Odd-frequency pairing inherent in a Bogoliubov Fermi liquid

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 nonideal 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.

Figure 1

Schematic phase diagram for the Bogoliubov Fermi liquid. The horizontal axis indicates the deviation from the ideal limit of the bogolons. The dashed line at $T_b$ indicates a possible ordering instability at very low temperature, which may be induced by the attractive interaction among bogolons as discussed in Refs. [19, 20]. Note that this paper focuses on $T>T_b$. We assume that the time-reversal symmetry $T$ is broken in the pairing state ($T<T_c$) but the inversion symmetry $P$ is preserved for $T>T_b$.

Figure 2

Diagrams for (a) normal and (b) anomalous self-energies which are relevant to the Bogoliubov Fermi liquid.

Figure 3

Feynman diagrams relevant to the self-consistent Born approximation.

Figure 4

Examples of the solutions of Eqs. (24) and (25). We define dimensionless self-energies and frequency as $\tilde{\mathrm{\Sigma }}=\mathrm{\Sigma }/(\pi n_{\mathrm{imp}}{D}_0|u_2{|}^2),{\tilde{S}}^{†}=S^{†}/(\pi n_{\mathrm{imp}}{D}_0|u_2{|}^2),{\tilde{\varepsilon }}_n={\varepsilon }_n/(\pi n_{\mathrm{imp}}{D}_0|u_2{|}^2)$. The parameters are chosen as (a) $u_1/\left|u_2\right|=3exp(\mathrm{i}\pi /6),arg{u}_2=\pi /12$ and (b) $u_1/\left|u_2\right|=3exp(\mathrm{i}\pi /3),arg{u}_2=\pi /12$. The inset shows the self-energy with ${\varepsilon }_n$ multiplied, indicating $-{\varepsilon }_n\mathrm{Im}\mathrm{\Sigma }\left(\mathrm{i}{\varepsilon }_n\right)={\varepsilon }_n\left|S^{†}\left(\mathrm{i}{\varepsilon }_n\right)\right|$ for ${\varepsilon }_n\to 0$.

Figure 5

(a) Density of states, (b) momentum distribution function, (c) pair potential for the solution of the first kind. (d), (e), and (f) are similar to (a), (b), and (c), respectively, but for the solution of the second kind. $D_0$ is a bare density of states. The pair amplitude in (c) is normalized by ${\mathrm{\Gamma }}_1{D}_0$ and in (f) by $\left|V\right|D_0$ where the phases are chosen as zero.

Figure 6

Wave-vector dependence of (a) $C_M\left(\mathit{k}\right)$ (multipoles), (b) $C_P\left(\mathit{k}\right)$, and (c) $C_{P^{†}}\left(\mathit{k}\right)$ (multiplet electron and hole pairs) along the Fermi surface in the $j=3/2$ fermion model. The inset of (a) shows the two Fermi surfaces near the $k_x$ axis located in the $k_y=0$ plane. The horizontal axis of (a)–(c) is a path shown by arrows in the inset of (a).

Figure 7

(a) Self-energy for bosons. (b) Wave-vector dependence of $h_{1,2}$ defined in Eqs. (A9) and (A13) at several temperatures. The cutoff wave vector is chosen as ${\omega }_{\mathrm{C}}/{\varepsilon }_{\mathrm{F}b}=0.3$.