Can electronic quantum criticality drive phonon-induced linear-in-temperature resistivity?

Optical phonons naturally generate linear-in-temperature ( T T ) resistivity in the equipartition regime, but their finite gap prevents this mechanism from surviving to asymptotically low temperatures. Here we analyze whether proximity to an electronic quantum critical point can remove this obstruction by strongly softening an optical phonon. We first derive a model-independent criterion for such softened phonons to control low-temperature transport. In addition to reducing the renormalized optical gap, the Landau-damped phonon must acquire a dynamical exponent z_p;gt d z p > d , where d d is the spatial dimension of the phonon, so that a sufficiently large thermally occupied phase space survives as T\to 0 T → 0 . We then analyze a concrete mechanism in which the phonon couples nonlinearly to long-wavelength electronic collective modes near a quantum critical point associated with an order parameter carrying zero center-of-mass momentum, and apply it to the Ising-nematic problem. Within a large- N N field theoretic formulation, the phonon softening is enhanced near criticality, but in the clean theory the resulting dynamics lies at or near the boundary for asymptotic T T -linear scattering. Including feedback from the softened phonon back onto the electronic critical sector further weakens the tendency toward robust low-temperature T T -linear transport. Our results sharpen both the promise and the limitations of phonon-based explanations of strange-metal transport near electronic criticality.