Pressure-Invariant Isotope Effect as Evidence for Electronically Driven Intertwined Order in Pr$_4$Ni$_3$O$_{10}$

We report muon-spin rotation measurements of the pressure dependence of the oxygen-isotope ($^{16}$O/$^{18}$O) effect on the spin-density wave (SDW) transition in the trilayer Ruddlesden-Popper nickelate Pr$_4$Ni$_3$O$_{10}$. At ambient pressure, the SDW transition shows a finite isotope shift, with $^{16}T_{\rm SDW}=158.04(5)$ K and $^{18}T_{\rm SDW}=159.81(6)$ K. Under hydrostatic pressure, $T_{\rm SDW}$ decreases linearly at nearly identical rates for the two isotope compositions, ${\rm d}\,^{16}T_{\rm SDW}/{\rm d}p=-4.93(5)$ K/GPa and ${\rm d}\,^{18}T_{\rm SDW}/{\rm d}p=-4.90(7)$ K/GPa, such that the isotope shift remains essentially unchanged under compression. The absence of pressure enhancement of the isotope effect points to a predominantly electronic origin of the SDW transition and is consistent with recent inelastic x-ray scattering results, suggesting a new regime of intertwined order in trilayer RP nickelates, which is stabilized by strong spin interactions.