Spin-orbit coupling in the kagome lattice with flux and time-reversal symmetry

We study the topological properties of a spin-orbit coupled tight-binding model with flux on the kagome lattice. The model is time-reversal invariant and realizes a ${\mathbb{Z}}_2$ topological insulator as a result of artificial gauge fields. We develop topological arguments to describe this system showing three inequivalent sites in a unit cell and a flat band in its energy spectrum in addition to the topological dispersive energy bands. We show the stability of the topological phase towards spin-flip processes and different types of on-site potentials. In particular, we also address the situation where on-site energies may differ inside a unit cell. Moreover, a staggered potential on the lattice may realize topological phases for the half-filled situation. Another interesting result is the occurrence of a topological phase for large on-site energies. To describe topological properties of the system we use a numerical approach based on the twisted boundary conditions and we develop a mathematical approach, related to smooth fields.

Figure 1

The schematic representation of the kagome lattice. The lattice contains three sites per unit cell, which we depict by red ($R$), blue ($B$), and green ($G$). ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ are the displacement vectors between the neighboring unit cells which form the triangular lattice. ${\mathbf{b}}_1, {\mathbf{b}}_2$, and ${\mathbf{b}}_3$ are the displacement vectors within the unit cell between $R$ and $B, R$ and $G$, and $B$ and $G$ sites, respectively.

Figure 2

Definition of the Brillouin zone into two domains (see Sec. 3b), each associated with a different gauge choice for the eigenvectors.

Figure 3

The spectra $E_{k_1}$ for vanishing on-site energies (${\lambda }_R={\lambda }_B={\lambda }_G=0$) and for different values of $\gamma$ and $\varphi$. Blue lines correspond to the edge states obtained by calculations where we consider periodic boundary conditions in the ${\mathbf{e}}_1$ direction and open boundary conditions in the ${\mathbf{e}}_2$ direction. Triangles right (left) indicate that the corresponding edge state is localized at the $x_2=N_2$ ($x_2=1$) edge of the system. Brown (gold) triangles indicate a state which dominantly contains down (up) spin fermions.

Figure 4

The phase diagram for $n=2/3$ filling and for ${\lambda }_R={\lambda }_B={\lambda }_G=0$. The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. White region in the lower panel corresponds to the parameter set when the gap is closed.

Figure 5

The edge states for $\gamma =0$ and $\varphi =\pi /2$ (a)–(d) and for $\gamma =0.05$ and $\varphi =\pi /2$ (e)–(h). Triangles pointing up correspond to spin up fermions and triangles pointing down correspond to spin down fermions. Red (solid), blue (dashed), and green (dashed-dotted) lines correspond to $R, B$, and $G$ sites, respectively. $l$ corresponds to the energy level. The number of unit cells along ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ directions are $N_1=500$ and $N_2=100$, respectively. Other parameters are ${\lambda }_R={\lambda }_B={\lambda }_G=0$.

Figure 6

Spin Hall conductivity ${\sigma }^H\left(\mu \right)$ as a function of chemical potential $\mu$ for $\gamma =0$ and $\gamma =0.05$. The shaded region corresponds to the values of $\mu$ where the chemical potential is inside the gap (blue for $\gamma =0$ and red for $\gamma =0.05$). Other parameters are $\varphi =\pi /2$ and ${\lambda }_R={\lambda }_B={\lambda }_G=0$.

Figure 7

The phase diagram for $n=2/3$ filling and $\varphi =\pi /2$. ${\lambda }_B=-{\lambda }_R=\lambda$ and ${\lambda }_G=0$. The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. White region in the lower panel corresponds to the parameter set when the gap is closed.

Figure 8

The phase diagram for $n=2/3$ filling and $\varphi =\pi /2$. ${\lambda }_R=\lambda$ and ${\lambda }_B={\lambda }_G=0$. The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. White region in the lower panel corresponds to the parameter set when the gap is closed.

Figure 9

Behavior of the gap $\mathrm{\Delta }$ as a function of $\gamma$ for filling $n=2/3$ and flux $\varphi =\pi /2$, with ${\lambda }_R=\lambda ={10}^{3}t$ and ${\lambda }_B={\lambda }_G=0$. The red curve with circles corresponds to our numerical calculations with the original Hamiltonian [see Eq. (2)]. Crosses correspond to numerical calculations obtained from the effective Hamiltonian Eq. (79) discussed in Sec. 4e and in Appendix pp3. Finally, the dashed curve is a fit function [see Eq. (63)].

Figure 10

The phase diagram for $n=2/3$ filling and $\varphi =\pi /2$. ${\lambda }_B=\lambda$ and ${\lambda }_R={\lambda }_G=0$. The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. White region in the lower panel corresponds to the parameter set when the gap is closed.

Figure 11

The critical value ${\gamma }_c$ as a function of $\lambda$ for $n=2/3$ filling and $\varphi =\pi /2$. ${\lambda }_B=\lambda$ and ${\lambda }_R={\lambda }_G=0$. The inset shows $2\pi \lambda {\gamma }_c$ as a function of $\lambda$.

Figure 12

The schematic representation of the lattice corresponding to the effective Hamiltonian in Eq. (79). The lattice contains two sites per unit cell, which we depict by aqua ($a$) and brown ($b$). Unit cells are arranged on a triangular lattice. Thick lines correspond to the hoppings existing also in the original Hamiltonian in Eq. (2), while thin lines correspond to the hoppings which arise in the effective Hamiltonian.

Figure 13

The phase diagram for filling $n=2/3, {\lambda }_{\alpha ,1}=-{\lambda }_{\alpha ,2}=\lambda$, and $\gamma =0$ and $\varphi =\pi /2$. (a) The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. (b) Topological phase for large values of $\lambda$ between red and blue curves. Red and blue points are obtained by our numerical calculations, while blue and red curves are obtained by fitting [see Eq. (89)]. Fitting parameters are $a^u=0.4335\pm 0.0001, {\lambda }^u/t=1.12\pm 0.09, a^l=0.4325\pm 0.0001$, and ${\lambda }^l/t=-0.79\pm 0.08$.

Figure 14

The phase diagram for half-filling ($n=1$), $\varphi =\pi /2$, and ${\lambda }_{\alpha ,1}=-{\lambda }_{\alpha ,2}=\lambda$. The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. White region in the lower panel corresponds to the parameter set when the gap is closed.

Figure 15

The phase diagram for filling $n=2/3, \varphi =\pi /2$, and ${\lambda }_{R,1}=-{\lambda }_{R,2}=\lambda$ and ${\lambda }_{B,s=1,2}={\lambda }_{G,s=1,2}=0$. The upper panel shows the size of the gap $\mathrm{\Delta }$, while the lower panel shows the ${\mathbb{Z}}_2$ number $\nu$. White region in the lower panel corresponds to the parameter set when the gap is closed.

Figure 16

Density of states for the spin up part of the Hamiltonian with $\gamma =0, 1/f=5$, and at $-{\lambda }_R={\lambda }_B=\lambda , {\lambda }_G=0$ (upper panel) and ${\lambda }_R=\lambda , {\lambda }_B={\lambda }_G=0$ (lower panel).