Superconductivity, pseudogap, and phase separation in topological flat bands

Superconductivity is a macroscopic quantum phenomenon that requires electron pairs to delocalize over large distances. A long-standing question is whether superconductivity can exist even if the electrons' kinetic energy is completely quenched, as is the case in a flat band. This is fundamentally a nonperturbative problem, since the interaction energy scale is the only relevant energy scale, and hence it requires going beyond the traditional Bardeen-Cooper-Schrieffer theory of superconductivity, which is perturbative by nature. In this work, we study a two-dimensional model of an isolated narrow band at partial filling with local attractive interactions, using numerically exact quantum Monte Carlo calculations. We focus on the case where the flat bands are topologically nontrivial, and hence the single-particle wave functions that span these bands cannot be completely spatially localized. Our calculations unambiguously demonstrate that the ground state is a superconductor; strikingly, the critical temperature scales nearly linearly with the interaction strength. Above the superconducting transition temperature, we find a broad pseudogap regime that exhibits strong pairing fluctuations and a tendency towards electronic phase separation. Introducing a small nearest-neighbor attraction suppresses superconductivity entirely and drives the system to phase separate. We discuss the possible relevance of superconductivity in this unusual regime to the physics of flat band moiré materials.

Figure 1

(a) Lattice model with $\pi$ flux through every plaquette, two orbitals ($A$ and $B$), and first, second, and fifth nearest-neighbor hopping of amplitude $t_1,t_2$, and $t_5$. (b) Top (bottom): band dispersions for $t_5=0.0$ ($-0.1$) with “flatness ratios” $\mathcal{F}=0.2$ ($0.009$). The lower bands have Chern numbers $C=+1$ ($-1$) for spin-up (spin-down) particles. Red (blue) indicates high (low) energy. (c) The superconducting $T_c$ (in blue) and the “pseudogap temperature,” $T_p$ [in red, defined through the maximum in the spin susceptibility; see panel (d)] as a function of $U$. Solid and dashed lines correspond to the band structures shown in (b), with $\mathcal{F}=0.2$ and $0.009$, respectively. (d) The orbital and spin magnetic susceptibilities for the dispersive (flat) band with $U=1$ ($U=3$).

Figure 2

The superfluid stiffness $D_s\left(T\right)$ [Eq. (4)] for the (a) flat-band ($\mathcal{F}=0.009,U=3$) and (b) dispersive ($\mathcal{F}=0.2,U=1$) cases, and different system sizes. The black line denotes the universal BKT jump, $D_s=2T/\pi$. (c) and (d) $D_s\left(T\right)$ for various coupling strength $U$ on the largest simulated lattice. In (c), $D_s\left(T\right)/U$ curves for different $U$'s collapse onto each other when plotted vs $T/U$, confirming that $U$ is effectively the only energy scale in the flat-band case. The shaded area marks the collapsed function in both (c) and (d) to guide the eye for comparison.

Figure 3

The quantity $\beta \tilde{G}\left(\mathit{k}\right)$, that serves as a proxy for the spectral function near the Fermi energy [see Eq. (5) and the following discussion], as a function of $\mathit{k}$. The green lines denote the Fermi surface in the noninteracting case. In (a), $U=1$ and the temperatures are $T=0.25,0.06,0.03$; in (b), $U=3$ and $T=0.4,0.1,0.05$.

Figure 4

(a) The reciprocal pairing and charge susceptibilities are shown for $\mathcal{F}=0.2,U=1t_1$ (solid) and $\mathcal{F}=0.009,U=3t_1$ (dashed). (b) Same as (a) but with an additional nearest-neighbor interaction $H_{int,nn}$ [see Eq. (8)], with $V=0.2t_1$.