Stabilization of topological insulator emerging from electron correlations on honeycomb lattice and its possible relevance in twisted bilayer graphene

Realization and design of topological insulators emerging from electron correlations, called topological Mott insulators (TMIs), is pursued by using mean-field approximations as well as multi-variable variational Monte Carlo (MVMC) methods for Dirac electrons on honeycomb lattices. The topological insulator phases predicted in the previous studies by the mean-field approximation for an extended Hubbard model on the honeycomb lattice turn out to disappear, when we consider the possibility of a long-period charge-density-wave (CDW) order taking over the TMI phase. Nevertheless, we further show that the TMI phase is still stabilized when we are able to tune the Fermi velocity of the Dirac point of the electron band. Beyond the limitation of the mean-field calculation, we apply the newly developed MVMC to make accurate predictions after including the many-body and quantum fluctuations. By taking the extrapolation to the thermodynamic and weak external field limit, we present realistic criteria for the emergence of the topological insulator caused by the electron correlations. By suppressing the Fermi velocity to a tenth of that of the original honeycomb lattice, the topological insulator emerges in an extended region as a spontaneous symmetry breaking surviving competitions with other orders. We discuss experimental ways to realize it in a bilayer graphene system.

Figure 1

Schematic picture of honeycomb lattice. Here, the on-site Coulomb interaction is denoted by $U$. Next-nearest-neighbor interaction $V_2$ and the loop current $\zeta$ flowing between the next-nearest neighbor are shown by red and blue arrows.

Figure 2

(a) Schematic picture of CDW state on honeycomb lattice, which is stabilized when $V_2$ is large. Sites labeled by 1 (Red), 2 (light red), 3 (light blue), and 4 (blue) correspond to the sites where the electron densities are very rich, rich, poor, and very poor, respectively. (b) The growth of order parameter of the CDW state. (c) Resulting phase diagram of honeycomb lattice in the parameter space of $U$ and $V_2$ at the level of mean-field approximation. We do not find the region of stable TMI phase.

Figure 3

(a) Nearest, next-nearest, and third-neighbor hoppings. (b) Relation between $t_3$ and Fermi velocity $v_f$ at Dirac point. Here, $v_f$ is shown by the ratio to the Fermi velocity at $t_3=0$.

Figure 4

Phase diagram calculated by mean-field approximation for $t_3=$ (a) 0.3, (b) 0.4, and (c) 0.45 in the parameter space of $U$ and $V_2$. SM denotes the semimetal.

Figure 5

Results of $\zeta$ calculated by the MVMC method. (a) MVMC results for several different values of $\lambda$ for $t_3=0.45t_1$ at $L=5$. (b) Size extrapolation at $\lambda =0.0002t_1$ for several different values of $V_2$. Here, four different sizes $\left(L=4,5,6,\text{and}7\right)$ are calculated. Lines are results of quadratic fittings. (c) Extrapolation to vanishing $\lambda$ is shown for different values of $V_2$. Lines are results of quadratic fitting.

Figure 6

MVMC results for $\zeta$. (a) Result of extrapolations for two different values of $t_3$ are shown. (b) Size extrapolation at $t_3=0.3t_1$ and $\lambda =0.0002t_1$ for several different values of $V_2$. Here, four different sizes $\left(L=4,5,6,\text{and}7\right)$ are calculated. Lines are results of quadratic fittings.

Figure 7

(a) Phase diagram of semimetal and TMI with respect to $V_2$ and $t_3$ at $U=0$ obtained by MVMC calculations. The crosses are points where the calculations were done. (b) Size dependence of mean-field calculation for several different values of $V_2$ at $t_3=0.45t_1$. (c) $V_2$ dependence of order parameter of TMI for $U=0.0t_1$ and $U=1.0t_1$ at $t_3=0.45t_1$. When $U$ is switched on, $\zeta$ decreases.