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 basis free-energy functional derived weak-coupling theory. consider a generic α1 + iα2 wave state, and α2 being different indices such as \((\alpha _{1},\alpha _{2}) = (\text{d},\text{s})\), \((\text{p}_{x},\text{p}_{y})\), \((\text{d},\text{d}')\). The state α1- α2-wave components is when phases these differ, called time-reversal-symmetry breaking (TRSB) state. However, their are equated by Cooper-pair scattering between if it occurs; i.e., off-diagonal elements \(S_{\alpha _{1}\alpha _{2}} S_{\alpha _{2}\alpha _{1}}\) matrix nonzero, they destabilize TRSB Hence, has often been believed that stable only systems spatial guarantees 0\). note that, contrary to this belief, can remain without relative phase shifts so 0\) restored, which results distorted (α1 α2) Here, restoration does not imply quasi-particle energy Ek recovered. This study shows stabilization occurs distortion sufficiently small \(\Delta _{\alpha _{1}}\Delta _{2}}\) large, where Δα amplitude α-wave component absence distortion. clarify manner shift eliminates prove yields minimum. also propose formula for upper bound degree lattice distortion, below be stable.