  We discuss a protocol for growing states with topological order in
interacting many-body systems using a sequence of flux quanta and particle
insertion. We first consider a simple toy model, the superlattice Bose Hubbard
model, to explain all required ingredients. Our protocol is then applied to
fractional quantum Hall systems in both, continuum and lattice. We investigate
in particular how the fidelity, with which a topologically ordered state can be
grown, scales with increasing particle number N. For small systems exact
diagonalization methods are used. To treat large systems with many particles,
we introduce an effective model based on the composite fermion description of
the fractional quantum Hall effect. This model also allows to take into account
the effects of dispersive bands and edges in the system, which will be
discussed in detail.
