a, Schematic of the spatial arrangements of CuO₂ electronic structure with Cu sites and orbitals indicated in blue and O sites and 2p_σ orbitals in yellow. E_F, Fermi energy. The inset shows the approximate energetics of the band structure when such a charge-transfer insulator is doped by removing electrons from the O atoms (the respective bands are indicated by the same colours as in the CuO₂ schematic). The ‘real’ part of any q-space electronic structure at the Bragg wavevector (Q,ω) is defined throughout this paper as being in phase with the Cu lattice (and therefore even about each Cu site). This definition is shown schematically as modulations in which would contribute to (Q,ω) at the two Bragg wavevectors Q_x and Q_y. DOS, density of states. b, Schematic copper-oxide phase diagram. Here T_c is the critical temperature circumscribing a ‘dome’ of superconductivity, T_φ is the maximum temperature at which phase fluctuations are detectable within the pseudogap phase, and T* is the approximate temperature at which the pseudogap phenomenology first appears. c, The two distinct classes of excitations as identified by multiple spectroscopies in underdoped copper-oxides (reproduced with permission from ref. 3) as a function of hole-density p. The excitations to energies Δ₁(p) are referred to as the ‘pseudogap states’ both because Δ₁(p) tracks T*(p) and because they exist unchanged in both the pseudogap and superconducting phases. Those excitations to energies ω < Δ₀(p) can be associated with the Bogoliubov quasiparticles^{6, 7, 8}. d, Evolution of the spatially averaged tunnelling spectra of Bi₂Sr₂CaCu₂O_8 + δ with diminishing p, here characterized by T_c(p). The energies Δ₁(p) (blue dashed line) are easily detected as the pseudogap edge while the energies Δ₀(p) (red dashed line) are more subtle but can be identified by the correspondence of the “kink” energy⁴ with the extinction energy of Bogoliubov quasiparticles, following the procedure in ref. 8.

a, Spatial image (R-map⁵) of the Bi₂Sr₂CaCu₂O_8 + δ pseudogap states ω ≈ Δ₁ at T ≈ 4.3 K for an underdoped sample with T_c = 35 K. The inset shows the Fourier transform upon which the inequivalent Bragg vectors Q_x = (1, 0)2π/a₀ and Q_y = (0, 1)2π/a₀ are identified by red arrows and circles. The inequivalent wavevectors S_x = (~3/4, 0)2π/a₀ and S_y = (0, ~3/4)2π/a₀ are identified by blue arrows and circles. b, Spatial image (R-map⁵) of the Bi₂Sr₂CaCu₂O_8 + δ pseudogap states ω ≈ Δ₁ at T ≈ 55 K for the same sample with T_c = 35 K. Again, the inset shows the Fourier transform with the inequivalent Bragg vectors Q_x = (1, 0)2π/a₀ and Q_y = (0, 1)2π/a₀ identified by red arrows and S_x = (~3/4, 0)2π/a₀ and S_y = (0, ~3/4)2π/a₀ identified by blue arrows and circles. The phenomenology of the ω ≈ Δ₁ pseudogap states, especially their broken spatial symmetries, appear indistinguishable whether in the superconducting phase (a) or in the pseudogap phase (b). c, A schematic representation of how electronic contributions from multiple sites within the CuO₂ unit cell could lead to global electronic nematicity in the copper oxides. Here the two O sites are labelled using different colours to represent the inequivalent electronic structure at those locations within each unit cell.

a, Topographic image T(r) of the Bi₂Sr₂CaCu₂O_8 + δ surface. The inset shows that the real part of its Fourier transform ReT(q) does not break C₄ symmetry at its Bragg points because plots of T(q) show its values to be indistinguishable at Q_x = (1, 0)2π/a₀ and Q_y = (0, 1)2π/a₀. Importantly, this means that neither the crystal nor the tip used to image it (and its Z(r, ω) simultaneously) exhibits C₂ symmetry (Supplementary Information section V). The bulk incommensurate crystal supermodulation is seen clearly here; as always, it is at 45° to, and therefore is the mirror plane between, the x and y axes. For this symmetry reason it has no influence on the electronic nematicity discussed in this paper. b, The Z(r, e = 1) image measured simultaneously with T(r) in a. The inset shows that the Fourier transform Z(q, e = 1) does break C₄ symmetry at its Bragg points because . This means that, on average throughout the FOV of a and b, the modulations of Z(r, ω ≈ Δ₁) that are periodic with the lattice have different intensities along the x axis and along the y axis. This is a priori evidence for electronic nematicity in the pseudogap states ω ≈ Δ₁. c, The value of defined in equation (3) computed from Z(r, e) data measured in the same FOV as a and b. Its magnitude is low for all ω < Δ₀ and then rises rapidly to become well established near e ≈ 1 or ω ≈ Δ₁. Thus the quantitative measure of intra-unit-cell electronic nematicity established in equations (1) and (3) reveals that the pseudogap states in this FOV of a strongly underdoped Bi₂Sr₂CaCu₂O₈ sample are nematic. d, Topographic image T(r) from the region identified by a small black box in a. It is labelled with the locations of the Cu atom plus both the O atoms within each CuO₂ unit cell (labels shown in the inset). Overlaid is the location and orientation of a Cu and four surrounding O atoms using a representation similar to that of Fig. 2c. e, The simultaneous Z(r, e = 1) image in the same FOV as d (the region identified by small white box in b) showing the same Cu and O site labels within each unit cell (see inset). Thus the physical locations at which the nematic measure of equation (4) is evaluated are labelled by the dashes. Overlaid is the location and orientation of a Cu atom and four surrounding O atoms using a representation similar to that of Fig. 2c. f, The value of in equation (4) computed from Z(r, e) data measured in the same FOV as a and b. As in c, its magnitude is low for all ω < Δ₀ and then rises rapidly to become well established at e ≈ 1 or ω ≈ Δ₁. If the function in equation (4) is evaluated using the Cu sites only, the nematicity is about zero (black diamonds), as it must be. This independent quantitative measure of intra-unit-cell electronic nematicity again shows that the pseudogap states are strongly nematic and, moreover, that the nematicity is due primarily to electronic inequivalence of the two O sites within each unit cell.

a, A large FOV T(r) image which preserves C₄ symmetry. b, Correlation lengths for the nematic ordering (red solid diamonds), and for the possibly smectic ordering (blue solid squares). (See Supplementary Information section VI for an evaluation of and .) The coarsening length scales 1/Λ_n and 1/Λ_s are one-third that of typical spatial variations, as determined from the 3σ radius of the respective peaks. The correlation lengths are determined from the full-width at half-maximum of the spatial auto-correlation functions. Comparison of the nematic order parameter evaluated from equation (3) (open diamonds) and smectic order parameter ) from equation (6) (open squares) shows that has low magnitude and is energy independent, whereas rises rapidly to become well established at the pseudogap energy. c, Image of ( Supplementary Information section VI) from the same FOV showing that has diverged to the size of the image. Thus, if there are Ising nematic domains (as there should be), they must be larger than 0.025 micrometres square. We note that the spatial resolution is limited by the cut-off scales shown in the figure. d, Images of (Supplementary Information section VI) from the same FOV as a showing that is spatially disordered with very short correlation length; this is equally true at all energies. The evolution of and as a function of e are available in Supplementary movies 1 and 2, respectively.