Phonon spectral functions of low-density polaron metals
We use the density matrix renormalization group (DMRG) to compute the phonon spectral function of a one-dimensional spinless Holstein model doped with a low but finite carrier concentration, $x \leq 0.15$, as a function of the electron-phonon coupling $\lambda$. To the best of our knowledge, these are the first such results in this regime, complementing extensive prior work at the single-polaron level ($x\to 0$). We find that significant phonon spectral weight is transferred both below, all the way down to $\omega=0$, and above the bare phonon energy $\Omega$, in stark contrast with the Kohn-anomaly phenomenology expected in the Migdal limit, where weight remains centered near $\Omega$ with a kink at $q=2k_F$. No signature of this $2k_F$ kink appears in our results. This behavior is captured qualitatively by the Random Phase Approximation (RPA), and semi-quantitatively, at negligible extra computational cost, by a ``dressed RPA'' scheme in which the electron addition propagator is renormalized using the Momentum Average (MA) approximation for the low-density electron-polaron. By contrast, adding the lowest-order vertex correction to this dressed scheme produces unphysical negative spectral weight, signaling that vertex and propagator dressings must be treated consistently once the propagators are dressed nonperturbatively. Our results provide an efficient approximation for the phonon spectral functions of low-density polaron metals, a regime relevant to weakly doped insulators.