Particle-hole symmetry and interaction effects in the Kane-Mele-Hubbard model

We prove that the Kane-Mele-Hubbard model with purely imaginary next-nearest-neighbor hoppings has a particle-hole symmetry at half filling. Such a symmetry has interesting consequences including the absence of charge and spin currents along open edges, and the absence of the sign problem in the determinant quantum Monte Carlo simulations. Consequentially, the interplay between band topology and strong correlations can be studied at high numeric precisions. The process that the topological band insulator evolves into the antiferromagnetic Mott insulator as interaction strength increases is studied by calculating both the bulk and edge electronic properties.

Figure 1

The comparison between the antiferromagnetic structure factors $S_{\mathit{AF}}^{zz}$ along the $z$ axis and $S_{\mathit{AF}}^{xx}$ in the $x,y$ plane at $\lambda =0.1$ for the size of $N=2\times L\times L$ with $L=6$. The easy-plane feature is clear.

Figure 2

The finite-size scaling of the $xx$ antiferromagnetic structure factors calculated at $\lambda =0.1$ for the sizes of $N=2\times L\times L$ ($L=3,6,9$, and 12), and the different values of $U$ indicated in the inset. Finite values of $S_{\mathit{AF}}^{xx}/N$ in the thermodynamic limit appear at $U⩾U_c$ with $U_c\approx 4.9$.

Figure 3

The QMC simulation of the phase diagram of the KMH model. The antiferromagnetically long-range-ordered phase appears at strong correlation regime. The paramagnetic phase is divided into two regimes: topological band insulator (TBI) with stable helical edges, and bulk paramagnetic phase with unstable edges (see further discussions in Sec. 4c). The two critical values of $U$ at $\lambda =0$ are from Ref. 37 by Meng et al., which are also confirmed in our QMC simulations.

Figure 4

The KMH model in the lattice with the zigzag edges. The boundary conditions are periodical and open along the $x$ and $y$ directions, respectively. The NNN bonds between two closest tips on the zigzag edges are removed.

Figure 5

The extrapolation of local single particle gap for the tip sites on the zigzag edges of a ribbon geometry with the size of $2\times L\times n_y$ with $n_y=8$. In the inset, the logarithms of on-site time-displaced Green's functions $\ln G(i,i;\tau )$ of the tip sites is depicted for $U=2$. The slopes of the long time tails measure the edge excitation gap ${\Delta }_{\mathrm{edge}}$. Here $\lambda$ is set to be $0.1$ in this calculation. We want to emphasize that the finite gap in the bulk paramagnetic phase is a limitation of system size, as the edge should be gapless due to $U\left(1\right)$ symmetry.

Figure 6

The row $xx$-AFSF defined in Eq. (16) for each zigzag rows parallel to the boundary. The parameters values are $\lambda =0.1$; the sample size $2\times L\times L$ with $L=8$; and the different values of $U$ are indicated in the inset. The row indices 1 and 8 correspond to the boundary rows, and those of 4 and 5 corresponds to the central rows.

Figure 7

The finite-size scaling of the $xx$-AFSF defined in Eq. (16) for the edge row with $\lambda =0.1$. The size of this ribbon is $2\times L\times 4$. We emphasize that due to the 1D nature of the edge and the $U\left(1\right)$ spin symmetry, this scaling actually shows the power-law correlation rather than the true long-range order. The finite intercepts are mainly due to small-size effects.

Figure 8

The two-point equal-time spin correlation functions along the zigzag edge with $\lambda =0.1$ at values of $U$ denoted in the insets. The size of the ribbon is $2\times 34\times 4$. Because the zigzag edge contains the sites of both A and B type, three different types of correlations are plotted in (a), (b), and (c), respectively. The Luttinger parameters are fitted from the correlation among A sites on the tips as $K\approx 0.8,0.5$, and $0.4$ for $U=1,1.5$, and 2, respectively.

Figure 9

The definition of the positive direction for the NNN bonds on the honeycomb lattice, based on which the NNN current form factors $Q_C^{\mathit{AF}}$ and $Q_S^{\mathit{AF}}$ are defined in Eq. (19).

Figure 10

The finite-size scaling of the form factors of the NNN currents in the charge sector $Q_c^{\mathit{AF}}$ and $Q_S^{\mathit{AF}}$ in the spin sector as defined in Eq. (19). The periodical boundary condition is employed. The sample size is $N=2\times L\times L$ with $L=3,6,9$, and 12. $\lambda$ is set 0 in this calculation, and $U$ $=$ 3, 4, and 5.