Topological properties of the bond-modulated honeycomb lattice

We study the combined effects of lattice deformation, $e\text{−}e$ interaction, and spin-orbit coupling in a two-dimensional (2D) honeycomb lattice. We adopt different kinds of hopping modulation—generalized dimerization and a Kekulé distortion—and calculate topological invariants for the noninteracting system and for the interacting system. We identify the parameter range (Hubbard $U$, hopping modulation, spin-orbit coupling) where the 2D system behaves as a trivial insulator or quantum spin Hall insulator.

Figure 1

Geometry of the two-dimensional (2D) honeycomb lattice with different hopping textures: (a) generalized dimerization with different nearest-neighbor hopping parameters; (b) Kekulé distortion. The unit cells in the two cases, containing two and six atoms, respectively, are also shown. Different values of second-neighbor interactions are indicated.

Figure 2

Upper panel: (a) Phase diagram for Kane-Mele model of the honeycomb lattice assuming the generalized bond dimerization of Fig. 1. The two phases, QSHI and TTI, correspond to different values of the ${\mathbb{Z}}_2$ invariant ($\Delta =1$ and $\Delta =0$, respectively; see text). (b) Gap value as a function of the hopping parameter $t_2$ assuming $t_1=t_3=1,t_{KM}=t_{KM}^{\prime }=t_{KM}^{\prime \prime }$. Continuous line is for $t_{KM}/t_1=0.1$, dotted line for $t_{KM}/t_1=0.3$. Lower panel: Density plots of the occupied energy states as a function of k point with $t_{KM}=0.1$. The evolution of the band structure is considered for $t_{KM}=t_{KM}^{\prime }=t_{KM}^{\prime \prime }$ in a subset of first nearest-neighbor hopping parameters with $t_1=t_3=1$ indicated as blue dots in panel (a): $t_2=1$ (c), $t_2=1.2$ (d), $t_2=2$ (e), $t_2=3\left(f\right)$. In the color scale, red indicates zero gap. Brillouin zones (thin red dotted lines) and high symmetry points are also shown.

Figure 3

(a) Phase diagram for the 2D honeycomb lattice with Kekulé distortion showing the topological behavior as a function of nearest-neighbor modulation $t^{\prime }$ and spin-orbit coupling $t_{KM}$. The gray area corresponds to $t_{KM}^{\prime }=t_{KM}$ while the dashed line is the phase separation for $t_{KM}^{\prime }=t^{\prime }/t\times t_{KM}$. (b) Gap value as a function of the hopping parameter $t^{\prime }$ for different values of $t_{KM}^{\prime }=t_{KM}$ as indicated in the inset. Lower panel: Density plots of the occupied energy states as a function of the k point with $t_{KM}^{\prime }=t_{KM}=0.1$. The evolution of the band structure is considered in a subset of first nearest-neighbor hopping parameters [$t^{\prime }/t=1$ (c), $t^{\prime }/t=1.5$ (d), $t^{\prime }/t=2$ (e)] indicated as blue dots in panel (a). In the color scale, red indicates zero gap. The reduced Brillouin zones for the 6-site unit cell (thin red dotted lines) and high symmetry points are shown.

Figure 4

Phase diagram and spectral functions for the Kane-Mele model of the honeycomb lattice in the presence of on-site $e\text{−}e$ interaction for the generalized bond dimerization (left panel) and for the Kekulé distortion (right panel). Panel (a): Phase diagrams for the generalized bond dimerization obtained with different values of $U$ as a function of $t_2$ and $t_3$ at $t_{KM}=0.1$. Panels (b)–(d): Spectral functions for the dimerized honeycomb lattice with $t_{KM}=0.1,U=3,t_2=t_3=1$, and (b) $t_1=1$, (c) $t_1=1.5$, and (d) $t_1=2$. Panel (e): Phase diagram for the Kekulé distortion at a fixed value $U=3$ as a function of $t^{\prime }$ and $t_{KM}$. The filled area corresponds in both cases to the noninteracting result reported in Figs. 2 and 3. The dashed line indicates the phase separation for $t_{KM}^{\prime }=t^{\prime }/t\times t_{KM}$. In blue and red are reported the results assuming $t_{KM}^{\prime }=t_{KM}$ and $t_{KM}^{\prime }=t^{\prime }/t\times t_{KM}$, respectively. Panels (f)–(h): Spectral functions for the honeycomb lattice in the Kekulé distortion with $U=3,t=1,t_{KM}^{\prime }=t_{KM}$, and (f) $t^{\prime }=1$, (g) $t^{\prime }=1.3$, and (h) $t^{\prime }=2$.