  Topological insulators are promising for spintronics and related technologies
due to their spin-momentum-locked edge states, which are protected by
time-reversal symmetry. In addition to the unique fundamental physics that
arises in these systems, the potential technological applications of these
protected states has also been driving TI research over the past decade.
However, most known topological insulator materials naturally contain spinful
nuclei, and their hyperfine coupling to helical edge states intrinsically
breaks time-reversal symmetry, removing the topological protection and enabling
the buildup of dynamic nuclear spin polarization through hyperfine-assisted
backscattering. Here, we calculate scattering probabilities and nuclear
polarization for edge channels containing up to $34$ nuclear spins using a
numerically exact analysis that exploits the symmetries of the problem to
drastically reduce the computational complexity. We then show the emergence of
universal scaling properties that allow us to extrapolate our findings to
vastly larger and experimentally relevant system sizes. We find that
significant nuclear polarization can result from relatively weak helical edge
currents, suggesting that it may be an important factor affecting spin
transport in topological insulator devices.
