  Topological crystalline insulators are materials in which the crystalline
symmetry leads to topologically protected surface states with a chiral spin
texture, rendering them potential candidates for spintronics applications.
Using scanning tunneling spectroscopy, we uncover the existence of
one-dimensional (1D) midgap states at odd-atomic surface step edges of the
three- dimensional topological crystalline insulator (Pb,Sn)Se. A minimal toy
model and realistic tight- binding calculations identify them as spin-polarized
flat bands connecting two Dirac points. This non-trivial origin provides the 1D
midgap states with inherent stability and protects them from backscattering. We
experimentally show that this stability results in a striking robustness to
defects, strong magnetic fields, and elevated temperature.
