Microscopic origin of the nematoelastic coupling and dynamics of hybridized collective nematic-phonon excitations

Electronically driven nematic order breaks the rotational symmetry of a system, e.g., through a Pomeranchuk instability of the Fermi surface, with a concomitant distortion of the lattice. As a result, in a metal, the nematic collective mode interacts with two different sets of gapless excitations: the particle-hole excitations of the metal and the lattice fluctuations that become soft at the induced structural transition, namely, the transverse acoustic phonons. However, the dynamics of these hybridized collective modes formed by the transverse acoustic phonons and the metallic electronic nematic fluctuations has remained largely unexplored. Here, we address this problem by developing a formalism in which the nematoelastic coupling is obtained microscopically from the direct coupling between electrons and transverse acoustic phonons enabled by impurities present in the crystal. We then demonstrate the emergence of hybrid nematoelastic modes that mix the characteristics of the transverse phonons and of the nematic fluctuations. Near the nematic quantum critical point in a metal, two massless modes emerge with intertwined dynamic behaviors, implying that neither mode dominates the response of the system. We systematically study the nontrivial dependence of these collective modes on the longitudinal and transverse momenta, revealing a rich landscape of underdamped and overdamped modes as the proximity to the quantum critical point and the strength of the electron-phonon coupling are changed. Since dynamics play an important role for determining superconducting instabilities, our results have implications for the study of pairing mediated by electronic nematic fluctuations.