Effects of short-ranged interactions on the Kane-Mele model without discrete particle-hole symmetry

We study the effects of short-ranged interactions on the $Z_2$ topological insulator phase, also known as the quantum spin Hall phase, in the Kane-Mele model at half-filling with staggered potentials, which explicitly breaks the discrete particle-hole symmetry. Within Hartree-Fock mean-field analysis, we conclude that the on-site repulsive interactions help stabilize the topological phase (quantum spin Hall) against the staggered potentials by enlarging the regime of the topological phase along the axis of the ratio of the staggered potential strength and the spin-orbit coupling. In sharp contrast, the on-site attractive interactions destabilize the topological phase. We also examine the attractive interaction case by means of the unbiased determinant projector quantum Monte Carlo and the results are qualitatively consistent with the Hartree-Fock picture.

Figure 1

Illustration of the honeycomb lattice. There are two sublattices per unit cell labeled as $A$ and $B$. ${\vec{e}}_{1/2}=(\pm 1/2,\sqrt{3}/2)$ are the two vectors connecting the same sublattices in different unit cells. We set the lattice constant to be 1.

Figure 2

The phase boundary between the topologically nontrivial phase (QSH) and the trivial phase within the Hartree-Fock picture. (a) The numerical data in the repulsive interaction case, during the numerics, we set $t=1$, ${\lambda }_{so}=0.4$ and $U>0$. The Hubbard repulsion tends to screen the stagger potentials to push the boundary of the nontrivial phase toward the trivial side, which widens the regime of the nontrivial phase. (b) The numerical date in the attractive interaction case, $U=-\left|U\right|$. Contrary to the result in (a), the attractive interactions enhances the staggered potential and shrink the regime of the topological phase.

Figure 3

The spin Chern number $C_{\sigma }$ vs the magnitudes of staggered potentials ${\lambda }_m/t$ using QMC at ${\lambda }_{SO}=0.4t$. The dot line indicates the noninteracting phase boundary ${\lambda }_m^c=3\sqrt{3}{\lambda }_{SO}=2.0784t$. The solid symbols denote the $6\times 6$ QMC results at $U=0,-2t$ and $-4t$. The hollow symbols denote the $12\times 12$ QMC at $U=-4t$.

Figure 4

Illustration of the phase boundary between trivial and nontrivial phase in the present of both U and V repulsion. For illustration, we take $t=1$, ${\lambda }_{so}=0.4$, and choose $V=U/2$ and $V=U/10$. The blue open dot squares represent the boundary in the case of $V=U/10$ and the open red diamonds line represents the boundary in the case of $V=U/10$. The black line represents the boundary in the present of on-site Hubbard $U$. (a) The repulsive interaction case, $U,V>0$. We can see that in this case. If the nearest-neighbor $V$ is much smaller than $U$, $V=U/10$, the interactions still tend to stabilize the topological phase. But due to the competition between $U$ and $V$ the topological widow is less widened by the interactions. If $V$ is more comparable to $U$,$V=U/2$ in this case, the effects of $V$ will be dominant over $U$ and will tend to destabilize the topological phase. According to the gap function defined around momentum $\mathbf{K}$ point, Eq. (B4), the transition point is at $U=6V$. (b) The attractive interaction case, $U,V<0$. The results in this case are qualitatively opposite to those in the repulsive case. When $V$ is more comparable to the $U$, the interactions tend to stabilize the topological phase.