The disorder and interaction effects on Bogoliubov-Fermi surfaces with preserved inversion symmetry are studied for a low-energy effective model coupled to bosonic degrees of freedom. It is shown that the non-ideal Bogoliubov quasiparticles (bogolons) generically induce the odd-frequency pair amplitude which reflects a Cooper pairing at different time. The self-energy of bogolons is mainly contributed by the disorder effects in the low frequency limit as in the usual electron liquid. Depending on the choice of the parameters, there are two kinds of solutions: one is frequency-independent (but with sign function of frequency) and the other is proportional to the inverse of the frequency, which exist in both the normal and anomalous parts of the self-energy. These characteristic self-energy structures are clearly reflected in the single-particle spectrum. Since the bogolons are originally composed of electrons, the connection between the two is also sought using the concrete $j=3/2$ fermion model, which reveals that the odd-frequency pairing of bogolons is mainly made of the electrons' odd-frequency pairing.
It was recently pointed out that Fermi surfaces can survive even in superconductors. Here, the authors study the instability of such systems toward electronic ordering. The ordered states are classified into diagonal and off-diagonal ones, which, respectively, indicate the Pomeranchuk instability and Cooper pairing not of original electrons but of Bogoliubov particles (bogolons). The corresponding order parameters are expanded by multipole moments and multiplet pair amplitudes of original electrons. These results provide insights into superconductors with multiple phase transitions.
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