  Bound states at sharp corners have been widely viewed as the hallmark of
two-dimensional second-order topological insulators and superconductors. In
this work, we show that the existence of sublattice degrees of freedom can
enrich the tunability of bound states on the boundary and hence lift the
constraint on their locations. We take the Kane-Mele model with
honeycomb-lattice structure to illustrate the underlying physics. With the
introduction of an in-plane exchange field to the model, we find that the
boundary Dirac mass induced by the exchange field has a sensitive dependence on
the boundary sublattice termination. We find that the sensitive sublattice
dependence can lead bound states to emerge at a specific type of boundary
defects named as sublattice domain walls if the exchange field is of
ferromagnetic nature, even in the absence of any sharp corner on the boundary.
Remarkably, this sensitive dependence of the boundary Dirac mass on the
boundary sublattice termination allows the positions of bound states to be
manipulated to any place on the boundary for an appropriately-designed sample.
With a further introduction of conventional s-wave superconductivity to the
model, we find that, no matter whether the exchange field is ferromagnetic,
antiferromagnetic, or ferrimagnetic, highly controllable Majorana zero modes
can be achieved at the sublattice domain walls. Our work reshapes the
understanding of boundary physics in second-order topological phases, and
meanwhile opens potential avenues to realize highly controllable bound states
for potential applications.
