Effects of phonons and nuclear spins on the tunneling of a domain wall

We consider the quantum dynamics of a magnetic domain wall at low temperatures, where dissipative couplings to magnons and electrons are very small, The wall motion is then determined by its, coupling to phonons and nuclear-spins, and any pinning potentials. In the absence of nuclear spins there is a dominant super Ohmic 1-phonon coupling to the wall velocity, plus a strongly T-dependent Ohmic coupling to pairs of phonons, There is also a T-independent Ohmic coupling between single phonons and the wall chirality, which suppresses "chirality tunneling". We calculate the effect of these couplings on the T-dependent tunneling rate of a wall out of a pinning potential. Nuclear spins have a very strong and hitherto unsuspected influence on domain, wall dynamics, coming from a hyperfine-mediated coupling to the domain wall position, For k(B)T much greater than omega(0) this coupling yields a spatially random potential, fluctuating at a late governed by the nuclear T-2. When k(B)T <much less than omega(0), the hyperfine potential fluctuates around a linear binding potential, The wall dynamics is influenced by the fluctuations of this potential, ie., by the nuclear spin dynamics. Wall tunneling can occur when fluctuations open an occasional "tunneling window". This changes the crossover to tunneling and also causes a slow "wandering", in time, of the energy levels associated with domain wall motion inside the pinning potential, This effect is fairly weak in Ni-and Fe-based magnets, and we give an approximate treatment of its effect on the tunneling dynamics, as well as a discussion of the relationship to recent domain wall tunneling experiments.