  We establish the theory of critical transport in amorphous Chern insulators
and show that it lies beyond the current paradigm of topological criticality
epitomized by the quantum Hall transitions. We consider models of Chern
insulators on percolation-type random lattices where the average density
determines the statistical properties of geometry. While these systems display
a two-parameter scaling behaviour near the critical density, the critical
exponents and the critical conductance distributions are strikingly
nonuniversal. Our analysis indicates that the amorphous topological criticality
results from an interpolation of a geometric-type transition at low density and
an Anderson localization-type transition at high density. Our work demonstrates
how the recently discovered amorphous topological systems display unique
phenomena distinct from their conventionally-studied counterparts.
