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 spatial-symmetry-broken systems, including chiral states, on the basis of the free-energy functional derived in the weak-coupling theory. We consider a generic α₁ + iα₂ wave state, with α₁ and α₂ being different symmetry indices such as (α₁,α₂)=(d,s), (p_x,p_y), and (d,d′). The time-reversal symmetry of the mixed-symmetry state with the α₁- and 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 E_k is recovered. This study shows that such stabilization of the TRSB state occurs when the distortion is sufficiently small and Δ_α1Δ_α2 is sufficiently large, where Δ_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. (author)