Anomalous softening of phonon-dispersion in cuprate superconductors

Abstract

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

Supplementary included

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• Giant Kohn anomaly and chiral phonons in the charge density wave phase of 1H-NbSe${}_2$
• Interplay of phonons, intertwined density waves, and induced spin density wave in trilayer nickelates Pr4-xLaxNi3O10
• Pole structure of the electronic self-energy with coexistence of Charge order and Superconductivity