Broken time-reversal symmetry through non-degenerate superconducting instabilities

Superconductivity is a manifestation of broken global gauge symmetry. In some materials other symmetries may also be broken at the superconducting transition. In the usual Ginzburg-Landau picture, time-reversal symmetry has a special status in that it can only be broken when the pairing instability has several degenerate channels. Higher-order terms in the free energy can then select a non-trivial phase factor between the different components of the order parameter, leading to a state with broken time-reversal symmetry. This is in contrast to other broken symmetries, such as C₄ symmetry in a d-wave superconductor, which are broken intrinsically at the level of the diverging pairing susceptibility. Here we show that time-reversal symmetry can also be broken in this way. We use simple models of a symmorphic and a non-symmorphic crystal, and show that their symmetry allows for non-degenerate instabilities into a superconducting state with intrinsically broken time-reversal symmetry. We consider singlet, on-site pairing involving several bands emerging from orbitals related by internal symmetries of the crystal. We apply these ideas to the recently discovered Re₆X (X=Zr, Hf) superconducting material.