The electronic structure of EuPd$_2$Si$_2$ in the vicinity of the critical endpoint
Hard X-ray angle-resolved photoemission spectroscopy reveals significant alterations in the valence band states of EuPd$_2$Si$_2$ at a temperature $T_V$, where the Eu ions undergo a temperature-induced valence crossover from a magnetic Eu$^{2+}$ state to a low-temperature valence-fluctuating state. The introduction of small amounts of Au on Pd lattice sites and Ge on Si sites, respectively, results in a decrease in $T_V$ and the emergence of an antiferromagnetic state at low temperatures without valence fluctuations. It has been proposed that the boundary between AFM order and valence crossover represents a first-order phase transition associated with a specific type of second-order critical end point. In this scenario, strong coupling effects between fluctuating charge, spin, and lattice degrees of freedom are to be expected. In the case of EuPd$_2$(Si$_{1-x}$Ge$_x$)$_2$ with x=0.13, which is situated close to the critical end point, a splitting of conduction band states and the emergence of flat bands with one-dimensional character have been observed. A comparison with ab initio theory demonstrates a high degree of correlation with experimental findings, particularly in regard to the bands situated in proximity to the critical end point.