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We study theoretically how electron-phonon interaction affects the energiesand level broadening (inverse lifetime) of Majorana bound states (MBSs) in aclean topological nanowire at low temperatures. At zero temperature, the energysplitting between the right and left MBSs remains exponentially small withincreasing nanowire length $L$. At finite temperatures, however, the absorptionof thermal phonons leads to the broadening of energy levels of the MBSs thatdoes not decay with system length, and the coherent absorption/emission ofphonons at opposite ends of the nanowire results in MBSs energy splitting thatdecays only as an inverse power-law in $L$. Both effects remain exponential intemperature. In the case of quantized transverse motion of phonons, thepresence of Van Hove singularities in the phonon density of states causesadditional resonant enhancement of both the energy splitting and the levelbroadening of the MBSs. This is the most favorable case to observe thephonon-induced energy splitting of MBSs as it becomes much larger than thebroadening even if the topological nanowire is much longer than the coherencelength. We also calculate the charge and spin associated with the energysplitting of the MBSs induced by phonons. We consider both a spinlesslow-energy continuum model, which we evaluate analytically, as well as aspinful lattice model for a Rashba nanowire, which we evaluate numerically.

Degeneracy lifting of Majorana bound states due to electron-phonon interactions