Pressure-Induced Split of the Density Wave Transitions in La$_3$Ni$_2$O$_{7-\delta}

The unveiling of superconductivity in La$_3$Ni$_2$O$_{7-\delta}$ under pressure, following the suppression of a high-temperature density wave (DW) state, has attracted considerable attention. Notably, the nature of this competing DW order remains elusive, presenting a crucial question that demands further investigation. Here, we employ the muon-spin rotation/relaxation ($\mu$SR) technique combined with dipole-field numerical analysis to probe the magnetic response of La$_3$Ni$_2$O$_{7-\delta}$ as a function of hydrostatic pressure. At ambient pressure, $\mu$SR experiments reveal commensurate static magnetic order below $T_{\rm N} \simeq 151$K. The comparison of the observed internal magnetic fields with dipole-field calculations reveals the magnetic structure's compatibility with a stripe-type arrangement of Ni moments ($\simeq0.3-0.7$$\mu_{\rm B}$), characterized by alternating lines of magnetic moments and non-magnetic stripes. Experiments under pressure (up to $p\simeq2.3$~GPa) demonstrate an increase of the magnetic ordering temperature at a rate ${\rm d}T_{\rm N}/{\rm d}p\simeq 2.8$ K/GPa. This trend is opposite in sign and significantly smaller in magnitude compared to the changes observed in the DW order of unknown origin reported by Wang et al. [arXiv:2309.17378]. Our findings reveal that the ground state of the La$_3$Ni$_2$O$_{7-\delta}$ system is characterized by the coexistence of two distinct orders -- the spin density wave and, most likely, charge density wave -- with a notable pressure-induced separation between them.