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.

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• Majorana bound states
• Majorana fermions
• Quantum control
• Quantum correlations in quantum information
• Quantum simulation
• Quantum spin models
• Topological superconductors
• Quantum spin chains