Pairing symmetry of interacting fermions on a twisted bilayer graphene superlattice

The pairing symmetry of the Hubbard Hamiltonian on a triangle lattice with a nearly flat low-energy band is studied with the determinant quantum Monte Carlo method (DQMC). We show that the low-temperature phase is insulating at half-filling, even for relatively weak interactions. The natures of the spin and pairing correlations upon doping are determined, and they exhibit an electron-hole asymmetry. Among the pairing symmetries allowed, we demonstrate that the dominating channels are $d$ wave, opening the possibility of condensation into an unconventional $d_{x^2-y^2}+i{d}_{xy}$ phase, which is characterized by an integer topological invariant and gapless edge states. The results are closely related to the correlated insulating phase and unconventional superconductivity discovered recently in twisted bilayer graphene.

Figure 1

(a) The effective lattice which results from treating each AA region as an effective “site” connected by modulated hopping amplitudes. Three elementary vectors ${\mathbf{e}}_{1,2,3}$, the $A,B,C$ sublattices are indicated. (b) The honeycomb lattice in the limit $t^{\prime }=0$. (c) The dice lattice in the limit $t=0$. (d) The band structure of the effective model in the first Brillouin zone. (e) The density of states corresponding to the band shown in panel (d). In panel (e), we also show the density of states of the limiting cases of the triangular ($t^{\prime }=t$) lattice and hexagonal ($t^{\prime }=0$) lattice. In the latter case, we do not show the $\delta$ function peak at $E/t=0$. In panels (d) and (e), the anisotropic factor $t^{\prime }/t=0.3$.

Figure 2

The pairing symmetries considered in this paper. A partner down-spin fermion is created on the six nearest-neighbor sites of the up-spin fermion placed at the hexagon center. For triplet pairing, there is an additional sign when the pairing is along the opposite direction of the arrow.

Figure 3

The quasiparticle spectrum on zigzag edge ribbon: (a) the $d+id$ chiral superconducting state and (b) the triplet $p+ip$ state. The parameters are $t^{\prime }/t=0.3,\mathrm{\Delta }=0.3,{\mathrm{\Delta }}^{\prime }=0.1$, and $\mu =0.22$ (corresponding to $\rho =0.94$).

Figure 4

The (a) density and (b) average sign as functions of $\mu$ for linear lattice sizes $L=6,8,10$. Here $U=2,T=1/12$, and $t^{\prime }/t=0.3$. From panel (a), we see a clear indication of the formation of a insulating gap at half-filling. Panel (b) indicates that accessible temperatures will be limited to $T\gtrsim t/12$ at $\mu \sim -0.5$.

Figure 5

The near-neighbor spin correlations $m_{t^{\prime }}$ and $m_t$ along the $t^{\prime }$ and $t$ bonds, respectively, (a) as a function of $t^{\prime }/t$ for fixed $U=2$ and (b) as a function of $U$ for fixed $t^{\prime }/t=0.3$. The temperature $T=0.2t$. Results for densities on either side of half-filling are shown.

Figure 6

The effective pairing susceptibility at $\rho =0.94$ as a function of temperature for different pairing channels. Here $U=2$ and the lowest temperature accessed by DQMC is $T=1/13$. Here ${\mathrm{\Delta }}^{\prime }/\mathrm{\Delta }=0.3$ and the results are similar for other ratios.

Figure 7

The evolution of the band structure as a function of the anisotropic ratio $t^{\prime }/t$.

Figure 8

The momentum dependence of ${\mathrm{\Delta }}_{\mathbf{k}}^t$ for $s^{*}, d_{x^2-y^2}, d_{xy}, f, p_x$, and $p_y$ pairing channels.

Figure 9

The effective susceptibility at different ${\mathrm{\Delta }}^{\prime }/\mathrm{\Delta }$ for $s^{*}$- and $d$-wave pairing channels. The filled (open) symbols represent $d$-($s^{*}$-) wave pairing phases. The $f$- and $p$-wave phases have smaller ${\chi }_{eff}$ and are not shown here. The filling is $\rho =0.94$.