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

Optical phonons naturally generate linear-in-$T$ resistivity in the high-temperature 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: besides reducing the renormalized optical gap, the Landau-damped phonon must acquire a dynamical exponent $z_p>d$, where $d$ is the spatial dimension of the phonon, so that a sufficiently large thermally occupied phase space survives as $T\to 0$. We then analyze a concrete mechanism in which the phonon couples nonlinearly to long-wavelength electronic collective modes near a $\vec{Q}=0$ quantum critical point, and apply it to the Ising-nematic case. Within a large-$N$ field theoretic formulation, the phonon softening is enhanced near criticality, but in the clean theory the resulting dynamics lies at or near the marginal boundary for asymptotic $T$-linear scattering. Including feedback from the softened phonon back onto the electronic critical sector further weakens the tendency toward robust low-temperature $T$-linear transport. Our results sharpen both the promise and the limitations of phonon-based explanations of strange-metal transport near electronic criticality.