Effect of Pressure and Oxygen-Isotope Substitution on Density-Wave Transitions in La$_4$Ni$_3$O$_{10}$
Understanding the interplay between magnetism and superconductivity in nickelate systems is a key objective in condensed matter physics. Here, we present a systematic muon-spin rotation/relaxation ($μ$SR) and resistivity study of the trilayer Ruddlesden-Popper nickelate La$_4$Ni$_3$O$_{10}$ under ambient and applied pressure, combined with oxygen-isotope substitution. At ambient pressure, two incommensurate spin-density-wave (SDW) transitions are identified at $T_{SDW}\simeq132$ K and $T^\ast\simeq80-90$ K. Comparison of the internal magnetic fields with dipole-field calculations reveals a magnetic structure consistent with antiferromagnetically coupled SDW order on the outer two Ni layers, with smaller moments on the inner layer. Above $T^\ast$, the moments lie mainly in the $ab$ plane, whereas below this temperature they develop a $c$-axis component. The internal fields at the muon stopping sites appear abruptly at $T_{SDW}$, suggesting a first-order-like SDW transition closely linked to the charge-density-wave (CDW) order occurring at the same temperature ($T_{SDW}=T_{CDW}$). Under pressure, all transition temperatures -- $T_{SDW}$, $T^\ast$, and $T_{CDW}$ -- are suppressed at a nearly uniform rate of $\simeq-13$ K/GPa. This contrasts with bilayer La$_3$Ni$_2$O$_7$, where pressure enhances the separation between the SDW and CDW transitions. Oxygen-isotope substitution ($^{16}$O $\rightarrow$ $^{18}$O) shifts $T_{CDW}$ to higher values. The isotope effect on $T_{SDW}$ and $T^\ast$ differs markedly: when CDW and SDW are intertwined, a notable isotope effect is observed on $T_{SDW}$, yielding nearly identical isotope shifts for $T_{CDW}$ and $T_{SDW}$, whereas no isotope effect is detected at $T^\ast$, where the SDW transition occurs independently of the CDW.