Atomic-scale Electronic Structure of the Cuprate d-Symmetry Form Factor Density Wave State

Extensive research into high temperature superconducting cuprates is now focused upon identifying the relationship between the classic 'pseudogap' phenomenon\(^{1,2}\) and the more recently investigated density wave state\(^{3-13}\). This state always exhibits wave vector \(Q\) parallel to the planar Cu-O-Cu bonds\(^{4-13}\) along with a predominantly \(d\)-symmetry form factor\(^{14-17}\) (dFF-DW). Finding its microscopic mechanism has now become a key objective\(^{18-30}\) of this field. To accomplish this, one must identify the momentum-space (\(k\)-space) states contributing to the dFF-DW spectral weight, determine their particle-hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization\(^{14}\) of electronic structure and show that the characteristic energy of the dFF-DW modulations is actually the 'pseudogap' energy \(\Delta_{1}\). Moreover, we demonstrate that the dFF-DW modulations at \(E=-\Delta_{1}\) (filled states) occur with relative phase \(\pi\) compared to those at \(E=\Delta_{1}\) (empty states). Finally, we show that the dFF-DW \(Q\) corresponds directly to scattering between the 'hot frontier' regions of \(k\)-space beyond which Bogoliubov quasiparticles cease to exist\(^{31,32,33}\). These data demonstrate that the dFF-DW state is consistent with particle-hole interactions focused at the pseudogap energy scale and between the four pairs of 'hot frontier' regions in \(k\)-space where the pseudogap opens.