Topological insulator on honeycomb lattices and ribbons without inversion symmetry

We study the Kane-Mele-Hubbard model with an additional inversion-symmetry-breaking term. Using the topological Hamiltonian approach, we calculate the ${\mathbb{Z}}_2$ invariant of the system as function of spin-orbit coupling, Hubbard interaction $U$, and inversion-symmetry-breaking onsite potential. The phase diagram calculated in that way shows that, on the one hand, a large term of the latter kind destroys the topological nontrivial state. On the other hand, however, this inversion-symmetry-breaking field can enhance the topological state since for moderate values the transition from the nontrivial topological to the trivial Mott insulator is pushed to larger values of interaction $U$. This feature of an enhanced topological state is also found on honeycomb ribbons. With inversion symmetry, the edge of the zigzag ribbon is magnetic for any value of $U$. This magnetic moment destroys the gapless edge mode. Lifting inversion symmetry allows for a finite region in interaction strength $U$ below which gapless edge modes exist.

Figure 1

Examples of WCCs according to the noninteracting KM model (${\lambda }_{S}O=0.5$) with nontrivial (top, ${\lambda }_{\nu }=2.5{\lambda }_{S}O$) and trivial (bottom, ${\lambda }_{\nu }=5.5{\lambda }_{S}O$) topology. Here, two unit cells are shown along the abscissa so that at least one WCC is continuously displayed. Thus, four instead of two WCCs are visible. $S_z$ is conserved, so the WCCs can be separated in spin up (blue) and spin down (red).

Figure 2

The left plot shows the full system, the right the reference system. Full symbols denote sublattice $A$, open symbols sublattice $B$. The bath sites (squares) are characterized only by an onsite energy. The impurity sites (circles), on which the Hubbard $U$ is acting, can additionally carry the symmetry-breaking Weiss fields.

Figure 3

Unit cell of the zigzag ribbon and the according reference system. The respective two-site clusters are identical, except for a different AF Weiss field.

Figure 4

Self-energy functional as a function of the antiferromagnetic Weiss fields $h_z$ and $h_x$ for ${\lambda }_{S}O=0.1,{\lambda }_{\nu }=0$, and $U=5$. The hybridization of the bath sites has been optimized for each grid point individually. The global minimum around $h_x\approx 1$ and $h_z=0$ can clearly be seen.

Figure 5

Phase diagram of the KMH model obtained from two-site DIA, as a function of the Hubbard interaction and the sublattice potential for a spin-orbit coupling of ${\lambda }_{S}O=0.1$.

Figure 6

Wannier charge centers of the topological Hamiltonian for ${\lambda }_{S}O=0.1,{\lambda }_{\nu }=0.25$, and $U=3$ (top) and $U=4$ (bottom).

Figure 7

Antiferromagnetic Weiss field of the first pair of sites $A$ and $B$ of the ribbon (solid line) and in the middle of the ribbon (dashed line) as a function of $U$ for ${\lambda }_{S}O=0.1,{\lambda }_{\nu }=0$, and $N=16$ pairs of sites. The inset shows how the moment at the edge decays across the ribbon to the midpoint for $U=2$.

Figure 8

Spectral functions of the KMH zigzag ribbon with parameters $U=2.5,{\lambda }_{S}O=0.1$, and $N=16$. Top panel: ${\lambda }_{\nu }=0.1$ leads to a magnetic solution with a Weiss field of about $h_{AFx}=0.4$, gapping the edge states. Bottom panel: ${\lambda }_{\nu }=0.2$, with a vanishing Weiss field and gapless edge states.

Figure 9

Chern insulator phase of the KMH model in mean-field approximation with antiferromagnetic moment in the $z$ direction for $U=3.2,{\lambda }_{S}O=0.1,{\lambda }_{\nu }=0.4$. The upper graph shows Wannier charge centers from the bulk calculations. The blue curve is the WCC of the spin-up band, the red curve the WCC of the spin-down band, resulting in $C_{\uparrow }=0$ and $C_{\downarrow }=-1$. The lower graph shows the bands of a ribbon ($N=32$) with one spin-down but no spin-up edge state.