Variational Monte Carlo study of a chiral spin liquid in the extended Heisenberg model on the kagome lattice

We investigate the extended Heisenberg model on the kagome lattice by using Gutzwiller projected fermionic states and the variational Monte Carlo technique. In particular, when both second- and third-neighbor superexchanges are considered, we find that a gapped spin liquid described by nontrivial magnetic fluxes and long-range chiral-chiral correlations is energetically favored compared to the gapless U(1) Dirac state. Furthermore, the topological Chern number, obtained by integrating the Berry curvature, and the degeneracy of the ground state, by constructing linearly independent states, lead us to identify this flux state as the chiral spin liquid with a $C=1/2$ fractionalized Chern number.

Figure 1

(a), (b) The variational ansatz with the NN hopping $t_1$ (black) and NNN hopping $t_2$ (red) is shown. Solid (dashed) lines indicate positive (negative) hoppings, which define the U(1) DSL. The phases ${\varphi }_1$ and ${\varphi }_2$ are added upon this ansatz to obtain a CSL. The direction of the arrows indicates one possible convention of phases. In each up and down triangle, the flux is ${\theta }_1=3{\varphi }_1$; in each hexagon, the flux is ${\theta }_2=\pi -6{\varphi }_1$. The triangle $abc$ has flux ${\theta }_3=\pi -2{\varphi }_1-{\varphi }_2$, and the triangle $acd$ has flux ${\theta }_4=3{\varphi }_2$. (c) Phase diagram: The red dots (CSL) and blue squares [U(1) DSL] are the calculated data. The red dashed line indicates the approximate phase boundary between the CSL and U(1) DSL.

Figure 2

The energy per site as a function of ${\varphi }_1$ on different lattices. Results for the $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg model at $J_2=0.5$ with (a) $J_3=0.6$ and (b) $0.5$, and (c) for the $J_1\text{−}{J}_{\chi }$ model at $J_{\chi }=0.15$. Insets: The energy per site as function of $1/L$ for the U(1) DSL (${\varphi }_1=0$) up to $L=28$, and the CSL with ${\varphi }_1=0.0505\pi$ on the $L=8$ lattice and ${\varphi }_1=0.0567\pi$ on larger clusters up to $L=36$ (a); the finite-size effect for different ${\varphi }_1$ (b) and with enlarged scale around ${\varphi }_1=0.0928\pi$ (c). The arrows in (a) and (c) show the energy minimum, and indicate the CSL stabilized in both models.

Figure 3

The chiral-chiral correlation function along the ${\vec{a}}_1$ direction on $L=8$ and 12 lattices for the $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg model (at $J_2=0.5$ and $J_3=0.6$) and the $J_1\text{−}{J}_{\chi }$ model (at $J_{\chi }=0.15$). The DMRG data are calculated on a cylinder with $L=6$ at $J_2=0.5$ and $J_3=0.6$.

Figure 4

The Berry curvature at $J_2=0.5$ and $J_3=0.6$ on a $L=12$ lattice. The Brillouin zone is divided into a mesh with 100 plaquettes. The summation between 0 and $2\pi$ gives $C=2.01$.