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The transverse field Ising model (TFIM) on the half-infinite chain possessesan edge zero mode. This work considers an impurity model -- TFIM perturbed by aboundary integrability breaking interaction. For sufficiently large transversefield, but in the ordered phase of the TFIM, the zero mode is observed todecay. The decay is qualitatively different from zero modes where theintegrability breaking interactions are non-zero all along the chain. It isshown that for the impurity model, the zero mode decays by relaxing to anon-local quasi-conserved operator, the latter being exactly conserved when theopposite edge of the chain has no non-commuting perturbations so as to ensureperfect degeneracy of the spectrum. In the thermodynamic limit, thequasi-conserved operator vanishes, and a regime is identified where the decayof the zero mode obeys Fermi's Golden Rule. A toy model for the decay isconstructed in Krylov space and it is highlighted how Fermi's Golden Rule maybe recovered from this toy model.

Slowly decaying zero mode in a weakly non-integrable boundary impurity model