Distortional weak-coupling instability of Bogoliubov Fermi surfaces

Centrosymmetric multiband superconductors which break time-reversal symmetry generically have two-dimensional nodes, i.e., Fermi surfaces of Bogoliubov quasiparticles. We show that the coupling of the electrons to the lattice always leads to a weak-coupling instability of such a state toward spontaneous breaking of inversion symmetry at low temperatures. This instability is driven by a Cooper logarithm in the internal energy but the order parameter is not superconducting but distortional. We present a comprehensive symmetry analysis and introduce a measure that allows us to compare the strengths of competing distortional instabilities. Moreover, we discuss the instability using an effective single-band model. This framework reveals a duality mapping of the effective model which maps the distortional order parameter onto a superconducting one, providing a natural explanation for the Cooper logarithm and the weak-coupling nature of the instability. Finally, we consider the possibility of a pair-density wave state when inversion symmetry is broken. We find that it can indeed exist but does not affect the instability itself.

Figure 1

Normal-state Fermi surface (semitransparent gray) and BFSs (yellow) for the specific model with superconducting chiral $T_{2g}$ order parameter $\mathrm{\Delta }(1,i,0)$. The parameters of the model are specified in Sec. 6c.

Figure 2

Sketches of the consequences of distortions: (a) shifts of quasiparticle bands, (b) splitting of band crossings, and (c) combination of both. The light blue dashed lines denote the quasiparticle bands of the undistorted superconductor. Their crossings with $E=0$ form the BFSs of the centrosymmetric superconductor. Note that the distorted system retains BFSs when the distortional gap is smaller than the band shift.

Figure 3

Enhancement $u$ of the coefficient of the quadratic term $\mathcal{O}\left({\mathrm{\Phi }}^2\right)$ in the internal energy due to odd-in-momentum band shifts. $\alpha$ is the odd part of the band shift in units of the distortional gap. For $\alpha >1$, residual BFSs exist.

Figure 4

The $\varphi$ averages $\overline{|{\mathbf{g}}^l{\times \mathbf{h}|}^2}$ characterizing the strength of the distortion terms $d_{E_u}^l$ that stabilize a distortion in the $z$ direction, as a function of the angle $\theta$. For details, see the text.