Localization and topology protected quantum coherence at the edge of hot matter.

Topological phases are characterized by edge states confined near the boundaries by a bulk energy gap. On raising temperature, these edge states are typically lost due to mobile thermal excitations. However, disorder can localize an isolated many-body system, potentially allowing for a sharply defined topological phase even in a highly excited state. We explicitly demonstrate this in a model of a disordered, one-dimensional magnet with spin one-half edge excitations. Furthermore, we show that the time evolution of a simple, highly excited state reveals quantum coherent edge spins. In particular, we demonstrate the coherent revival of an edge spin over a time scale that grows exponentially with system size. This is in sharp contrast to the general expectation that quantum bits strongly coupled with a hot many-body system will rapidly lose coherence. This result opens new directions in the study of topologically protected quantum dynamics.