Zero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization

TL;DR

This paper develops a second-quantized formalism to analyze zero modes and topological order in Laughlin states, providing systematic construction methods, connections to bosonization, and explicit order parameter derivations.

Contribution

It introduces a novel second-quantized approach to study Laughlin states, linking zero modes, bosonization, and topological order without relying on first-quantized wave functions.

Findings

Systematic construction of zero modes from dominance patterns.

Proof of equivalence between bosonic and fermionic Fock spaces.

Explicit derivation of second-quantized non-local order parameter.

Abstract

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-pseudo-potential-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 pseudo-potential Hamiltonian in "lattice form", stripped of all information about the analytic structure of Landau levels (dynamical momenta). Secondly, 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 non-local (string) order parameter for the Laughlin state, extending an earlier description by Stone. Commutation relations between the local quasi-hole operator and the local electron operator are generalized to various geometries.