Hidden Massive Spectators in the Effective Field Theory for Integral Quantum Hall Transitions

Integral quantum Hall plateau transitions in a planar lattice system due to gap collapse can be described by an effective field theory with Dirac fermions. We discuss how to reproduce the correct integral values for the Hall conductance, $\sigma_{xy}$, before and after the plateau transition, which are dictated by the microscopic topological invariant. In addition to the massless Dirac fermions that appear at gap closing, the matching condition of $\sigma_{xy}$ requires the introduction of massive Dirac fermions, as "spectators", in the effective field theory. For non-interacting electrons on the lattice, we give a general prescription to determine these massive "spectators", based on microscopic information on the vortices in the magnetic Brillouin zone which are closely related to edge states. Our description is demonstrated in a model with both nearest-neighbor and next-nearest-neighbor hoppings.