Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state

Research on high-temperature superconducting cuprates is at present focused on identifying the relationship between the classic ‘pseudogap’ phenomenon^1,2 and the more recently investigated density wave state^{3,4,5,6,7,8,9,10,11,12,13}. This state is generally characterized by a wavevector Q parallel to the planar Cu–O–Cu bonds^{4,5,6,7,8,9,10,11,12,13} along with a predominantly d-symmetry form factor^14,15,16 (dFF-DW). To identify the microscopic mechanism giving rise to this state^{17,18,19,20,21,22,23,24,25,26,27,28,29}, one must identify the momentum-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 visualization¹⁴ of electronic structure and reveal that the characteristic energy of the dFF-DW modulations is actually the ‘pseudogap’ energy Δ₁. Moreover, we demonstrate that the dFF-DW modulations at E  =   −Δ₁ (filled states) occur with relative phase π compared to those at E  =  Δ₁ (empty states). Finally, we show that the conventionally defined dFF-DW Q corresponds to scattering between the ‘hot frontier’ regions of momentum-space beyond which Bogoliubov quasiparticles cease to exist^30,31,32. These data indicate that the cuprate dFF-DW state involves particle–hole interactions focused at the pseudogap energy scale and between the four pairs of ‘hot frontier’ regions in momentum space where the pseudogap opens.