Fig. 1.

Intraunit-cell electronic structure symmetry in the CuO₂ plane. (A) Schematic of translationally invariant electronic structure pattern with opposite-sign density on O_x and O_y (d symmetry), as discussed in refs. 12, 13, and 32. The inactive Cu sites are indicated by gray circles. Inset shows elementary Cu, O_x, and O_y orbitals, on the three sublattices in the CuO₂ plane; r_Cu sublattice contains only the Cu sites, ${\mathit{r}}_{{\text{O}}_x}$ sublattice contains only the O_x sites, and ${\mathit{r}}_{{\text{O}}_y}$ sublattice contains only the O_y sites. Referring to this arrangement as having “d symmetry” does not, at this stage, imply any specific relationship to the d-symmetry Cooper pairing of the superconducting state. (B) Fourier transform of the Q = 0 C₄-breaking pattern of opposite-sign density on O_x and O_y in A. The Bragg peaks have the opposite sign, indicating the IUC states have d symmetry (12, 32). (C) Schematic of a d-form factor DW ${\rho }_D\left(\mathit{r}\right)=D\hspace{0.17em}\mathrm{Cos}\left({\mathit{Q}}_{\mathit{x}}\cdot \mathit{r}+{\varphi }_D\left(\mathit{r}\right)\right)$ with the origin chosen at a Cu site and modulating only along the x axis. The colors represent the density ${\rho }_D\left(\mathit{r}\right)$ at every oxygen site. Two thin lines are shown; one labeled O_x traverses parallel to ${\mathit{Q}}_{\mathit{x}}$ and passes only through oxygen sites oriented along the x axis, and the second one labeled O_y again traverses parallel to ${\mathit{Q}}_{\mathit{x}}$ but passes only through oxygen sites oriented along the y axis. The color scale demonstrates how the amplitude of the DW is exactly π out of phase along these two trajectories. (D) The real component of the Fourier transform of the pattern in C. For this d-form factor DW, the Bragg-satellite peaks at Q′ and Q′′ exhibit opposite sign. More profoundly, because they are out of phase by π, the modulations of O_x and O_y sites cancel, resulting in the disappearance of the DW modulation peaks at ±Q within the BZ (dashed box). (E) The real component of the Fourier transform of a d-form factor DW having modulations at Q = (Q,0);(0,Q) (SI Text, section 3); the DW satellites of inequivalent Bragg peaks at Q′ and Q′′ exhibit opposite sign, and the basic DW modulation peaks at Q have disappeared from within the BZ, as indicated by open circles. (F) The real component of the Fourier transform of an s′-form factor DW having modulations at Q = (Q,0);(0,Q) (SI Text, section 3); the DW satellites of inequivalent Bragg peaks at Q′ and Q′′ now cancel whereas the basic DW modulation peaks at Q are robust within the BZ.