  Strong zero modes are edge-localized degrees of freedom capable of storing
information at infinite temperature, even in systems with no disorder. To date,
their stability has only been systematically explored at the physical edge of a
system. Here, we extend the notion of strong zero modes to the boundary between
two systems, and present a unifying framework for the stability of these
boundary strong zero modes. Unlike zero-temperature topological edge modes,
which are guaranteed to exist at the interface between a trivial and
topological phase, the robustness of boundary strong zero modes is
significantly more subtle. This subtlety is perhaps best illustrated by the
following dichotomy: we find that the interface between a trivial and ordered
phase does not guarantee the existence of a strong zero mode, while the
interface between two ordered phases can, in certain cases, lead to an exact
strong zero mode.
