Fragile Topology and Flat-Band Superconductivity in the Strong-Coupling Regime

In flat bands, superconductivity can lead to surprising transport effects. The superfluid “mobility”, in the form of the superfluid weight $D_s$, does not draw from the curvature of the band but has a purely band-geometric origin. In a mean-field description, a nonzero Chern number or fragile topology sets a lower bound for $D_s$, which, via the Berezinskii-Kosterlitz-Thouless mechanism, might explain the relatively high superconducting transition temperature measured in magic-angle twisted bilayer graphene (MATBG). For fragile topology, relevant for the bilayer system, the fate of this bound for finite temperature and beyond the mean-field approximation remained, however, unclear. Here, we numerically use exact Monte Carlo simulations to study an attractive Hubbard model in flat bands with topological properties akin to those of MATBG. We find a superconducting phase transition with a critical temperature that scales linearly with the interaction strength. Then, we investigate the robustness of the superconducting state to the addition of trivial bands that may or may not trivialize the fragile topology. Our results substantiate the validity of the topological bound beyond the mean-field regime and further stress the importance of fragile topology for flat-band superconductivity.

Figure 1

(a) Kagome-3 lattice model with the unit-cell area shaded in gray. The inset shows the details of one plaquette. The red circle highlights the $1a$ Wyckoff position of the lattice, while the yellow circle the $1b$ Wyckoff position. Note that, although the full point group of the model is $p6mm$, we refer to the real space high-symmetry points of its subgroup $p2$ [51]. The basis vectors ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$ are represented by the green arrows. The three inequivalent sublattices $A$, $B$, and $C$ are also highlighted. (b) Single-particle spectrum of the model along high-symmetry lines of $p6mm$. The high-symmetry lines and the first Brillouin zone (BZ) are shown in the inset. The flat bands at $\epsilon =-2$ are doubly degenerate. (c) Wilson loop spectrum of the two flat bands of the kagome-3 model. (d) Wilson loop spectrum of the three lowest bands of the model with an additional $s$ orbital at $1a$ Wyckoff position. (e) Same as (d) but with an $s$ orbital at $1b$ Wyckoff position.

Figure 2

Superfluid weight $D_s(T)$ for the attractive Hubbard model with interaction strength $|U|$. The crossing of $D_s$ with the dashed line $2T/\pi$ indicates the BKT transition, where the superconducting transition occurs. (a) Different system sizes $L\times L$, with $L=4$, 6, 8 and $|U|=2$. The arrow on the $y$ axis represents the mean-field topological lower bound for $D_s(T=0)$ [51]. (b) Different interaction strengths $|U|=1$, 1.5, 2 in a $6\times 6$ system. In both (a) and (b) the kagome-3 model is considered. (c) Results for the four-band models for $|U|=2$ and $L=6$. The trivial model has an additional $s$ orbital at Wyckoff position $1a$, cf. Fig. 1. The model with fragile topology has an additional $s$ orbital at Wyckoff position $1b$, instead, cf. Fig. 1.

Figure 3

(a) Spin susceptibility ${\chi }_S$ as a function of temperature $T$ for the Hubbard model on the kagome-3 lattice with $|U|=2$ and $L=6$. (b) Single-particle density of states $N({\epsilon }_F)$ as a function of temperature $T$ for the same model. The shaded areas correspond to the superconducting state in dark orange, the range below ${\chi }_S$ peaks in yellow and that below the peak of $N({\epsilon }_F)$ in blue. (c) Scaling of the critical temperatures $T_c$, $T_S$, and $T_N$ as a function of the interaction strength $|U|$. All these quantities show a linear scaling.