Intertwined Charge and Spin Density Waves in Trilayer Nickelate La$_4$Ni$_3$O$_{10}$ Revealed by $^{139}$La NQR

The discovery of superconducting transitions in pressurized La$_3$Ni$_2$O$_{7}$ and La$_4$Ni$_3$O$_{10}$ has highlighted the pivotal role of density wave (DW) orders in nickelate superconductors. To gain a comprehensive understanding of the superconducting state, it is essential to elucidate the nature of the DW order. In this study, we utilized $^{139}$La nuclear quadrupole resonance (NQR) to investigate the charge density wave (CDW) and spin density wave (SDW) orders in both single-crystal and polycrystalline La$_4$Ni$_3$O$_{10}$. Near $T_{\rm{DW}} \approx 133$ K, an abrupt change in both the linewidth and frequency of the La(2) site in the single-crystal sample provides compelling evidence for a first-order-like phase transition. The pronounced broadening of the NQR lines indicates the incommensurate nature of the DW order. Furthermore, the spin-lattice relaxation rate divided by temperature 1/$T_1$$T$ exhibits a strong enhancement at $T_{\rm{DW}}$, indicating the strong spin fluctuations above the first-order DW transition. These observations suggest an intricate interplay between incommensurate CDW and SDW orders. Our findings offer critical insights into the microscopic mechanisms of the DW state in La$_4$Ni$_3$O$_{10}$ and establish an essential framework for exploring the interplay between DW and superconducting phases in nickelate superconductors.