Bogoliubov Fermi surfaces from pairing of emergent $j=\frac{3}{2}$ fermions on the pyrochlore lattice

We examine the appearance of superconductivity in the strong-coupling limit of the Hubbard model on the pyrochlore lattice. We focus upon the limit of half-filling, where the normal-state band structure realizes a $j=\frac{3}{2}$ semimetal. Introducing doping, we show that the pairing is favored in a $J=2$ quintet $E_g$ state. The attractive interaction in this channel relies on the fact that $E_g$ pairing on the pyrochlore lattice avoids the detrimental onsite repulsion. Our calculations show that a time-reversal symmetry-breaking superconducting phase is favored, which displays Bogoliubov Fermi surfaces.

Figure 1

(a) Range of parameters $\beta$ and $\gamma$ in the Luttinger-Kohn Hamiltonian where a $j=\frac{3}{2}$ semimetal is realized (shaded). (b) Range of parameters $t_{\sigma }$ and $t_{\pi }$ where a $j=\frac{3}{2}$ semimetal state is realized for the pyrochlore lattice: The pink (light gray) region indicates where a semimetallic quadratic band touching is found at the $\mathrm{\Gamma }$ point and is obtained by mapping the region in (a) to the pyrochlore lattice using Eqs. (18, 19, 20). Within the enclosed dark blue (dark gray) region, there are no other states elsewhere in the Brillouin zone at the same energy as the quadratic band-touching point. (c) Typical band structure along high-symmetry directions showing the presence of a $j=\frac{3}{2}$ semimetal state. The parameter choice ($t_{\sigma }=-0.795t_O$, $t_{\pi }=0.53t_O$) is indicated by the yellow dot in (b). The dashed red lines show the low-energy bands in the equivalent Luttinger-Kohn model.

Figure 2

Condensation energy $F_n-F_s$ per unit cell for the three $E_g$ pairing states as a function of (a) the interaction strength $V_0$ and (b) the Hubbard repulsion $U_0$. The insets show closeups of regimes with weak and strong interaction. Note that the energetically preferred state corresponds to the largest value. Here, parameters $t_1=0.321$, $t_2=-0.021$, $t_1^{\prime }=-0.013$, $t_2^{\prime }=0.004$, $t_3^{\prime }=0.008$ are used in the numerical calculation. This corresponds to the band structure presented in Fig. 1.

Figure 3

Ratios of the condensation energies $\mathrm{\Delta }F\equiv F_n-F_s$ for the $E_g$ pairing states (0,1), (1,0), and $(1,i)$ to the condensation energy $\mathrm{\Delta }{F}_{(1,i)}$ for the $(1,i)$ state. The parameters are the same as in Fig. 2.

Figure 4

Positions of sublattice $a=1,2,3,4$ in the elementary tetrahedron.