Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of twisted bilayer graphene

We study superconductivity of twisted bilayer graphene with local and nonlocal attractive interactions. We obtain the superfluid weight and Berezinskii-Kosterlitz-Thouless (BKT) transition temperature for microscopic tight-binding and low-energy continuum models. We predict qualitative differences between local and nonlocal interaction schemes which could be distinguished experimentally. In the flat-band limit where the pair potential exceeds the band width we show that the superfluid weight and BKT temperature are determined by multiband processes and quantum geometry of the band.

Figure 1

(a) The moiré superlattice of TBG depicted with a twist angle $\theta$ and the choice of the $x$ and $y$ axes. (b), (c) Single-particle energy band structures of the RM and DP methods, respectively, plotted within the moiré Brillouin zone along the path connecting the high symmetry points shown in the inset of (b). In the DP model (c) the bands coming from the valley $\mathbf{K}\left({\mathbf{K}}^{\prime }\right)$ are drawn as solid (dashed) lines.

Figure 2

(a), (b) Spatial profiles of the order parameter for local and RVB interaction schemes, respectively, computed with RM. The DP model for the local interaction yields a similar spatial distribution [51]. In the case of RVB, ${\mathrm{\Delta }}_{i\alpha j\beta }$ are plotted at ${\mathbf{r}}_{i\alpha }$. Red parallelograms represent the moiré unit cell. The maximum order parameter in both cases is $max\left|\mathrm{\Delta }\right|\approx 3.4\mathrm{meV}$. (c), (d) $max\left|\mathrm{\Delta }\right|$ as a function of the interaction strength at $\nu \approx -2$ for the RM and DP methods, respectively. Here $a$ is the graphene lattice constant. (e) Spatial components of $D^s$ as a function of $max\left|\mathrm{\Delta }\right|$ at $\nu \approx -2$ for local and RVB pairing. For local interaction $D_{xx}^s=D_{yy}^s=D^s$. Inset of (e) shows the total density of states (DOS) for RVB (blue curve) and local interaction (red) at $max\left|\mathrm{\Delta }\right|\approx 3.4\mathrm{meV}$ computed with RM. The dashed curve is the DOS for local interaction obtained with DP at $max\left|\mathrm{\Delta }\right|\approx 3.5\mathrm{meV}$. From the DOS we see the nematic phase being gapless, while the $s$-wave state is gapped. The RM results are evaluated at $T\approx 0.1\mathrm{K}$, whereas the DP results at $T=0$.

Figure 3

(a), (b) $T_{\text{BKT}}$ and $k_B{T}_{\text{BKT}}/max\left|\mathrm{\Delta }(T=0)\right|$, respectively, as a function of $max\left|\mathrm{\Delta }\right(T=0\left)\right|$ at $\nu \approx -2$. This result is almost independent of the filling [29]. (c), (d) $T_{\text{BKT}}$ as a function of $\nu$ at $max\left|\mathrm{\Delta }\right|\approx 0.4\mathrm{meV}$ and $max\left|\mathrm{\Delta }\right|\approx 3\mathrm{meV}$ at CNP, respectively.

Figure 4

Various superfluid components as a function of $max\left|\mathrm{\Delta }\right|$ at $\nu \approx -2$ and $T=1.5\mathrm{K}$ for the (a) RVB and (b) local interaction obtained from the RM model. Blue curve is $D^s$ and blue (pink) area depicts $D_{\mathrm{geom}}^s\left(D_{\mathrm{conv}}^s\right)$. Results for $D^s$ and $D_{\mathrm{conv}}^s$ computed by considering only four and eight Bloch bands are also shown, labeled as $D_4^s,D_8^s$, and $D_{4/8,\mathrm{conv}}^s$. We have numerically checked that $D_{\mathrm{conv}}^s\approx D_{4,\mathrm{conv}}^s$.