Anomalous softening of phonon dispersion in cuprate superconductors

Abstract

A softening of phonon dispersion has been observed experimentally in underdoped 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_{CDW}. An understanding of the “anomalous” nature of the phonon softening and its connection to the pseudogap phase in underdoped cuprates remain open questions. Employing a perturbative approach, we find that a complex interplay among the ubiquitous CDW order, SC order, and an unusually connected thermal fluctuations of these orders 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.