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The pressure-temperature-phase diagram of the new heavy-fermionsuperconductor CeAu$_2$Si$_2$ is markedly different from those studiedpreviously. Indeed, superconductivity emerges, not on the verge, but deepinside the magnetic phase. In this context, we have carried out ac-calorimetry,resistivity and thermoelectric power measurements on a CeAu$_2$Si$_2$-singlecrystal under high-pressure. The principal novelties of this experiment are theobservation of a new transition line, presumably a magnetic structurerearrangement at $T_{\mathrm{M}}^{\mathrm{mod}}$ inside the antiferromagneticphase, and the occurrence of quantum critical behavior in resistivity linked tosuperconductivity. Strong non-Fermi-liquid behavior is observed around themaximum of superconductivity and enhanced scattering rates are observed closeto both the emergence and the maximum of superconductivity. Intriguingly,$T_{\mathrm{M}}^{\mathrm{mod}}$ almost coincides with the onset of thesuperconducting transition over a broad pressure range, where mysteriously Tcincreases with the strengthening of magnetism. A comparison of the features inCeAu$_2$Si$_2$ and its parent compounds CeCu$_2$Si$_2$ and CeCu$_2$Ge$_2$plotted as function of the unit-cell volume leads us to promote that criticalfluctuations of a valence crossover play a crucial role in the superconductingparing mechanism. Our study reveals new intriguing features of magnetism andillustrates a complex interplay between magnetism, quantum criticality andsuperconductivity.

High-pressure study of the ground- and superconducting-state properties of CeAu$_2$Si$_2$