Orbital Chern Insulator and Quantum Phase Diagram of a Kagome Electron System with Half-Filled Flat Bands

We study the quantum phase diagram of electrons on kagome lattice with half-filled lowest flat bands by considering the antiferromagnetic Heisenberg interaction $J$, and short-range Coulomb interaction $V$. In the weak $J$ regime, we identify a fully spin-polarized phase. The presence of finite $V$ drives a spontaneous chiral current, which makes the system an orbital Chern insulator by contributing an orbital magnetization. Such an out-of-plane orbital magnetization allows the presence of a Chern insulating phase independent of the spin orientation in contrast to the spin-orbit coupling induced Chern insulator that disappears with in-plane ferromagnetism constrained by symmetry. Such a symmetry difference provides a criterion to distinguish the physical origin of topological responses in kagome systems. The orbital Chern insulator is robust against small coupling $J$. By further increasing $J$, we find that the ferromagnetic topological phase is suppressed, which first becomes partially polarized and then enters a nonmagnetic phase with spin and charge nematicity. The frustrated flat band allows the spin and Coulomb interaction to play an essential role in determining the quantum phases.

Figure 1

(a) Phase diagram vs repulsion-interaction strength $V$ and exchange-interaction strength $J$. Phases I, II, and III are fully spin polarized spontaneous Chern insulator phase, partially polarized intermediate phase, and nonpolarized phase with spin and charge nematicity, respectively. (b) Lowest energy level at each sector of azimuthal spin polarization $S_z$ vs exchange-interaction strength $J$ at $V=0.6$. Circle, triangle, and square stand for the system with $N_s=36$, 48, and 72, separately. The phase diagram is calculated for $N_s=3\times 3\times 4=36$ site system. The first and second nearest neighbor interactions are $V_1=2V_2=V$. The dashed lines indicate the phase boundaries.

Figure 2

Measurements of the ground state in phase region III with $J=1.5$ and $V=0.5$. (a) Charge density at each site. Circle size represents the magnitude of density. (b) Scaling behavior of the charge gap. Inset: intraunit cell charge density imbalance $\delta n_i$ as the unit-cell position along $x$ direction $i_x$. (c) Dependence of electron hopping function $G_{ij}$ and spin-spin correlation between two sites on their distance.

Figure 3

(a) Variation of ground state energy due to the presence of nonzero $t_C$ at different $J$ for $N_s=36$ sites. The slope of each curve at $t_C=0$ represents the average current $⟨j⟩$ of the system as labeled by the number nearby. (b) Size scaling of current at the corresponding $J$. The repulsion interaction $V=0.6$ here.

Figure 4

(a),(b) Electronic band structure and orbital magnetization in the presence of spontaneous chiral current and ferromagnetism. (b) Orbital magnetization vs chemical potential. The green shaded region shows the band gap. Unit of magnetization: $et/(2\pi {)}^2\hslash$. (c),(d) Band structure and orbital magnetization in the presence of Kane-Mele type spin-orbit coupling and ferromagnetism. (d) Orbital magnetization vs azimuthal angle $\theta$ that characterizes the spin orientation as illustrated in the inset.