Fig. 1

Phase diagram and correlation between p* and p _FS. a Temperature-doping phase diagram, showing the pseudogap temperature T* measured by ARPES (triangles), electrical resistivity (circles) and Nernst effect (squares), in LSCO (black), Nd-LSCO (red) and Eu-LSCO (green) (adapted from ref. ⁶; see references therein). The diamonds mark the critical doping p* at which the pseudogap phase ends at T = 0 (in the absence of superconductivity), for LSCO (black; p* = 0.18, refs. ^8,9) and Nd-LSCO (red; p* = 0.23, refs. ^4,5). The short solid lines mark the transition into the low-temperature tetragonal structure (LTT) for Eu-LSCO (green; ref. ³⁶ and Nd-LSCO (red; ref. ³⁴), in the interval where it crosses T*. The grey line is the superconducting critical temperature T _c of LSCO. The dashed line is a linear extension of the T* line (blue). The red arrow illustrates the effect of applying hydrostatic pressure to Nd-LSCO: it shifts p* down in doping. b, c, d Sketch of the Fermi surface in the first Brillouin zone (frame), as measured by ARPES in LSCO (ref. ¹¹) at p = 0.17 (b) and in Nd-LSCO (ref. ³) at p = 0.20 (c) and p = 0.24 (d). The red dashed line is the anti ferromagnetic zone boundary (AFZB), which intersects the hole-like Fermi surface of Nd-LSCO at p = 0.20, but not its electron-like surface at p = 0.24. e Correlation between p* and p _FS in single-layer cuprates. p* is measured in the normal state at T = 0, by high-field transport in Nd-LSCO (p* = 0.23 ± 0.01, ref. ⁵) and LSCO (p* = 0.18 ± 0.01, refs. ^8,9), and by high-field NMR in Bi2201 (p* = 0.38 ± 0.02, ref. ¹⁰). p _FS is measured by ARPES in LSCO (p _FS = 0.19 ± 0.01, refs. ^2,11), Nd-LSCO (p _FS = 0.22 ± 0.01; ref. ³) and Bi2201 (p _FS = 0.41 ± 0.02, ref. ¹²). The red arrow shows the effect of applying pressure to Nd-LSCO (red square). We find that p* and p _FS decrease in tandem, preserving the equality p* = p _FS and thereby showing that p* is constrained by the condition p* ≤ p _FS. The grey shading marks the forbidden region