Correlation effects in two-dimensional topological insulators

We investigate correlation effects in two-dimensional topological insulators (TI). In the first part, we discuss finite size effects for interacting systems of different sizes in a ribbon geometry. For large systems, there are two pairs of well separated massless modes on both edges. For these systems, we analyze the finite size effects using a standard bosonization approach. For small systems, where the edge states are massive Dirac fermions, we use the inhomogeneous dynamical mean-field theory (DMFT) combined with iterative perturbation theory as an impurity solver to study interaction effects. We show that the finite size gap in the edge states is renormalized for weak interactions, which is consistent with a Fermi-liquid picture for small size TIs. In the second part, we investigate phase transitions in finite size TIs at zero temperature focusing on the effects of possible interedge umklapp scattering for the edge states within the inhomogeneous DMFT using the numerical renormalization group. We show that correlation effects are effectively stronger near the edge sites because the coordination number is smaller than in the bulk. Therefore the localization of the edge states around the edge sites, which is a fundamental property in TIs, is weakened for strong coupling strengths. However, we find no signs for “edge Mott insulating states” and the system stays in the topological insulating state, which is adiabatically connected to the noninteracting state for all interaction strengths smaller than the critical value. Increasing the interaction further, a nearly homogeneous Mott insulating state is stabilized.

Figure 1

Ribbon geometry: square lattice with open boundaries in $y$ direction.

Figure 2

The dispersion relation for $M_0=-1.0$. The edge states spectrum is enlarged in the lower panel so that the finite size gap can be seen.

Figure 3

The finite size gap as a function of $M_0$ for $L_y=10,20,40$ from the top to the bottom. For $0<M_0<4$, ${\Delta }_{\mathrm{f}}$ depends on $M_0$ in the same way as for $-4<M_0<0$.

Figure 4

Self-energy for $U=4$ and $L_y=10$. Red, green, and blue curves are ${\Sigma }_{11},{\Sigma }_{12,21}$, and ${\Sigma }_{22}$, respectively.

Figure 5

The local Green's functions for $y=1\sim 5$ from the top to the bottom when $U=4$ and $L_y=10$. The red, green, and blue curves are for $G_{11},G_{12,21}$, and $G_{22}$, respectively.

Figure 6

The energy spectrum $A(k_x,\omega )$ for $U=4$ and $L_y=10$.

Figure 7

(Upper panel) The finite size gap ${\Delta }_{\mathrm{f}}$, the bulk gap ${\Delta }_{\mathrm{TI}}$, and the $y$-averaged renormalization factor $z$ as functions of $U$ for $L_y=10$. (Lower panel) The ratio of the finite size gap ${\delta }_{\mathrm{f}}$ and $z$ as functions of $\mathcal{M}$.

Figure 8

The elementary excitations in two-dimensional TIs for different system sizes when the Coulomb interactions are weak.

Figure 9

A schematic picture of the NRG chain Hamiltonian of the effective impurity Anderson model.

Figure 10

Self-energy for $U=8$ and $L_y=10$. Red, green, and blue curves are ${\Sigma }_{11},{\Sigma }_{12,21}$, and ${\Sigma }_{22}$, respectively.

Figure 11

The renormalization factor $z\left(y\right)$ for several values of $U$ for $L_y=10$.

Figure 12

The local density of states at $y=1$ (red curve) and $y=5$ (blue curve) for several values of $U<U_c$.

Figure 13

The imaginary parts of the local Green's functions for several values of $U$ at $y=1$ (upper panel) and $y=5$ (lower panel) when $L_y=10$. Red, green, and blue curves are $G_{11},G_{12,21}$, and $G_{22}$, respectively.

Figure 14

A schematic picture of the $U$ dependence of ${\Delta }_{\mathrm{TI}}$ and ${\Delta }_{\mathrm{MI}}$. At finite temperatures or if spatial correlations are taken into account, ${\Delta }_{\mathrm{TI}}$ would not close and the transition would be discontinuous.

Figure 15

Truncation and discretization effects for the metal insulator transition in the ordinary one-orbital Hubbard model.[83] We show the renormalization $z$ for $U=1.4W\approx 0.94U_c$. The lines are meant as guide to the eye. The blue line is the reference value calculated for $\Lambda =2$, $N_K=1000$ (independent of the discretization parameter).

Figure 16

Real part of the self-energy for $U=8$ between two discretization parameters in NRG and IPT as impurity solver. The upper panel shows $y$ site 1, and the lower panel $y$ site 5.