  We provide a comprehensive quantitative analysis of localized and extended
topological defects in the steady state of 2D passive and active repulsive
Brownian disk systems. We show that, both in and out-of-equilibrium, the
passage from the solid to the hexatic is driven by the unbinding of
dislocations, in quantitative agreement with the KTHNY singularity. Instead,
although disclinations dissociate as soon as the liquid phase appears, extended
clusters of defects largely dominate below the solid-hexatic critical line. The
latter percolate in the liquid phase very close to the hexatic-liquid
transition, both for continuous and discontinuous transitions, in the
homogeneous liquid regime. At critical percolation the clusters of defects are
fractal with statistical and geometric properties that, within our numerical
accuracy, are independent of the activity and compatible with the universality
class of uncorrelated critical percolation. We also characterize the spatial
organization of different kinds of point-like defects and we show that the
disclinations are not free, but rather always very near more complex defect
structures. At high activity, the bulk of the dense phase generated by
Motility-Induced Phase Separation is characterized by a density of point-like
defects, and statistics and morphology of defect clusters, set by the amount of
activity and not the packing fraction. Hexatic domains within the dense phase
are separated by grain-boundaries along which a finite network of topological
defects resides, interrupted by gas bubbles in cavitation. The fractal
dimension of this network diminishes for increasing activity. This structure is
dynamic in the sense that the defect network allows for an unzipping mechanism
that leaves free space for gas bubbles to appear, close, and even be released
into the dilute phase.
