Anomalous softening of phonon dispersion in cuprate superconductors

A softening of phonon-dispersion has been observed experimentally in under-doped cuprate superconductors at the charge-density wave (CDW) ordering vector. Interestingly, occurs below superconducting (SC) transition temperature T$_{c}$, contrast to metallic systems, where usually CDW onset T$_{\text{CDW}}$. An understanding `anomalous' nature phonon-softening and its connection pseudo-gap phase cuprates remain open questions. Within a perturbative approach, we show that complex interplay among ubiquitous CDW, SC orders life-time quasi-particles associated thermal fluctuations, can explain anomalous T$_{c}$. Furthermore, our formalism captures different characteristics low depending on material specificity.