Reduction of $\mathbb{Z}$ classification of a two-dimensional weak topological insulator: Real-space dynamical mean-field theory study

One of the remarkable interaction effects on topological insulators is the reduction of topological classification in free-fermion systems. We address this issue in a bilayer honeycomb lattice model by taking into account temperature effects on the reduction. Our analysis, based on the real-space dynamical mean-field theory, elucidates the following results. (i) Even when the reduction occurs, the winding number defined by the Green's function can take a nontrivial value at zero temperature. (ii) The winding number taking the nontrivial value becomes consistent with the absence of gapless edge modes due to Mott behaviors emerging only at the edges. (iii) Temperature effects can restore the gapless edge modes, provided that the energy scale of interactions is smaller than the bulk gap. In addition, we observe the topological edge Mott behavior only in some finite-temperature region.

Figure 1

Sketch of the model. We consider the system of an A-A stacked honeycomb lattice under the open (periodic) boundary condition for the $x$ ($y$) direction. The parameter $t$ ($rt$) denotes hopping of electron between sites connected with a brown (gray) line.

Figure 2

Phase diagrams for interaction $U$ vs temperature $T$ with $U=J$. (a) The dashed line denotes the region where hysteresis behaviors are observed in the bulk and the edge. In the colored region, the Mott behavior is observed only around the edges. (b) The colors denote the spectral weight of single-electron excitations at zero energy along the edge, $i_x=0$.

Figure 3

(a) and (b) Interaction dependence of double occupancy and spin correlations, respectively. (c) Winding number as a function of interaction strength. The winding number is obtained from the single-particle Green's function at $i_x=20$ for $T=0.05t$. (d) The imaginary part of the self-energy at each site for $U=J=t$.

Figure 4

(a) and (b) LDOS for the edge ($i_x=0$) and the bulk ($i_x=20$). (a) [(b)] shows the data for $U=J=0$ ($U=J=0.25t$). (c) The LDOS at the edge for $U=J=0.25t$ and several values of temperature.

Figure 5

(a) and (b) Color plots of the spectral weight of LDOS and $\langle S_{a{i}_x}^{+}{S}_{b{i}_x}^{-}\rangle$, respectively, at $\omega =0$ and $i_x=0$. The solid lines denote the crossover temperature where spin excitations become gapless. The dashed lines denote the crossover temperature where single-particle excitations become gapless. In the region sandwiched by green lines, the topological edge Mott behaviors are observed. (c) and (d) LDOS at $i_x=0$ for several values of temperature. (e) and (f) Spectral weight of spin excitations $\langle S_{a{i}_x=0}^{+}{S}_{b{i}_x=0}^{-}\rangle$ at $i_x=0$ for several values of temperature. In (c) and (e) [(d) and (f)], the data for $(U,J)=(t,0.05t)\left[\right(t,0.5t\left)\right]$ are plotted.