Phonon Softening Due to the Coupling with Charge Density Fluctuations in High-Temperature Superconducting Cuprates

Bond-stretching phonons in hole-doped cuprates exhibit pronounced anomalies in momentum regions where charge correlations are observed, indicating a strong coupling between lattice dynamics and the charge sector. Motivated by this phenomenology, we develop a minimal theoretical framework to clarify how the bond-stretching phonon is renormalized by the proximity to charge-order instabilities and why short-ranged dynamical charge density fluctuations (CDFs) can provide a dominant contribution even in the presence of nearly static charge density wave (CDW) correlations. Starting from a correlated Fermi-liquid description with Coulomb-frustrated charge ordering and electron–phonon coupling, we formulate a two-mode random-phase-approximation treatment in which a low-energy phonon channel involved in the charge-order instability is coupled, through the electronic polarization, to the higher-energy bond-stretching branch. The resulting off-diagonal phonon self-energy transfers the low-energy charge softening to the bond-stretching phonon. We then introduce an average self-energy description to account for the coexistence of CDW and CDF components. The model shows that, although a nearly static CDW component produces a stronger local softening, a broader CDF component can dominate the effective phonon self-energy because of its larger reciprocal-space volume. The analysis identifies two key parameters controlling the bond-stretching anomaly: the characteristic energy of charge correlations and the effective electron–phonon coupling.