Long coherence times for edge spins

We show that in certain one-dimensional spin chains with open boundary conditions, the edge spins retain memory of their initial state for very long times, even at infinite temperature. The long coherence times do not require disorder, only an ordered phase. In the integrable Ising and XYZ chains, the presence of a strong zero mode means the coherence time is infinite. When Ising is perturbed by interactions breaking the integrability, the coherence time remains exponentially long in the perturbing couplings. We show that this is a consequence of an edge 'almost' strong zero mode that almost commutes with the Hamiltonian. We compute this operator explicitly, allowing us to estimate accurately the plateau value of edge spin autocorrelator.