Quantum Spin Liquid with Even Ising Gauge Field Structure on Kagome Lattice

Employing large-scale quantum Monte Carlo simulations, we study the extended $XXZ$ model on the kagome lattice. A ${\mathbb{Z}}_2$ quantum spin liquid phase with effective even Ising gauge field structure emerges from the delicate balance among three symmetry-breaking phases including stripe solid, staggered solid, and ferromagnet. This ${\mathbb{Z}}_2$ spin liquid is stabilized by an extended interaction related to the Rokhsar-Kivelson potential in the quantum dimer model limit. The phase transitions from the staggered solid to a spin liquid or ferromagnet are found to be first order and so is the transition between the stripe solid and ferromagnet. However, the transition between a spin liquid and ferromagnet is found to be continuous and belongs to the 3D $X{Y}^{*}$ universality class associated with the condensation of spinons. The transition between a spin liquid and stripe solid appears to be continuous and associated with the condensation of visons.

Figure 1

(a) The kagome lattice with lattice vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, sublattices $A$, $B$, and $C$, and all the interactions $J_{\pm }$, $J_z$, and $J_z^{\prime }$ of the Hamiltonian in Eq. (1) are depicted. (b) Brillouin zone of the kagome lattice, with the reciprocal space vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$, the high-symmetry points $\mathrm{\Gamma }=(0,0)$, and $K=(4\pi /3,0)$. (c) $J_{\pm }/J_z-J_z^{\prime }/J_z$ phase diagram of the Hamiltonian in Eq. (1) with magnetization $m^z=1/6$ (or $1/3$ filling factor in boson language). The yellow, green, violet, and blue shaded areas are ${\mathbb{Z}}_2$ spin liquid (SL), stripe solid (SS), staggered solid (ST), and ferromagnetic (FM) phases, respectively. The arrangements of the spin configuration in the SS and ST phases are schematically shown in the insets, with the red (gray) ball denoting spin up (down). The red dimers stand for the effective dimer covering in the SS and ST phases. The phase transition SL-FM is continuous, SL-SS is seemingly continuous, and SL-ST, ST-FM, and SS-FM are first order.

Figure 2

The structure factor $S_{\mathbf{q}=\mathrm{\Gamma }}/N$ and spin stiffness ${\rho }_s$ of the system as a function of $J_{\pm }/J_z$ at $J_z^{\prime }/J_z=-0.005$ [$S_{\mathbf{q}=\mathrm{\Gamma }}/N$ (a) and ${\rho }_s$ (b)] and $J_z^{\prime }/J_z=0$ [$S_{\mathbf{q}=\mathrm{\Gamma }}/N$ (c) and ${\rho }_s$ (d)]. The inverse temperature is set to $\beta J_{\pm }=2L$, and the initial spin configuration is set to be inside the SS phase.

Figure 3

The structure factor $S_{\mathbf{q}=K}/N$ and spin stiffness ${\rho }_s$ of the system as a function of $J_z^{\prime }/J_z$ at $J_{\pm }/J_z=0.07$ [$S_{\mathbf{q}=K}/N$ (a) and ${\rho }_s$ (b)] and as a function of $J_{\pm }/J_z$ at $J_z^{\prime }/J_z=0.03$ [$S_{\mathbf{q}=K}/N$ (c) and ${\rho }_s$ (d)]. The inverse temperature is set to $\beta J_{\pm }=2L$, and the initial spin configuration is set to be inside the ST phase.

Figure 4

Data collapse of the spin stiffness ${\rho }_s{L}^z$ as a function of $(J_{\pm }/J_z-J_{\pm }^c/J_z)L^{1/\nu }$ at (a) $J_z^{\prime }/J_z=0.005$ with 3D $XY$ exponents. The inverse temperature is set to $\beta J_{\pm }=2L$. The critical point is determined as $J_{\pm }^c/J_z=0.0775$; see the inset. (b) Equal-time spin-spin correlation function $G(r)=⟨S_0^{+}{S}_r^{-}⟩$ as a function of the distance $r$ at critical points $J_{\pm }^c/J_z=0.0775$ for $J_z^{\prime }/J_z=0.005$. The log-log plot gives rise to $G(r)\sim r^{-(1+\eta )}=r^{-2.53(4)}$, and large anomalous dimension $\eta =1.53(4)$ is consistent with the 3D $X{Y}^{*}$ transition; the $r^{-1.04}$ line stands for the conventional 3D $XY$ behavior with $\eta =0.04$.

Figure 5

The magnetic structure factor $S_{\mathbf{q}=\mathrm{\Gamma }}/N$ of the system as a function of $J_z^{\prime }$ at $J_{\pm }/J_z=0.06$. The inverse temperature is set to $\beta J_{\pm }=2L$, and the initial spin configuration is set to be inside the SS phase.