Stability of Bogoliubov Fermi surfaces within BCS theory

It has recently been realized that the gap nodes of multiband superconductors that break time-reversal symmetry generically take the form of Fermi surfaces of Bogoliubov quasiparticles. However, these Fermi surfaces lead to a nonzero density of states (DOS) at the Fermi energy, which typically disfavors such superconducting states. It has thus not been clear whether they can be stable for reasonable pairing interactions or are in practice preempted by time-reversal-symmetric states with vanishing DOS. In this Letter, we show within BCS theory applied to a paradigmatic model that the time-reversal-symmetry-breaking states are indeed stabilized over broad parameter ranges at weak coupling. Moreover, we introduce a fast method that involves solving the inverse BCS gap equation, does not require iteration, does not suffer from convergence problems, and can handle metastable solutions.

Figure 1

(a) Pairing amplitudes $\mathrm{\Delta }$ as functions of the coupling strength $V_E$ at $T=0$ for the $E_g$ pairing states (1,0), (0,1), and $(1,i)$. At small $V_E, \mathrm{\Delta }\left(V_E\right)$ shows weak-coupling scaling, $ln\mathrm{\Delta }\sim A-B/V_E$. The dotted curve is a fit of this scaling form to the numerical results for the $(1,i)$ state at small $V_E$. The inset shows the ratios of $\mathrm{\Delta }$ for the three states to $\mathrm{\Delta }$ for the $(1,i)$ state. (b) Free-energy differences $\mathrm{\Delta }F=F_s-F_n$ between the superconducting and normal states as functions of the coupling strength $V_E$ at $T=0$. The state with minimal (most negative) $\mathrm{\Delta }F$ is stable. (c) The same on a logarithmic scale, for the weak-coupling branches. Note that $-\mathrm{\Delta }F>0$ is plotted. At small $V_E$, we observe the weak-coupling scaling $ln(-\mathrm{\Delta }F)\sim A^{\prime }-B^{\prime }/V_E$. The dotted curve shows the weak-coupling scaling function for $\mathrm{\Delta }F$ obtained from the fit in (a). The inset shows the ratios of $\mathrm{\Delta }F$ for the three states to $\mathrm{\Delta }F$ for the $(1,i)$ state. Numerical parameters are given in the SM [39].

Figure 2

Free-energy differences $\mathrm{\Delta }F=F_s-F_n$ between the superconducting and normal states as functions of the coupling strengths $V_E$ and $V_T$. A pure $E_g$ state, a pure $T_{2g}$ state, and a mixed-irrep state are compared (see text). Note that $-\mathrm{\Delta }F>0$ is plotted on a logarithmic scale so that the largest value corresponds to the stable state.