Edge dynamics of the Ising chain in the presence of a static boundary field

We study dynamics of the one-dimensional Ising model in the presence of static boundary field (h_B) via the two-time correlation function of the boundary spin, │〈σ^z₁  (0)σ^z₁ (t)〉_c │. We find that the correlations decay as a power law. Interestingly, we unveil a dynamical phase diagram, previously uncharted, where upon tuning the strength of the boundary field and distance from bulk critical point, we observe qualitatively distinct temporal behavior in correspondence to the change of number of edge modes as h_B and the bulk transverse field vary. These dynamical boundary phase transitions are separated by critical lines. In the presence of large boundary and bulk field, correlations are infinitely long lived and can be trace back to stability of non-zero energy fermionic edge mode and a zero energy Majorana edge mode, yielding a stable qubit. We suggest how the universal physics can be demonstrated in Rydberg chains.