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Results are presented for the dynamics of an almost strong edge mode which isthe quasi-stable Majorana edge mode occurring in non-integrable spin chains.The dynamics of the edge mode is studied using exact diagonalization, andcompared with time-evolution with respect to an effective semi-infinite modelin Krylov space obtained from the recursion method. The effective KrylovHamiltonian is found to resemble a spatially inhomogeneous SSH model where thehopping amplitude increases linearly with distance into the bulk, typical ofthermalizing systems, but also has a staggered or dimerized structuresuperimposed on it. The non-perturbatively long lifetime of the edge mode isshown to be due to this staggered structure which diminishes the effectivenessof the linearly growing hopping amplitude. On taking the continuum limit of theKrylov Hamiltonian, the edge mode is found to be equivalent to the quasi-stablemode of a Dirac Hamiltonian on a half line, with a mass which is non-zero overa finite distance, before terminating into a gapless metallic bulk. Theanalytic estimates are found to be in good agreement with the numericallyobtained lifetimes of the edge mode.

Dynamics of almost strong edge modes in spin chains away from integrability