Chiral spin liquid in a correlated topological insulator

In this paper, we investigate the topological Hubbard model, the spinful Haldane model with on-site interaction on honeycomb lattice with spin-rotation symmetry by using the slave-rotor approach, and find that chiral spin liquid exists in such a correlated electron system of the intermediate coupling region. By considering the anyon nature of excitations, chiral spin liquid may be the ground state of the topological Hubbard model. The low-energy physics is basically determined by its Chern-Simons gauge theory.

Figure 1

The scheme of the phase diagram of the topological Hubbard model.

Figure 2

Illustration of a honeycomb lattice with $A$ and $B$ sublattices for the Haldane model. $a_0$ is the length of the hexagon side and is chosen to be the unit of measure.

Figure 3

The edge states of QAH. The blue lines denote the electron edge states.

Figure 4

Phase diagram at $T=0$. There are four regions: I is the TI state, II is the quantum spin liquid, III is an AF-SDW order with QAH effect, and IV is the trivial AF-SDW order. The blue and black lines are $(U/t){}_{c1}$ and $(U/t){}_{c2},$ respectively. The red line denotes the phase boundary between the (trivial) AF-SDW order and the AF-SDW order with quantized anomalous Hall effect.

Figure 5

The solution of the parameter $Q_X$.

Figure 6

The solution of the parameter $Q_f$.

Figure 7

The solution of the parameter $Q_X^{\prime }$.

Figure 8

The solution of the parameter $Q_f^{\prime }$.

Figure 9

The solution of the parameter $\rho$.

Figure 10

The energy gaps of fermionic spinon ${\Delta }_f$.

Figure 11

The energy gaps of rotor ${\Delta }_X$.

Figure 12

Scheme of an anyon as a composite object of a fermionic spinon and a $\pi$ flux.

Figure 13

The edge states of CSL. The blue lines denote the spinon edge states.

Figure 14

$1,$$2,$ and $3$ correspond to three vertices of an equilateral triangle to define the spin chiral order parameter in Eq. (48).

Figure 15

The spin chiral order parameter ${\chi }_{{}_{\langle 123\rangle }}$.