state in the second Landau level. Indeed, even the notion that these states are described by a ``wavefunction'' is misleading, because non-locality after projection into (any) Landau level removes the possibility of using a local basis that defines $\Psi(\bm x)$ = $\langle \bm x|\Psi\rangle$, and a Heisenberg formalism must be used. I have found this reinterpretation of the holomorphic states is not just a ``debating point'' but leads to powerful new identities on the torus that were previously missed. I present a new discretized and modular-invariant formulation based on a ``lattice'' of $(N_{\Phi})^2$ points in the fundamental region through which $N_{\Phi}$ flux quanta pass. This mathematically-exact reformulation allows a greatly-improved (faster) formulation for Monte-Carlo studies of model FQH states, as well as explicitly implementing the physical requirement of modular invariance, and expansions in an orthonormal Landau basis.