Instability of $j= 3/2$ Bogoliubov Fermi-surfaces

Exotic quantum phases including topological states and non-Fermi liquids may be realized by quantum states with total angular momentum $j=3/2$, as manifested in HgTe and pyrochlore iridates. Recently, an exotic superconducting state with non-zero density of states of zero energy Bogoliubov (BG) quasiparticles, Bogoliubov Fermi-surface (BG-FS), was also proposed in a centrosymmetric $j=3/2$ system, protected by a Z$_2$ topological invariant. Here, we consider interaction effects of a centrosymmetric BG-FS and demonstrate its instability by using mean-field and renormalization group analysis. The Bardeen-Cooper-Schrieffer (BCS) type logarithmical enhancement is shown in fluctuation channels associated with inversion symmetry. Thus, we claim that the inversion symmetry instability is an intrinsic characteristic of a BG-FS under generic attractive interactions between BG quasiparticles. In drastic contrast to the standard BCS superconductivity, a Fermi-surface may generically survive even with the instability. We propose the experimental setup, a second harmonic generation experiment with a strain gradient, to detect the instability. Possible applications to iron based superconductors and heavy fermion systems including FeSe are also discussed.