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Phenomenological theory of the superconducting state inside the hidden-order phase of URu$_{2}$Si$_{2}$

Recent experiments have unveiled important properties of the ground state of the elusive heavy fermion $\mathrm{URu_{2}Si_{2}}$. While tetragonal symmetry-breaking was reported below the hidden-order (HO) transition at $T_{HO}\approx17.5$ K, time-reversal symmetry-breaking was observed below the superconducting transition temperature $T_{c}<T_{HO}$. Although the latter results have been used to argue in favor of a chiral $d+id$ superconducting state, such an order parameter is incompatible with broken tetragonal symmetry. Here, we employ a phenomenological model to investigate the properties of a chiral superconducting state that develops inside the hidden-order phase. In this case, there are actually two superconducting transition temperatures: while $T_{c}$ marks a normal-state to superconducting transition, $T_{c}^{*}<T_{c}$ signals a superconducting-to-superconducting transition in which time-reversal symmetry is broken. In the phase $T_{c}^{*}<T<T_{c}$, the low-energy density of states $ρ\left(ω\right)$ is enhanced due to the crossing of two nodal lines, giving rise to an unusual $ω\log(ω)$ dependence of $ρ\left(ω\right)$, which is manifested in several thermodynamic properties. We also investigate the emergence of a soft amplitude gap mode near $T_{c}^{*}$. In contrast to the usual amplitude mode near a regular normal-state to superconducting transition, this mode becomes soft near a superconducting-to-superconducting transition, which in principle allows for its detection by Raman spectroscopy. Finally, we investigate the impact of twin domains on the anisotropic properties of the superconducting state, and propose experiments in mechanically strained samples to explore the interplay between hidden order and superconductivity in $\mathrm{URu_{2}Si_{2}}$.