Zero modes, bosonization, and topological quantum order: The Laughlin state in second quantization

We introduce a "second-quantized" representation of the ring of symmetric functions to further develop a purely second-quantized, or "lattice," approach to the study of zero modes of frustration-free Haldane-pseudopotential-type Hamiltonians, which in particular stabilize Laughlin ground states. We present three applications of this formalism. We start demonstrating how to systematically construct all zero modes of Laughlin-type parent Hamiltonians in a framework that is free of first-quantized polynomial wave functions, and show that they are in one-to-one correspondence with dominance patterns. The starting point here is the pseudopotential Hamiltonian in "lattice form," stripped of all information about the analytic structure of Landau levels (dynamical momenta). Second, as a by-product, we make contact with the bosonization method, and obtain an alternative proof for the equivalence between bosonic and fermionic Fock spaces. Finally, we explicitly derive the second-quantized version of Read's nonlocal (string) order parameter for the Laughlin state, extending an earlier description by Stone. Commutation relations between the local quasihole operator and the local electron operator are generalized to various geometries.