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CeRh_{2}As_{2} is an unconventional superconductor with multiple superconducting phases and T_{c}=0.26 K. When H∥c, it shows a field-induced transition at μ_{0}H^{*}=4 T from a low-field superconducting state SC1 to a high-field state SC2 with a large critical field of μ_{0}H_{c2}=14 T. In contrast, for H⊥c, only the SC1 with μ_{0}H_{c2}=2 T is observed. A simple model based on the crystal symmetry was able to reproduce the phase diagrams and their anisotropy, identifying SC1 and SC2 with even and odd parity superconducting states, respectively. However, additional orders were observed in the normal state which might have an influence on the change of the superconducting state at H^{*}. Here, we present a comprehensive study of the angle dependence of the upper critical fields using magnetic ac susceptibility, specific heat, and torque on single crystals of CeRh_{2}As_{2}. The experiments show that the state SC2 is strongly suppressed when rotating the magnetic field away from the c axis and it disappears for an angle of 35°. This behavior agrees perfectly with our extended model of a pseudospin triplet state with d[over →] vector in the plane and hence allows us to conclude that SC2 is indeed the suggested odd-parity state.

CeRh_{2}As_{2} has recently been reported to be a rare case of a multiphase unconventional superconductor close to a quantum critical point (QCP). Here, we present a comprehensive study of its normal-state properties and of the phase (I) below T_{0}≈0.4 K which preempts superconductivity at T_{c}=0.26 K. The second-order phase transition at T_{0} presents signatures in specific heat and thermal expansion but none in magnetization and ac susceptibility, indicating a nonmagnetic origin of phase I. In addition, an upturn of the in-plane resistivity at T_{0} points to a gap opening at the Fermi level in the basal plane. Thermal expansion indicates a strong-positive-pressure dependence of T_{0}, dT_{0}/dp=1.5 K/GPa, in contrast to the strong-negative-pressure coefficient observed for magnetic order in Ce-based Kondo lattices close to a QCP. Similarly, an in-plane magnetic field shifts T_{0} to higher temperatures and transforms phase I into another nonmagnetic phase (II) through a first-order phase transition at about 9 T. Using renormalized band-structure calculations, we find that the Kondo effect (T_{K}≈30 K) leads to substantial mixing of the excited crystalline-electric-field states into the ground state. This allows quadrupolar degrees of freedom in the resulting heavy bands at the Fermi level which are prone to nesting. The huge sensitivity of the quadrupole moment on hybridization together with nesting causes an unprecedented case of phase transition into a quadrupole-density-wave state at a temperature T_{0}≪T_{K}, which explains the nature of phases I and II.