Metal-insulator transition and dominant $d+id$ pairing symmetry in twisted bilayer graphene

Motivated by recent experimental studies that have found signatures of a correlated insulator phase and tuning superconductivity in twisted bilayer graphene, we study the temperature-dependent conductivity, the spin correlation, and the superconducting pairing correlation within a two-orbital Hubbard model on an emergent honeycomb lattice. The evaluation of the temperature dependence of the conductivity demonstrates that there is a metal-insulator transition and the Mott phase at strong coupling is accompanied by antiferromagnetic order. The electronic correlation drives a $d+id$ superconducting pairing to be dominant over a wide filling region. All of the dc conductivity, the spin correlation, and the superconductivity are suppressed as the interlayer coupling strength increases, and the critical $U_c$ for the metal-insulator transition is also reduced. Our intensive numerical results reveal that twisted bilayer graphene should be a uniquely tunable platform for exploring strongly correlated phenomena.

Figure 1

(a) The geometry of TBG for the effective model [18, 19] with $L=4$ and (b) the band structure of the corresponding tight-binding model. Solid lines are $t_1^{\prime }=t_2^{\prime }=0.10$ and dotted lines are $t_1^{\prime }=t_2^{\prime }=0.05$. The insert shows the definition of hoppings.

Figure 2

(a) The AFM spin structure factor $S_{\mathrm{AFM}}$ depends on $\beta =1/T$ with different interaction strength and lattice size; (b) the scaling behavior of the normalized AFM spin structure factor $S_{\mathrm{AFM}}/N_s$ for different values of $U$ at $\beta =12$. Solid lines are fit on the third-order polynomial in $1/\sqrt{N_s}$. (c)The AFM spin structure factor $S_{\mathrm{AFM}}$ depends on $\beta =1/T$ at $L=4$ and $U=4$ for different $t_1^{\prime }=t_2^{\prime }$. The average sign is shown in (d) for the simulation shown in (a).

Figure 3

The dc conductivity ${\sigma }_{\mathrm{dc}}$ vs temperature $T$ computed at various interaction strengths for (a) $L=4$ and (b) 5. The conductivity ${\sigma }_{\mathrm{dc}}$ vs the interaction $U$ for three different temperatures at $L=3$, 4, and 5 are shown in (c). The extracted $U_c$ for Mott-insulator transition with different lattices are shown in (d).

Figure 4

The dc conductivity ${\sigma }_{\mathrm{dc}}$ vs temperature $T$ computed at (a) various interaction strengths for $t_1^{\prime }=t_2^{\prime }=0.05$ and (b) at $U=3.6$ for different $t_1^{\prime }=t_2^{\prime }$.

Figure 5

(a) The effective pairing interaction $P_{\alpha }$ with different pairing symmetry as a function of electronic fillings for $t_1^{\prime }=t_2^{\prime }=0.10$ and (b) the effective pairing interaction $P_{d+idNN}$ as a function of temperature for different $t_1^{\prime }=t_2^{\prime }$. The effective pairing interaction $P_{d+idNN}$ as a function of temperature for $L=3,$ 4, and 5 are shown for $t_1^{\prime }=t_2^{\prime }=0.10$ (c) and 0.08.

Figure 6

Pairing correlation $C_{\alpha }$ as a function of distance $r$ (a) for different inter-orbital Coulomb of $d+id$-wave with lattice of $L=4$ at $〈n〉\approx 0.896$ and (b) with lattices of $L=4$, 5, and 6 at $〈n〉\approx 0.979$. The inset of (b) shows the scaling analysis on the average of the long-range $d+id$ pairing correlation. Solid lines are fit on the third-order polynomial in $1/\sqrt{N_s}$.