  We study theoretically the electronic structure of three-dimensional (3D)
higher-order topological insulators in the presence of step edges. We
numerically find that a 1D conducting state with a helical spin structure,
which also has a linear dispersion near the zero energy, emerges at a step edge
and on the opposite surface of the step edge. We also find that the 1D helical
conducting state on the opposite surface of a step edge emerges when the
electron hopping in the direction perpendicular to the step is weak. In other
words, the existence of the 1D helical conducting state on the opposite surface
of a step edge can be understood by considering an addition of two
different-sized independent blocks of 3D higher-order topological insulators.
On the other hand, when the electron hopping in the direction perpendicular to
the step is strong, the location of the emergent 1D helical conducting state
moves from the opposite surface of a step edge to the dip ($270^{\circ}$ edge)
just below the step edge. In this case, the existence at the dip below the step
edge can be understood by assigning each surface with a sign ($+$ or $-$) of
the mass of the surface Dirac fermions. These two physical pictures are
connected continuously without the bulk bandgap closing. Our finding paves the
way for on-demand creation of 1D helical conducting states from 3D higher-order
topological insulators employing experimental processes commonly used in
thin-film devices, which could lead to, e.g., a realization of high-density
Majorana qubits.
