Flat-band-induced superconductivity in synthetic bilayer optical lattices

Stacking two layers of graphene with a relative twist angle gives rise to Moiré patterns, which can strongly modify electronic behavior and may lead to unconventional superconductivity. A synthetic version of twisted bilayers can be engineered with cold atoms in optical lattices. Here, the bilayer structure is mimicked through coupling between atomic sublevels, and the twist is achieved by a spatial modulation of this coupling. In the present paper, we investigate the superconducting behavior of fermionic atoms in such a synthetic twisted bilayer lattice. Attractive interactions between the atoms are treated on the mean-field level, and the superconducting behavior is analyzed via the self-consistently determined pairing gap. A strong enhancement of the pairing gap is found when a quasi-flat band structure occurs at the Fermi surface, reflecting the prominent role played by the twist on the superconductivity. The tunability of interactions allows for the switching of superconducting correlations from intra (synthetic) layer to inter (synthetic) layer. This includes also the intermediate scenario, in which the competition between inter- and intra-layer coupling completely destroys the superconducting behavior, resulting in re-entrant superconductivity upon tuning of the interactions.

Figure 1

(a) Square lattice with unit cell of eight physical sites due to synthetic coupling which is different on the black ($AA$ and $AB$ - depending on the amount of A-type neighbors), green ($B$), and red ($C$) sites. The lattice is covered by unit cells in a brick wall arrangement, as indicated in the inset. (b) Schematic representation of the three interaction types appearing in the system. (c) Evolution of the average energy of the bands as a function of the modulation parameter $\alpha$ at ${\mathrm{\Omega }}_0=100$ (in units of $t$). The panel shows the outer bands (two blue lines and two read ones) of each, positive and negative manifold being shifted with respect to the two inner subsets of bands marked with black lines. At the critical value of ${\alpha }_c\approx 0.67$ the blue band from the upper manifold reaches lower energy than black, negative manifold. This results with change of the position of the Fermi surface. (d) Full spectrum of the system under periodic modulation ${\mathrm{\Omega }}_0=100$ and $\alpha =0.2$. (e) Zoomed plot of (d) highlighting the negative six-fold subset of bands including the central quasi-flat bands. (f) Spectrum of the system at ${\mathrm{\Omega }}_0=100$ and $\alpha =0.7$, after the flipping of the bands occurred (dashed line in panel (c)).

Figure 2

(a), (b) Expansion of superconducting gaps ${\mathrm{\Delta }}_1^{AA}$ and ${\mathrm{\Delta }}_1^{AB}$ as a function of the $U=U_1=U_2\le 16t$ under different modulation values $\alpha$ resulting in specific cutoff value $U_c$ (${\mathrm{\Delta }}_2$ has an identical behavior). (c) Critical temperature dependence on the modulation strength $\alpha$ at $U_{1,2,3}=16t$. Inset plot represents the energetic width ${\delta }_F$ of the quasi-flat bands as a function of the modulation parameter $\alpha$.

Figure 3

Superconducting order parameter ${\mathrm{\Delta }}_1^{AA}$ [panels (a) and (c)] and ${\mathrm{\Delta }}_2^{AA}$ [panels (b) and (d)] as functions of relative amplitude of $U_1$ and $U_2$ interactions at $\alpha =0.67$. Existence of SC phase in this $SU\left(4\right)$ symmetric system is not present until $U=U_1=U_2$ reaches a cut-off value $U_c$, at which the non-SC valley narrows to zero. The width of the valley as well as the critical interaction strength depends on the modulation parameter $\alpha$ (as shown in Fig. 2). Panels (c) and (d) represent the regime of small interaction strengths that are not covered on the panels (a) and (b) due to limited colormap resolution. Sites of (AB) type reveal qualitatively identical behavior, however, with much smaller pairing amplitude [See Fig. 4].

Figure 4

${\mathrm{\Delta }}_{AA,AB}$ (dashed) and ${\mathrm{\Delta }}_2$ (solid) as functions of $U_2$ (in units of $U_1$) for four different values of $U_1$ at modulation strength $\alpha =0.67$. One can observe the shift of the SC gap decay towards $U_2/U_1=1$ with the increase of interaction strength. For $U_1=6t$, the decay of SC gap occurs at $U_2/U_1=1$ marking the $U_C$ for this particular value of $\alpha$.

Figure 5

Superconducting gap in the system of $U_1=2t$ and $U_1\gg U_2$. (a) The order parameter ${\mathrm{\Delta }}_1$ associated with interaction type $U_1$ as a function of the inter-layer hopping strength ${\mathrm{\Omega }}_0$ at $\alpha =0.2$. We have set $U_2=0$ as its presence weakens the order parameter ${\mathrm{\Delta }}_1$, as shown on Fig. 4. The green line represents the situation where the $\mathrm{\Omega }\left(\mathbf{r}\right)$ is homogeneous, i.e. $\alpha =0$. The separate plot for ${\mathrm{\Delta }}_C$ has been omitted since it's behavior is identical to ${\mathrm{\Delta }}_B$, which has been plotted. (b) Dependence of the order parameter ${\mathrm{\Delta }}_1^{AA}$ on the modulation strength $\alpha$ for ${\mathrm{\Omega }}_0=100t$. We have omitted the plot of ${\mathrm{\Delta }}_1^{AB}$ due to its negligible amplitude. Sudden drop of pairing (marked by dashed gray line) occurs around critical value of $\alpha \approx 2/3$ at which the Fermi level does not reside at quasi-flat bands any more. Such situation has been shown on Fig. 1.