Mixed-valence state approaching the single-impurity limit in La-substituted SmB$_6$

One of the most intriguing aspects of $f$-electron systems is their homogeneous mixed valence (MV) behavior. Despite extensive efforts, a fundamental aspect which remains unsettled is the determination of the limiting cases for which MV emerges, as highlighted by Haldane's theoretical proposal of the occurrence of a MV state for a single $f$-type impurity in a Fermi sea of $d$-electrons. Here we address this open question for SmB$_6$, a prototypical system characterized by the interplay between two nearly-degenerate Sm$^{2+}$ and Sm$^{3+}$ configurations. By combining La substitution with angle resolved photoemission (ARPES) and x-ray absorption spectroscopy (XAS), we track the evolution of the mean Sm valence for the $f$-shell, $v_{Sm}$, in the Sm$_x$La$_{1-x}$B$_6$ series. Upon substitution of Sm ions by trivalent La$^{3+}$, we observe a linear decrease of valence fluctuations to an almost complete suppression at $x$$\,$=$\,$0.2, with $v_{Sm}$$\,$$\sim$$\,$2; surprisingly, by further reducing $x$, a re-entrant increase of $v_{Sm}$ develops for $x$$\,$<$\,$0.2, approaching a single-impurity-limit value of $v_{imp}$$\,$$\sim$$\,$2.35. Such observation marks a stark deviation from a monotonic evolution of $v_{Sm}$ across the entire series, as well as from the expectation of its convergence to an integer value for $x$$\,$$\rightarrow$$\,$0. Our study suggests the realization of a $\textit{mixed-valence single-impurity state}$ in Sm$_x$La$_{1-x}$B$_6$, paving the way to further considerations on the concept of MV and its influence on the macroscopic properties of rare-earth compounds in the dilute-to-intermediate impurity regime.