Interaction-Driven Spontaneous Quantum Hall Effect on a Kagome Lattice

Topological states of matter have been widely studied as being driven by an external magnetic field, intrinsic spin-orbital coupling, or magnetic doping. Here, we unveil an interaction-driven spontaneous quantum Hall effect (a Chern insulator) emerging in an extended fermion-Hubbard model on a kagome lattice, based on a state-of-the-art density-matrix renormalization group on cylinder geometry and an exact diagonalization in torus geometry. We first demonstrate that the proposed model exhibits an incompressible liquid phase with doublet degenerate ground states as time-reversal partners. The explicit spontaneous time-reversal symmetry breaking is determined by emergent uniform circulating loop currents between nearest neighbors. Importantly, the fingerprint topological nature of the ground state is characterized by quantized Hall conductance. Thus, we identify the liquid phase as a quantum Hall phase, which provides a “proof-of-principle” demonstration of the interaction-driven topological phase in a topologically trivial noninteracting band.

Figure 1

Phase diagram of an extended fermion-Hubbard model [Eq. (1)] plotted in (a) $V_1=V_2$ and $V_3$ parameter space and in (b) $V_1$ and $V_2=V_3$ parameter space, obtained by DMRG calculations on a cylinder of circumference $L_y=6$. The QAH phase is characterized by the long-range current-current correlations and integer quantized Hall conductance. The phase boundary between the QAH phase and the other phase is determined by the emergent loop current which signals spontaneous TRS breaking. Contour plots of the static density structure factor are shown for (c) the charge density wave $q=(0,0)$ phase, (d) the stripe phase, and (e) the QAH phase. The white dashed line shows the first Brillouin zone.

Figure 2

(a) Energy spectra from ED versus momentum quantum numbers ($K_x$, $K_y$) on the $N_s=3\times 3\times 3=27$ (the red cross) and $N_s=3\times 3\times 4=36$ (the black square) sites cluster, by setting $V=V_1=V_2=V_3$ and $V=3.95$. The ground state degeneracy is labeled with numbers. (b) Finite-size extrapolations of the single-particle gap $\mathrm{\Delta }(N_s)=E(N_s,N_e-1)+E(N_s,N_e+1)-2E(N_s,N_e)$ obtained by the DMRG [$E(N_s,N_e)$, the ground state energy on $N_s=3\times N_x\times N_y$ with $N_e$ electrons] on several lattice clusters: $3\times 3\times 4$, $3\times 4\times 4$, $3\times 4\times 5$, $3\times 5\times 6$.

Figure 3

(a) Real-space plot of the emergent current pattern ${\mathcal{J}}_{ij}$ (we only show a section of three columns on the $L_y=4$ cylinder), for a $|{\mathrm{\Psi }}^L⟩$ with left chirality. The width of the bond is proportional to the absolute value (shown on the bond as a number) and arrows point to current directions. The red arrow indicates the current direction in each hexagon. (b) Log-linear plot of current-current correlations versus distance for a QAH phase with system width $L_y=4$ (the blue circle), $L_y=5$ (the black square), and $L_y=6$ (the red diamond). All correlations demonstrate long-range order (and they are also positive) for the QAH phase, while $⟨{\mathcal{J}}_{ij}{\mathcal{J}}_{i_0{j}_0}⟩$ decay exponentially in stripe phase and charge density wave phase. The real-space plot of the current-current correlations is shown in Ref. [38].

Figure 4

Hall conductance ${\sigma }_H$ obtained from the Laughlin flux insertion gedanken experiment, where ${\sigma }_H$ is equal to the charge transfer $\mathrm{\Delta }Q$ from one edge to the other edge. (a) Net charge transfer $\mathrm{\Delta }Q$ for the QAH phase (the blue circle), the stripe phase (the navy triangle), the charge density wave (the green cross), and the ${\mathrm{QAH}}^{*}$ phase (the purple dot). The system size is an $L_y=4$ cylinder. (Inset) The adiabatic threading of a $U(1)$ charge flux in the hole of cylinder. (b) Net charge transfer $\mathrm{\Delta }Q$ of the QAH phase for the system sizes $L_y=4$ (the blue circle), $L_y=5$ (the black square), and $L_y=6$ (the red diamond).