Correlation effects on a topological insulator at finite temperatures

We analyze the effects of the local Coulomb interaction on a topological band insulator (TBI) by applying the dynamical mean-field theory to a generalized Bernevig-Hughes-Zhang model having electron correlations. It is elucidated how the correlation effects modify electronic properties in the TBI phase at finite temperatures. In particular, the band inversion character of the TBI inevitably leads to the large reduction of the spectral gap via the renormalization effect, which results in the strong temperature dependence of the spin Hall conductivity. We clarify that a quantum phase transition from the TBI to a trivial Mott insulator, if it is nonmagnetic, is of first order with a hysteresis. This is confirmed via the interaction dependence of the double occupancy and the spectral function. A magnetic instability is also addressed. All these results imply that the spectral gap does not close at the transition.

Figure 1

(a) Sketch of the BHZ model on a square lattice. The hopping integral between orbitals depends on $\theta$ which defines the angle between hopping direction and $x$ axis. (b) Sketch of the DMFT. The lattice model is mapped to an effective impurity model.

Figure 2

The spin Hall conductivity as a function of the interaction strength. Solid circles, open triangles, and open circles denote the conductivity at $T=0.04$, $0.05$, and $0.08$, respectively. Note that the data plotted in this figure are those calculated with increasing interaction from the small $U$ region. The data with decreasing $U$ give a different transition point, giving rise to a hysteresis; an example of the hysteresis behavior is shown in the upper inset for the $T/t=0.05$ case. For the detailed discussion on the hysteresis, see Sec. 3b. Lower inset: Spin Hall conductivity as a function of temperature. Open triangles show the data at $U=8$. The solid red line shows the data calculated with the renormalized parameters which are obtained by low-energy expansion at $U=8$.

Figure 3

Upper panel: Local density of states for each value of $U$. Panels (a), (b), and (c) correspond to the TBI phase, while panels (d), (e), and (f) correspond to the MI phase, respectively. Note that in $11.5<U/t<12.125$, a hysteresis is found. Lower panel: Renormalized energy splitting between the two orbitals.

Figure 4

Double occupancy of each orbital as a function of interaction: (a) $T/t=0.05$, (b) $T/t=0.12$. Note that at $T/t=0.12$, no hysteresis is found.

Figure 5

(a) Sketch of the spin configuration in the AF phase. Electron spins are denoted by the blue arrows. The numbers 1, 2 specify the orbital indices. (b) Magnetic moment as a function of the interaction at $T/t=0.05$.

Figure 6

The temperature vs interaction phase diagram. The dotted blue line denotes the second-order transition between the TBI phase (including the high-temperature region connected via crossover) and the AF phase. If the AF phase is suppressed, a first-order Mott transition can be observed (see text). Solid red lines denote the Mott transition line, where the coexistence phase accompanied by the hysteresis is found in the region surrounded by red lines.