  We consider an experimentally realizable model of non-interacting but
randomly coupled atoms in a two-dimensional optical lattice. By choosing
appropriate real or complex-valued random fields and species-dependent energy
offsets, this system can be used to analyze effects of disorder in four
different classes: The chiral BDI and AIII, and the A and AI symmetry classes.
These chiral classes are known to support a metallic phase at zero energy,
which here, due to the inevitable finite size of the system, should also
persist in a neighborhood of non-zero energies. As we discuss, this is of
particular interest for experiments involving quenches. Away from the centre of
the spectrum, we find that excitations appear as domain walls in the cases with
time-reversal symmetry, or as vortices in the cases where time-reversal
symmetry is absent. Therefore, a quench in a system with uniform density would
lead to the formation of either vortices or domain walls depending on the
symmetry class. For the non-chiral models in the A and AI classes, a population
imbalance between the two atomic species naturally occurs. In these cases, one
of the two species is seen to favour a more uniform density. We also study the
onset of localization as the disorder strength is increased for the different
classes, and by deriving an effective model for the non-chiral cases we show
how their eigenstates remain extended for larger values of the coupling with
the disorder, if compared to the non-chiral ones.
