Fig. 1: Crystal structure, phase diagram, resistance curves, and magnetic torque curves of La₄Ni₃O₁₀.

a, b The crystal structure of orthogonal La₃Ni₂O₇ at ambient pressure and tetragonal La₃Ni₂O₇ under pressure. There is a small tilting angle between the adjacent NiO₆ octahedron in the orthogonal La₃Ni₂O_7, which disappears in the tetragonal La₃Ni₂O₇. c, d The crystal structure of monoclinic La₄Ni₃O₁₀ at ambient pressure and tetragonal La₄Ni₃O₁₀ under pressure. A small tilting angle between the adjacent NiO₆ octahedron also exists in the monoclinic La₄Ni₃O₁₀ and disappears in the tetragonal La₄Ni₃O_10. e The Temperature-Pressure (T-P) phase diagram of La₄Ni₃O₁₀ determined by the data in Fig. 3a, b. The information on structure under different pressure comes from ref. ^15,16. The circular, triangle, and square data dots represent the T_c (onset) for sample S1, the T_c (onset) for sample S2 and density wave transition temperature (T_DW) for sample S1, respectively. These data are obtained from Figs. 3a, b and Supplementary Fig. 7. There is a phase boundary between the density wave state and superconductivity, accompanied with a structure transition from monoclinic phase to tetragonal phase. f The temperature-dependent resistance curve of monoclinic (black curve, Sample S1) and tetragonal (red curve, Sample S3) La₄Ni₃O₁₀ at near-ambient pressure (below 1 GPa). There is a density-wave transition evidenced by an up-turn in the resistivity at around 135 K for the monoclinic La₄Ni₃O₁₀, which is absent in the tetragonal La₄Ni₃O₁₀. g The temperature-dependent magnetic torque (τ(T)) data of monoclinic and tetragonal La₄Ni₃O₁₀, respectively. More magnetic torque data is shown in Supplementary Fig. 9. There is a kink in τ(T) at 135 K for monoclinic La₄Ni₃O₁₀, which is consistent with the anomaly in resistance due to density-wave transition in Fig. 1f. Similar evidence for the density-wave transition in monoclinic La₄Ni₃O₁₀ has been also observed by specific heats and magnetic susceptibility measurements¹⁶. In contrast, both resistivity and magnetic torque results indicate that such a density-wave transition is absent in the tetragonal La₄Ni₃O₁₀. a–d The green, blue, and red circles represent the atoms of La, Ni, and O, respectively.