Non-Abelian $S=1$ chiral spin liquid on the kagome lattice

We study $S=1$ spin liquid states on the kagome lattice constructed by Gutzwiller-projected $p_x+i{p}_y$ superconductors. We show that the obtained spin liquids are either non-Abelian or Abelian topological phases, depending on the topology of the fermionic mean-field state. By calculating the modular matrices $S$ and $T$, we confirm that projected topological superconductors are non-Abelian chiral spin liquid (NACSL). The chiral central charge and the spin Hall conductance we obtained agree very well with the $SO{\left(3\right)}_1$ (or, equivalently, $SU{\left(2\right)}_2$) field-theory predictions. We propose a local Hamiltonian which may stabilize the NACSL. From a variational study, we observe a topological phase transition from the NACSL to the $Z_2$ Abelian spin liquid.

Figure 1

Laughlin's gauge invariant argument of quantum spin Hall effect. The calculation is performed on a cylinder with $10\times 10$ unit cells (300 sites). (a) the structure of kagome lattice, (b) the pumping process, (c) projected topological superconductor (with $\chi =-1,\mathrm{\Delta }=1,\lambda =1$) has quantized spin Hall conductance $\frac{1}{2\pi }$, (d) projected trivial superconductor (with $\chi =-1,\mathrm{\Delta }=1,\lambda =10$) has no spin Hall effect.

Figure 2

The spin-spin correlations of the projected weak pairing state with parameters $\chi =-1,\mathrm{\Delta }=1,\lambda =1$ are calculated on a cylinder with $20\times 10$ unit cells. The $x$ direction has a periodic boundary condition and the $y$ direction is open. (a) The spin-spin correlation function on the upper boundary has a power-law decaying behavior. (b) The spin-spin correlation function in the bulk (at the center of the cylinder) decays exponentially.

Figure 3

One step of the Dehn twist ${\widehat{T}}_y$. The kagome lattice with $L_x=L_y$ is deformed to a square. The blue dashed lines represent the $x$-boundary couplings before the twist and the red solid lines represent the $x$-boundary couplings after one step of the Dehn twist. Some of the red lines across the $y$ boundary and the signs of the associated couplings are affected by the $y$-boundary condition. After a full Dehn twist, the $x$-boundary condition will be changed by the $y$-boundary condition.

Figure 4

Scaling of the Berry phases (a) ${\phi }_{y,xy}$ and (b) ${\phi }_{x,x}$ for the Dehn twist ${\widehat{T}}_y$ versus the system size. The system has $L_x$ and $L_y$ unit cells along the two directions and they are chosen to be equal for this calculation. The subscript $x,y$ and $xy$ in $\phi$ denotes the three degenerate ground states $|P_G(\pi ,0)\rangle , |P_G(0,\pi )\rangle$, and $|P_G(\pi ,\pi )\rangle$, respectively.

Figure 5

(a) Two $\pi$-flux vortices are connected with a string. All the mean-field couplings across the string reverse their signs; (b) correlation functions of the “cluster spin” $S_3^z\left(i\right)={\sum }_{j\in \mathrm{\Delta }\left(i\right)}{S}_j^z$, i.e., the total spin of the three vertices of a triangle. The red (blue) line is the case where a vortex is present (absent) at the center of each cluster.

Figure 6

Tentative phase diagram of the Hamiltonian Eq. (15). We choose $J_1=1$ and only consider the region where $J_{\chi }$ is not too small and $K$ is not too large.

Figure 7

Energy spectra on the kagome lattice with $L_x=3$ and $L_y=2$. (a) $J=1, K=0.2, J_{\chi }=0.3$; (b) $J=1, K=0.25, J_{\chi }=0.4$; (c) $J=1, K=0.3, J_{\chi }=0.5$. All the states are spin singlet.