Cellular dynamical mean-field theory study of an interacting topological honeycomb lattice model at finite temperature

We use cellular dynamical mean-field theory with the continuous-time quantum Monte Carlo solver to study the Kane-Mele-Hubbard model supplemented with an additional third-neighbor hopping term. For weak interactions, the third-neighbor hopping term drives a topological phase transition between a topological insulator and a trivial insulator, consistent with previous fermion sign-free quantum Monte Carlo results [Hung et al., Phys. Rev. B 89, 235104 (2014)]. At finite temperatures, the Dirac cones of the zero-temperature topological phase boundary give rise to a metallic regime of finite width in the third-neighbor hopping. Furthermore, we extend the range of interactions into the strong-coupling regime and find an easy-plane antiferromagnetic insulating state across a wide range of third-neighbor hopping at zero temperature. In contrast to the weak-coupling regime, no topological phase transition occurs at strong coupling, and the ground state is a trivial antiferromagnetic insulating state. A comprehensive finite-temperature phase diagram in the interaction-third-neighbor hopping plane is provided.

Figure 1

(a) The lattice structure of a honeycomb lattice with various hopping parameters described in Eq. (1). (b) The bulk density of states for various values of third-neighbor hopping $t_{3n}$ for $\lambda =0.4,U=0.0$. (c)–(e) The noninteracting energy bands of Eq. (1) with the zigzag strip geometry for (c) $t_{3n}=0.0$, (d) $t_{3n}=1/3$, and (e) $t_{3n}=0.6$. The width is $N_s=120$, and the spin-orbit coupling is $\lambda =0.4$. A single Dirac cone is present in (c), indicating a topological state, whereas two are present in (e), a trivial state. The bulk band gap is closed in (d) as in (b) for $t_{3n}=1/3$.

Figure 2

(a) The bulk density of states for different $t_{3n}$'s when $\lambda =0.4,T=0.05,U=1.0$. (b) The evolution of the single-particle gap $\Delta E$ (left vertical axis) and spin Chern number $C_{\sigma }$ (right vertical axis) as a function of $t_{3n}$ when $\lambda =0.4,T=0.05,U=1.0$. (c) and (d) The momentum-dependent spectral function $A(k,\omega )$ in a strip geometry with $N_s=80$ for (c) $t_{3n}=0.0$ and (d) $t_{3n}=0.6$ when $\lambda =0.4,T=0.05,U=1.0$. By comparison with Fig. 1, it is clear that (c) is a TBI and (d) is a TTI.

Figure 3

The evolution of (a) bulk single-particle gap $\Delta E$, (b) staggered magnetic moment $m_z$, and transverse magnetism $m_x$ as a function of interaction $U$ when $\lambda =0.4,T=0.05$ for different $t_{3n}$'s. The dashed lines show the critical interaction $U_c$, which depends on $t_{3n}$. (c) The evolution of double occupancy as a function of $U$; the arrows show the critical points for different $t_{3n}$'s when $\lambda =0.4,T=0.05$. (d) and (e) The momentum-dependent spectral function $A(k,\omega )$ obtained from the zigzag boundary condition for (d) $t_{3n}=0.1$ and (e) $t_{3n}=0.6$ when $\lambda =0.4,U=6.0,T=0.05$. Note that $U=6.0>U_c$ and the spectrum is fully gapped.

Figure 4

The $U\text{−}{t}_{3n}$ phase diagram for $T=0.05.\lambda =0.4$ is considered. TBI: topological band insulator; TTI: topological trivial insulator; AFI: antiferromagnetic insulator; $M$: metal. The inset shows the temperature dependence of the metallic state at $U=1.0,\lambda =0.4$.

Figure 5

The cluster geometry for (a) bulk CDMFT and (b) real-space CDMFT.