Absence of superconductivity and density-wave transition in ambient-pressure tetragonal La$_4$Ni$_3$O$_{10}$

The recent discovery of superconductivity in La$_3$Ni$_2$O$_7$ and La$_4$Ni$_3$O$_{10}$ under high pressure stimulates intensive research interests. These nickelates crystallize in an orthogonal/monoclinic structure with tilted NiO$_6$ octahedra at ambient pressure and enter a density-wave-like phase at low temperatures. The application of pressure suppresses the octahedral tilting and triggers a transition to tetragonal structure (I4/mmm), which is believed to be a key prerequisite for the emergence of superconducting state. Here, by developing a high oxidative environment growth technology, we report the first tetragonal nickelates La$_4$Ni$_3$O$_{10}$ microcrystals without octahedral tilting at ambient pressure. In tetragonal La$_4$Ni$_3$O$_{10}$, transport measurements find that both density-wave and superconducting transitions are absent up to 160 GPa, indicating a robust tetragonal metallic ground state. Density functional theory calculations reveal that the band structure of ambient-pressure tetragonal La$_4$Ni$_3$O$_{10}$ involves more $d_{z2}$ orbital contribution to the Fermi surface, compared to the monoclinic phase or the high-pressure superconducting tetragonal phase. The concurrent absence of density-wave state and high-pressure superconductivity in our ambient-pressure tetragonal crystals of La$_4$Ni$_3$O$_{10}$ suggests an underlying correlation between these two orders. It suggests that the tetragonal structure is not necessary, while the density-wave state is crucial for the superconductivity in nickelates. Our findings impose important constraints on the mechanism of pressure-induced superconductivity in nickelates and sheds new light on exploring ambient pressure high-temperature Ni-based superconductors.