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 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. 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 order parameter for the Laughlin state, extending an earlier description by Stone.