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An appropriate description of the state of matter that appears as a secondorder phase transition is tuned toward zero temperature, {\it viz.}quantum-critical point (QCP), poses fundamental and still not fully answeredquestions. Experiments are needed both to test basic conclusions and to guidefurther refinement of theoretical models. Here, charge and entropy transportproperties as well as AC specific heat of the heavy-fermion compoundCeRh$_{0.58}$Ir$_{0.42}$In$_5$, measured as a function of pressure, reveal twoqualitatively different QCPs in a {\it single} material driven by a {\itsingle} non-symmetry-breaking tuning parameter. A discontinuous sign-changejump in thermopower suggests an unconventional QCP at $p_{c1}$ accompanied byan abrupt Fermi-surface reconstruction that is followed by a conventionalspin-density-wave critical point at $p_{c2}$ across which the Fermi surfaceevolves smoothly to a heavy Fermi-liquid state. These experiments areconsistent with some theoretical predictions, including the sequence ofcritical points and the temperature dependence of the thermopower in theirvicinity.

Unconventional and conventional quantum criticalities in CeRh$_{0.58}$Ir$_{0.42}$In$_5$