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

Topological phases are often characterized by special edge states confined near the boundaries by an energy gap in the bulk. On raising temperature, these edge states are lost in a clean system due to mobile thermal excitations. Recently, however, it has been established that disorder can localize an isolated many-body system, potentially allowing for a sharply defined topological phase even in a highly excited state. Here we show this to be the case for the topological phase of a one-dimensional magnet with quenched disorder which features spin one-half excitations at the edges. The time evolution of a simple, highly excited initial state is used to reveal quantum coherent edge spins. In particular, we demonstrate, using theoretical arguments and numerical simulation, the coherent revival of an edge spin over a time scale that grows exponentially larger with system size. This is in sharp contrast to the general expectation that quantum bits strongly coupled to a `hot' many body system will rapidly lose coherence.