  Many phenomena in condensed matter are thought to result from competition
between different ordered phases. Palladium is a paramagnetic metal close to
both ferromagnetism and superconductivity, and is therefore a potentially
interesting material to consider. Nanoscale structuring of matter can modify
relevant physical energy scales leading to effects such as locally modified
magnetic interactions. We present transport measurements in electromigrated
palladium break junction devices showing the emergence at low temperatures of
anomalous sharp features in the differential conductance. These features appear
symmetrically in applied bias and exhibit a temperature dependence of their
characteristic voltages reminiscent of a mean field phase transition. The
systematic variation of these voltages with zero-bias conductance, together
with density functional theory calculations illustrating the relationship
between the magnetization of Pd and atomic coordination, suggest that the
features may result from the onset of spontaneous magnetization in the
nanojunction electrodes. We propose that the characteristic conductance
features are related to inelastic tunneling involving magnetic excitations.
