  The concept of ferroelectricity is now often extended to include continuous
inversion symmetry-breaking transitions in various metals and doped
semiconductors. Paraelectric metals near ferroelectric quantum criticality,
which we term `quantum paraelectric metals,' typically possess soft transverse
optical phonons that have Rashba-type coupling to itinerant electrons in the
presence of spin-orbit coupling. We find through the Kubo formula calculation
that such Rashba electron-phonon coupling has a profound impact on electron
spin transport. While the spin Hall effect arising from non-trivial electronic
band structures has been studied extensively, we find here the presence of the
Rashba electron-phonon coupling can give rise to spin current, including spin
Hall current, in response to an inhomogeneous electric field even with a
completely trivial band structure. Furthermore, this spin conductivity displays
unconventional characteristics, such as quadrupolar symmetry associated with
the wave vector of the electric field and a thermal activation behavior
characterized by scaling laws dependent on the phonon frequency to temperature
ratio. These findings shed light on exotic electronic transport phenomena
originating from ferroelectric quantum criticality, highlighting the intricate
interplay of charge and spin degrees of freedom.
