Universal relationship between the energy scales of the pseudogap phase, the superconducting state, and the charge-density-wave order in copper oxide superconductors

We report the hole doping dependencies of the pseudogap phase energy scale $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{PG}}$, the antinodal (nodal) superconducting energy scales $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{SC}}^{\text{AN}}$ ($2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{SC}}^{\text{N}}$), and the charge-density-wave energy scale $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{CDW}}$ extracted from the electronic Raman responses of several copper oxide families. We show for all the cuprates studied that the three energy scales $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{PG}}, 2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{SC}}^{\text{AN}}$, and $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{CDW}}$ display the same decreasing monotonic behavior with doping. In particular, $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{SC}}^{\text{AN}}$ and $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{CDW}}$ have nearly equal values. This suggests a universal scenario in which $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{PG}}, 2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{SC}}^{\text{AN}}$, and $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{CDW}}$ are governed by common microscopic interactions that become relevant well above the superconducting transition at ${T}_c$. This is to be contrasted with the behavior of the nodal superconducting energy scale $2{\mathrm{\ensuremath{\Delta}}}_{\mathrm{SC}}^{\text{N}}$, which tracks the doping dependence of ${T}_c$ and, hence, seems to be controlled by different interactions.