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Stability of Mixed-Symmetry Superconducting States with Broken Time-Reversal Symmetry against Lattice Distortions

We examine the stability of mixed-symmetry superconducting states with broken time-reversal symmetry in spatialsymmetry-broken systems, including chiral states, on the basis of the free-energy functional derived in the weakcoupling theory. We consider a generic α1 + iα2 wave state, with α1 and α2 being different symmetry indices such as (α1, α2) = (d, s), (px, py), and (d, d ′). The time-reversal symmetry of the mixed-symmetry state with the α1and α2-wave components is broken when the phases of these components differ, and such a state is called the time-reversal-symmetry breaking (TRSB) state. However, their phases are equated by Cooper-pair scattering between these components if it occurs; i.e., when the off-diagonal elements S α1α2 = S α2α1 of the scattering matrix are nonzero, they destabilize the TRSB state. Hence, it has often been believed that the TRSB state is stable only in systems with a spatial symmetry that guarantees S α1α2 = 0. We note that, contrary to this belief, the TRSB state can remain stable in systems without the spatial symmetry when the relative phase shifts so that S α1α2 = 0 is restored, which results in a distorted TRSB (α1 + α2) + iα2 wave state. Here, note that the restoration of S α1α2 = 0 does not imply that the symmetry of the quasi-particle energy Ek is recovered. This study shows that such stabilization of the TRSB state occurs when the distortion is sufficiently small and ∆α1∆α2 is sufficiently large, where ∆α is the amplitude of the α-wave component in the TRSB state in the absence of the distortion. We clarify the manner in which the shift in the relative phase eliminates S α1α2 and prove that such a state yields a free-energy minimum. We also propose a formula for the upper bound of the degree of lattice distortion, below which the TRSB state can be stable.