Interaction-induced anomalous quantum Hall state on the honeycomb lattice

We examine the existence of the interaction-generated quantum anomalous Hall phase on the honeycomb lattice. For the spinless model at half-filling, the existence of a quantum anomalous Hall phase (Chern insulator phase) has been predicted using mean-field methods. However, recent exact diagonalization studies for small clusters with periodic boundary conditions have not found a clear sign of an interaction-driven Chern insulator phase. We use the exact diagonalization method to study properties of small clusters with open boundary conditions, and contrary to previous studies, we find clear signatures of the topological phase transition for finite-size clusters. We also examine the applicability of the entangled-plaquette-state (correlator-product-state) ansatz to describe the ground states of the system. Within this approach the lattice is covered with plaquettes and the ground-state wave function is written in terms of the plaquette coefficients. Configurational weights can then be optimized using a variational Monte Carlo algorithm. Using the entangled-plaquette-state ansatz we study the ground-state properties of the system for larger system sizes and show that the results agree with the exact diagonalization results for small clusters. This confirms the validity of the entangled-plaquette-state ansatz to describe the ground states of the system and provides further confirmation of the existence of the quantum anomalous Hall phase in the thermodynamic limit, as predicted by mean-field theory calculations.

Figure 1

Mean-field phase diagram for the half-filling case obtained in Ref. [15]. The various phases are discussed in the text. SM, semimetal phase; QAH, quantum anomalous Hall phase; CMs, charge-modulated phase; CDW, charge-density-wave phase; K, Kekulé phase.

Figure 2

Illustration of the cluster of 18 sites studied in Sec. 4. A and B labels correspond to two triangular sublattices of the honeycomb lattice.

Figure 3

Ground-state and first two excited-state energies as functions of the NNN interaction strength ${\overline{V}}_2=V_2/t$. Insets: Closeup views of the level crossings at ${\overline{V}}_2={\overline{V}}_{2c}^{\left(1\right)}$ and ${\overline{V}}_2={\overline{V}}_{2c}^{\left(2\right)}$. The level crossing at ${\overline{V}}_2={\overline{V}}_{2c}^{\left(1\right)}$ marks the second-order transition from the SM to the QAH state, where the value of the topological index (Chern number) changes from 0 to 1. The level crossing at ${\overline{V}}_2={\overline{V}}_{2c}^{\left(2\right)}$ corresponds to the first-order phase transition from the QAH to the CMs phase accompanied by a change in the value of the Chern number from 1 to 0.

Figure 4

Excitation gaps for the first three excited states as functions of the NNN interaction strength ${\overline{V}}_2=V_2/t$. Closing of the excitation gap at ${\overline{V}}_2={\overline{V}}_{2c}^{\left(1\right)}$ marks the transition from the SM to the QAH state; at ${\overline{V}}_2={\overline{V}}_{2c}^{\left(2\right)}$, the transition from the QAH to the CMs state.

Figure 5

Fidelity susceptibility ${\chi }_F({\overline{V}}_2,\delta {\overline{V}}_2)$ with $\delta {\overline{V}}_2=0.003$ as a function of the interaction strength ${\overline{V}}_2=V_2/t$ for the 18-site cluster shown in Fig. 2.

Figure 6

Density profile of a hole created out of the ground state ($\Delta n_j$) at half-filling for the 18-site cluster shown in Fig. 2 and for the interaction strength (a) ${\overline{V}}_2=0$, (b) ${\overline{V}}_2=1.5$, (c) ${\overline{V}}_2=2$, and (d) ${\overline{V}}_2=3$. Here the radius of each filled circle is proportional to the magnitude of $|\Delta n_j|$ and open circles denote lattice sites where $\Delta n_j<0$.

Figure 7

Illustration of the eight-site plaquette correlators used in the calculation of the ground-state properties of the system for larger system sizes. Here the underlying triangular lattice has a two-site basis, where the basis sites are labeled A and B.

Figure 8

Ground-state energy at half-filling as a function of the NNN interaction strength $V_2/t$ ($V_1=0$) for a 128-site cluster of the same shape as the cluster in Fig. 7 and with open boundary conditions, obtained using the EPS ansatz with eight-site plaquettes (Fig. 7) and stochastic minimization method. Statistical errors are smaller than the symbol size.

Figure 9

Density-functional fidelity susceptibility ${\chi }_{F_n}({\overline{V}}_2,\delta {\overline{V}}_2)$, with $\delta {\overline{V}}_2=0.05$, as a function of the interaction strength ${\overline{V}}_2=V_2/t$ ($V_1=0$) for a 128-site cluster of the same shape as the cluster in Fig. 7 and with open boundary conditions. Statistical errors are smaller than the symbol size. The SM-to-QAH phase transition (a) and QAH-to-CMs phase transition (b) are characterized by a singular peak (discontinuity) in ${\chi }_{F_n}({\overline{V}}_2,\delta {\overline{V}}_2)$.

Figure 10

Ground-state density distribution for a 128-site cluster with open boundary conditions at $V_2/t=1.8$ ($V_1=0$), when the system is in the QAH phase. Here the radius of each circle is proportional to the magnitude of the density.

Figure 11

Ground-state density distribution for a 128-site cluster with open boundary conditions at $V_2/t=2.8$ ($V_1=0$), when the system is in the CMs phase. Here the radius of each circle is proportional to the magnitude of the density.