Competing chiral orders in the topological Haldane-Hubbard model of spin-$\frac{1}{2}$ fermions and bosons

Motivated by experiments on ultracold atoms which have realized the Haldane model for a Chern insulator, we consider its strongly correlated Mott limit with spin-$\frac{1}{2}$ fermions. We find that slave rotor mean-field theory yields gapped or gapless chiral spin liquid Mott insulators. To study competing magnetic orders, we consider the strong coupling effective spin Hamiltonian which includes chiral three-spin exchange. We obtain its classical phase diagram, uncovering various chiral magnetic orders including tetrahedral, cone, and noncoplanar spiral states which can compete with putative chiral quantum spin liquids. We study the effect of thermal fluctuations on these states, identifying crossovers in the spin chirality, and phase transitions associated with lattice symmetry breaking. We also discuss analogous effective spin Hamiltonians for correlated spin-$\frac{1}{2}$ bosons. Finally, we point out possible experimental implications of our results for cold atom experiments.

Figure 1

(a) Honeycomb lattice showing the two sublattices with hoppings $t_1,t_2$ and phase $\varphi$ defining the noninteracting Haldane model, with large and small triangles enclosing fluxes $\pm 3\varphi$ and $-\varphi$, respectively. (b) Phase diagram at half-filling for each spin species showing Chern band metal and Chern band insulator, and the Hall conductance per spin ${\sigma }_{xy}$ in units of $e^2/h$.

Figure 2

Slave rotor mean-field phase diagram of the Haldane-Hubbard model showing the topological Chern metal (CM, green), Chern insulator (CI, brown), and topological spin liquid Mott insulators (black) at $U/t_1=10$. The topological Mott insulator could have gapless or gapped spinon excitations depending on whether it descends from a CM or CI phase, leading to gapless or gapped chiral spin liquids. An example of the parent Fermi surface of the noninteracting model, which is inherited by the spinons at mean-field level, is shown at a particular point. The slave rotor result has been rescaled by $\eta =0.65$ (see text).

Figure 3

(Top) Phase diagram of $H_{\mathrm{spin}}$ showing various classical magnetically ordered ground states at $U/t=10$: Néel, Spiral, Tetrahedral, Cantellated Tetrahedral (CT), Cone-I, and Cone-II. Below the dashed (white) line, the slave rotor result shows that we are in the Mott insulator, so the strong coupling expansion is expected to be valid. Stars indicate points where we numerically study the effect of thermal fluctuations. (Bottom) Spin configurations in the Tetrahedral and Cone states (viewing the honeycomb lattice as a “brickwall” for visual convenience). The indicated Triad states act as parent states for Cone-I and Cone-II, which are obtained by spiraling the Triads about the indicated ${\widehat{n}}_s$ axis. The CT descends from the Tetrahedral by a $\sqrt{3}\times \sqrt{3}$ canting pattern.

Figure 4

(a) Temperature dependence of the spin chirality ${\chi }_{▵}$ on small triangles for $U/t=10$ and $\varphi =\pi /3$, showing its low temperature saturation in the Tetrahedral phase $\left(t_2/t_1=1\right)$ and its vanishing in the $T=0$ Néel phase $\left(t_2/t_1=0.4\right)$. We have plotted $5\times {\chi }_{▵}$ in the Néel state for clarity. (b) Log plot of finite size scaling of critical susceptibility for the $Z_3$-clock ordering transition in the Cone-II state at $U/t_1=10,\varphi =9\pi /20,t_2/t_1=1.5$ showing ${\chi }_3\sim L^{\gamma /\nu }$, with $\gamma /\nu =1.80\left(8\right)$.

Figure 5

Common origin plots for the Néel, Tetrahedral, Cantellated Tetrahedral, Spiral, Cone-I, and Cone-II states, with red and black dots representing spins on the two sublattices of the honeycomb (or “brickwall”) lattice.