  We study a variety of Hofstadter ladders in order to probe the interplay
between interactions, an applied magnetic field and crystalline or
quasiperiodic geometries. Rotational motion will be induced on charged
particles when a magnetic field is present, which can result in exotic
distributions of current on a lattice. Typically, the geometry of a ladder
lattice is assumed to be homogeneous. In this work, however, we will also study
ladders that possess non-uniform bond lengths, in order to study the formation
of localised currents. By using Density Matrix Renormalisation Group (DMRG) to
characterise the quantum phases, we confirm the presence of the usual vortex
and Meissner distributions of current, in which particles circulate within the
bulk and around the edge respectively. Furthermore, it is also possible to
observe variations to these patterns; which combine both vortex and Meissner
order, and the onset of incompressible domains for certain fillings of the
lattice. If the bond lengths of a ladder fluctuate, we find substantial
differences to the structure of currents. This is a consequence of an
inhomogeneous, effective magnetic flux, resulting in preferential localisation
of currents throughout the lattice bulk, towards the smaller bond lengths. We
then find that incompressible domains can significantly grow in size extent
across the parameter space, with currents no longer possessing an extended
structure across the longitudinal direction of the ladder.
