Microscopic theory of strain-controlled split superconducting and time-reversal symmetry-breaking transitions in $s+id$ superconductor

We study conditions of the appearance of $U(1)\times {\mathbb{Z}}_2$ superconducting states that spontaneously break time-reversal symmetry (BTRS) on a square lattice as a function of applied stress. Calculations show that if critical temperatures coincide at zero stress, they exhibit a linear kink and no kink otherwise for uniaxial and isotropic strain. Linear kink is absent for shear strain. We find that in general, the microscopic calculations show a complex phase diagram, for example, non-monotonic behavior of BTRS transition. Another beyond-Ginzburg-Landau theory result is that $U(1)$ critical temperature can decrease under compressional [100] uniaxial strain for small Poisson ratio materials. In the second part of the paper, we consider the effects of boundaries and finiteness of the sample on the strain-induced splitting of $T_c^{U(1)}$ and $T_c^{{\mathbb{Z}}_2}$ transitions. A finite sample has BTRS boundary states with persistent superconducting currents over a wide range of band filling. Overall, the BTRS dome occupies a larger band filling–temperature phase space region for a mesoscopic sample with [110] surface compared to an infinite system. Hence, the presence of boundaries helps to stabilize the BTRS phase.