Proximate quantum spin liquid state in the frustrated HoInCu$_4$ metal

We conducted a comprehensive and comparative muon-spin relaxation and rotation ($\mu$SR) investigation on two fcc-lattice metallic compounds, HoCdCu$_4$ ($T_\mathrm{N}\approx 8$ K) and HoInCu$_4$ ($T_\mathrm{N}\approx 0.76$ K), to elucidate the nature of their magnetic ground states and the role of frustration in stabilizing them. Our $\mu$SR results reveal that, in contrast to HoCdCu$_4$, strong magnetic frustration exists in HoInCu$_4$. Notably, in HoInCu$_{4}$, only 30% of the Ho-moments participate in the static magnetic ordering below $T_\mathrm{N}$, while the remaining 70% of the Ho-moments exhibit dynamic correlations and persistent spin dynamics down to 0.3 K, resembling a quantum spin-liquid (QSL) behavior. By contrast, in HoCdCu$_{4}$, all the Ho-moments contribute to the magnetic order below $T_\mathrm{N}$. Furthermore, in HoInCu$_{4}$, the temperature dependence of the relaxation rate indicates the presence of quantum critical fluctuations in the paramagnetic state near $T_\mathrm{N}$, suggesting the proximity to a quantum critical point (QCP). These observations suggest that the ground state of HoInCu$_{4}$ is a proximate quantum spin liquid (PQSL), a state that has not been reported before in frustrated metallic systems. Our $\mu$SR findings are further corroborated by recent inelastic neutron results on HoInCu$_4$, which show similarities to other insulating PQSL candidates, thus reinforcing our conclusions.