  Robustness to disorder - the defining property of any topological state - has
been mostly tested in low-disorder translationally-invariant materials systems
where the protecting underlying symmetry, such as time reversal, is preserved.
The ultimate disorder limits to topological protection are still unknown,
however, a number of theories predict that even in the amorphous state a
quantized conductance might yet reemerge. Here we report a directly detected
robust spin response in structurally disordered thin films of the topological
material Sb2Te3 free of extrinsic magnetic dopants, which we controllably tune
from a strong (amorphous) to a weak crystalline) disorder state. The magnetic
signal onsets at a surprisingly high temperature (~ 200 K) and eventually
ceases within the crystalline state. We demonstrate that in a strongly
disordered state disorder-induced spin correlations dominate the transport of
charge - they engender a spin memory phenomenon, generated by the
nonequilibrium charge currents controlled by localized spins. The negative
magnetoresistance (MR) in the extensive spin-memory phase space is isotropic.
Within the crystalline state, it transitions into a positive MR corresponding
to the weak antilocalization (WAL) quantum interference effect, with a 2D
scaling characteristic of the topological state. Our findings demonstrate that
these nonequilibrium currents set a disorder threshold to the topological
state; they lay out a path to tunable spin-dependent charge transport and point
to new possibilities of spin control by disorder engineering of topological
materials
