Evidence for spin-triplet odd-parity superconductivity close to type-II van Hove singularities

Searching for unconventional Cooper pairing states has been at the heart of superconductivity research since the discovery of BCS superconductors. In particular, spin-triplet odd-parity pairing states were recently revisited due to the possibility of tuning towards topological superconductors. In this context, it is interesting to note a recent proposal that such a spin-triplet pairing instability occurs when the band filling is near van Hove singularities (vHS) associated with momenta away from time-reversal invariant momenta named type-II vHS. However, this result was obtained within a weak coupling renormalization group with Fermi surface patch approximation. To explore superconducting instabilities beyond this weak coupling Fermi surface patch approximation, we perform systematic study on the Hubbard model in a two-dimensional square lattice using three different methods: random phase approximation, large-scale dynamical mean-field theory simulations with continuous time quantum Monte Carlo (CTQMC) impurity solver, and large-scale dynamical cluster simulations with the CTQMC cluster solver. We find, in a wide doping range centered around the type-II van Hove filling, a twofold degenerate, spin-triplet, odd-parity $p$-wave pairing state emerges due to repulsive interaction, when the Fermi surface is not sufficiently nested. Possible relevance of our findings to the recently discovered superconductors ${\mathrm{LaO}}_{1-x}{\mathrm{F}}_x{\mathrm{BiS}}_2,{\mathrm{Ir}}_{1-x}{\mathrm{Pt}}_x{\mathrm{Te}}_2$, and proposed doped ${\mathrm{BC}}_3$ are also discussed.

Figure 1

(a) Energy dispersion of the noninteracting system, with parameters $t_2=-0.5t_1,t_3=0.1t_2,\mu =-2t_1$. The Fermi surface is highlighted by the black contour line and its projection to BZ at the bottom. Blue circles represent the position of vH momenta at $\mathbf{K}=(\pm \pi /2,0)$ and $(0,\pm \pi /2)$. (b) Density of the states, $\text{DOS}\left(\omega \right)$, the VHS is located at the Fermi level, and the corresponding band filling $n_V=0.376$, as indicated by red dashed line. (c) The BZ patches of $N_c=16$ cluster for DCA simulations. The self-energy in each patch is approximated by the self-energy at the cluster momentum point (solid pink dot). The vH momenta (blue circles) are captured by the $N_c=16$ cluster.

Figure 2

Distribution of the bare susceptibilities $|{\chi }_0^{ph}\left(\mathbf{q}\right)|$ in the BZ for the filling of (a) $n_V=0.376$ and (b) $n=0.5$. The ${\chi }_0^{ph}\left(\mathbf{q}\right)$ is peaked close to $\Gamma =\mathbf{q}=(0,0)$.

Figure 3

RPA diagrams of the two second-order processes which contribute the effective interaction Eq. (5), wherein a cooper pair with momenta $(\mathbf{k},-\mathbf{k})$ are scattered to the momenta $({\mathbf{k}}^{\prime },-{\mathbf{k}}^{\prime })$ by the magnetic fluctuation represented by the renormalized magnetic susceptibility (shaded ellipses).

Figure 4

The doping dependence of the LEV $\lambda$ of Eq. (8) for the four possible pairing symmetries in the square lattice, i.e., the $s$ wave, the $d_{xy}$, the $d_{x^2-y^2}$, and the degenerate $p$ wave (i.e., $p_x$ and $p_y$). Here, a weak interaction parameter $U=0.5t_1$ and a low finite temperature $k_{B}T={10}^{-4}{t}_1$ are adopted to avoid the divergence of the magnetic susceptibility near the vHS. The pink dashed lines point out the fillings at which the leading eigenvectors are presented in Fig. 5.

Figure 5

The leading pairing gap function $\Delta \left(\mathbf{k}\right)$ of the $p_x$ symmetry for the vH filling $n_V=0.376$ (a) and filling $n=0.5$ (b). The other one of $p_y$ symmetry can be obtained from the present one by a ${90}^{\circ }$ rotation.

Figure 6

(a) Bethe-Salpeter equations in the particle-hole charge/magnetic channels. ${\chi }_{ph}^{c/m}(P,P^{\prime },Q)$ are the two-particle correlation functions and ${\Gamma }_{ph}^{c/m}(P^{\prime },P^{\prime \prime },Q)$ are the irreducible vertex functions; ${\chi }_0^{ph}(P,Q)$ is the bare bubble. (b) Parquet equation of the irreducible particle-particle singlet vertex function, ${\Gamma }_{pp}^s(P,P^{\prime },Q)$. It is decomposed into fully irreducible vertex ${\Lambda }_{pp}^s$ and cross channel contributions from particle-hole charge/magnetic vertex ladders ${\Phi }_{ph}^{c/m}$. The vertex ladders are products between irreducible vertex functions $\Gamma$, and two-particle correlation function $\chi$, in the same channel, with the internal momentum-frequency indices integrated out, but the momentum-frequency transfer $\tilde{Q}$ and $\overline{Q}$ determined by the indices of the irreducible vertex functions in the left-hand side of the equation.

Figure 7

LEVs close to the type-II vHS with $t_2=-0.5t_1,t_3=0.1t_1,n=0.4$, and $U=2t_1$. As temperature goes down, triplet pairing wins over the singlet pairing, ferromagnetic instability, and becomes the leading instability of the system. The corresponding leading eigenvectors are twofold degenerate, one has $p_x^{\prime }=-p_x-p_y$ symmetry (left inset) and the other has $p_y^{\prime }=-p_x+p_y$ symmetry (right inset). The regions with larger amplitude of $\varphi \left(P\right)$ are close to the vH momenta $\mathbf{K}=(\pm \pi /2,0)$ and $(0,\pm \pi /2)$.

Figure 8

LEVs close to the type-II vHS with $t_2=-0.5t_1,t_3=0.1t_1,n=0.4$, and $U=4t_1$. As temperature goes down, the ferromagnetic instability becomes the subleading below the leading twofold degenerate, triplet pairing.

Figure 9

LEVs close to the type-II vHS with $U=2t_1,t_2=-0.5t_1,t_3=0.1t_1$, and filling $n=0.4$. Results are obtained from 16-site DCA simulations. As temperature goes down, triplet pairing wins over the singlet pairing and ferromagnetic instabilities to become the leading instability. The corresponding leading eigenvectors are twofold degenerate; one has $p_x^{\prime }=p_x+p_y$ symmetry (upper inset) and the other has $p_y^{\prime }=-p_x+p_y$ symmetry (lower inset).