Lattice Chern-Simons model for FQHE

Motivated by [60][61] and using topological invariance and cycle homology, we construct an effective gauge invariant lattice Chern-Simons (CS) model for the fractional quantum Hall effect (FQHE) with applications to (i) the Laughlin state on 3D crystals, (ii) Haldane hierarchy, and (iii) composite fermions. First, we give a new derivation of the lattice CS action and demonstrate its gauge symmetry and invariance under lattice duality. Then, we use this lattice construction to study the filling factors of the above mentioned QH states. We show amongst others that the νlattL of Laughlin state on lattice is given by the ratio Φ0/ΦWL with ΦWL designating the flux of CS link variables through Wilson loops WL. For composite fermions, we use the adb formalism to show that the factor νlattCF is given by the charge Qb of the dumbbell field under the dual gauge link variables. Our model gives explicitly the effective filling fraction of composite fermions on the lattice as νlatt⁎=νlattCF/(1−2nνlattCF) with νlattCF being the bare lattice filling factor in agreement with the continuum limit result.