Spin-fluctuation-mediated chiral $d+id'$-wave superconductivity in the $α$-$\mathcal{T}_3$ lattice with an incipient flat band

TL;DR

The study investigates spin-fluctuation-mediated chiral $d+id'$-wave superconductivity in the $\alpha$-$\mathcal{T}_3$ lattice near quarter-filling, highlighting the impact of an incipient flat band. Using an extended Hubbard model with off-site attraction and a self-consistent mean-field BdG approach, it identifies two topological superconducting phases with Chern numbers $|\mathcal{C}|=4$ and $|\mathcal{C}|=8$. Complementarily, FLEX calculations on the repulsive Hubbard model reveal a spin-fluctuation–driven $d$-wave gap corresponding to a $d+id'$ state with $|\mathcal{C}|=8$, mediated by finite-energy spin fluctuations at $\mathbf{q}=0$ arising from the flat band. Together, these results connect flat-band physics to topological superconductivity in pseudospin-1 lattices and suggest avenues for realizing $d+id'$ topological superconductivity in engineered lattices and multilayer graphene systems.

Abstract

We study anisotropic superconductivity in the nearly quarter-filled $α$-$\mathcal{T}_3$ lattice. We analyze an extended Hubbard model with off-site attractive interactions within the mean-field framework and find two distinct chiral $d+id'$-wave superconducting phases characterized by different Chern numbers. We further investigate the superconducting mechanism mediated by spin fluctuations arising from purely repulsive interactions by applying the fluctuation-exchange (FLEX) approximation to the Hubbard model. The gap symmetry obtained by solving the linearized Eliashberg equation is $d$-wave, which corresponds to a $d+id'$-wave superconducting state with a Chern number of $8$, including the spin degree of freedom. The $\mathbf{q}=\mathbf{0}$ antiferromagnetic spin fluctuation, which possesses the largest spectral weight at finite energies arising from the incipient flat band, gives rise to an effective spin-singlet pairing glue between rim sites.

Figures (9)

• Figure 1: (a) Real-space structure of the $\alpha$--$\mathcal{T}_3$ lattice. $\bm{\delta}_\ell$ ($\ell=1,2,3$) is given in Eq. (\ref{['eq:AB-vectors']}). The lattice vectors are $\bm{a}_1 = \bm{\delta}_1-\bm{\delta}_3$, $\bm{a}_2 = \bm{\delta}_2-\bm{\delta}_3$. Hopping is allowed only between the hub (B) and rim (A and C) sites. (b) Tight-binding band structure of the $\alpha$--$\mathcal{T}_3$ lattice. The band structure is independent on the hopping ratio $\alpha=t_{\rm AC}/t_{\rm AB}$.
• Figure 2: (a) Band and sublattice representations of the pairing potentials $\Delta(\bm{k})$ obtained by self-consistently solving the BdG equation using an extended Hubbard model with nearest-neighbor attractive interactions. Two representative parameter sets of the attractions ($V_{\rm AB}=V_{\rm BC}=V_{\rm CA}$ and $V_{\rm CA}\gg V_{\rm AB}=V_{\rm BC}$) are shown. The region enclosed by the dashed lines denotes the first Brillouin zone (BZ), and the black solid contour in the ($--$) panels indicates the Fermi surface of the single-particle band. (b) Distribution of the nodal points and the corresponding winding numbers of the $(--)$ component of the $d+id'$-wave paring potential $\Delta_{--}(\bm{k})$ in the first BZ. As the interaction strengths between sublattices are varied, the total Chern number $\mathcal{C}$ of the occupied BdG bands switches between $\mathcal{C}=4$ (SC1 phase) and $\mathcal{C}=-8$ (SC2 phase). (c) Superconducting phase diagram for $V_{\rm CA}$ versus $V_{\rm AB}=V_{\rm BC}$. When $V_{\rm CA}$ is much larger than $V_{\rm AB}$ and $V_{\rm BC}$, the system is in the SC2 phase. The band-representation component with the largest amplitude of $\Delta$ is $(--)$ for the SC1 phase and $(00)$ for the SC2 phase. The total Chern numbers were calculated using Fukui--Hatsugai--Suzuki's method Fukui_2005.
• Figure 3: Superconducting phase diagrams for (a) $V_{\rm CA}$ versus $V_{\rm AB}=V_{\rm BC}$ and (b) $V_{\rm AB}$ versus $V_{\rm BC}$ with $V_{\rm CA}$ fixed, calculated at $\alpha=0.5$. All other parameters are the same as in Fig. \ref{['fig:BdG_plot']}.
• Figure 4: Band filling $n$ dependence of the eigenvalue $\lambda$ of the linearized Eliashberg equation (\ref{['eq:Eliash_eq']}) for (a) various values of $\alpha$ at $T/t=0.002$ and (b) various values of $T$ at $\alpha=1$. Band fillings $n=0.75$ and $n=1$ correspond to the $1/4$- and $1/3$-filling, respectively. The gap functions $\Delta(\bm{k})$ are the $d$-wave for all $n$. The maximum eigenvalue of $U^{\rm s}\chi^0(\bm{q})$ (Stoner factor) approaches to $1$ at $\bm{q}=(0,0)$ near the $1/3$-filling.
• Figure 5: Band and sublattice representations of the gap functions $\Delta(\bm{k},\omega\!=\!0)$ obtained at $\alpha=1$ and band fillings (a) $n=0.75$ and (b) $n=0.9$. The hue represents the phase of the gap function, while the brightness indicates its amplitude. The black solid contour in the ($--$) panels indicates the Fermi surface of the single-particle band. For all the parameters we examined, the phase winding number around the $\textrm{K}$($\textrm{K}'$) point in the band representation is $|w|=2$, which is consistent with the SC2 phase ($|\mathcal{C}|=8$) obtained from the mean-field analysis of the extended Hubbard model.