  The search for topological systems has recently broadened to include random
substitutional alloys, which lack the specific crystalline symmetries that
protect topological phases, raising the question whether topological properties
can be preserved, or are modified by disorder. To address this question, we
avoid methods that assumed at the outset high (averaged) symmetry, using
instead a fully-atomistic, topological description of alloy. Application to
PbSe-SnSe alloy reveals that topology survives in an interesting fashion: (a)
spatial randomness removes the valley degeneracy (splitting larger than 150
meV), leading to a sequential inversion of the split valley components over a
range of compositions; (b) absence of inversion lifts spin degenerates, leading
to a Weyl semimetal phase without the need of external magnetic field, an
unexpected result, given that the alloy constituent compounds are
inversion-symmetric. (a) and (b) underpin the topological physics at low
symmetry and complete the missing understanding of possible topological phases
within the normal-topological insulator transition.
