Global phase diagram for the spin-1 antiferromagnet with uniaxial anisotropy on the kagome lattice

The phase diagram of the $XXZ$ spin-1 quantum magnet on the kagome lattice is studied for all cases where the $z$-component nearest-neighbor spin interaction $J_z$ is antiferromagnetic. Besides $J_z$ and the nearest-neighbor in-plane spin interaction $J_{xy}$, the system is also parametrized by an on-site anisotropic term $D{\left(S^z\right)}^2$. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the “superfluid” (ferromagnetic) phase with an in-plane global U(1) symmetry broken when $J_{xy}<0$; the $\sqrt{3}\times \sqrt{3}$ order when $J_{xy}>0$; the nematic phase when $D<0$ and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\widehat{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{−}\mathrm{MPYNN}\bullet \mathrm{B}{\mathrm{F}}_4$, or optical lattices of atoms with three degenerate states on every site.

Figure 1

(Color online) The $q=0$ order.

Figure 2

(Color online) The $\sqrt{3}\times \sqrt{3}$ order.

Figure 3

(Color online) The classical ground states of Eq. (1) when $J_{xy}=0$. Every triangle has configuration $(1,-1,0)$, which can be mapped to the three-color model on the dual honeycomb lattice.

Figure 4

(Color online) The effect of the ring-exchange term (3). It can flip ${\mathcal{A}}_1$ $\left({\mathcal{B}}_1\right)$ to ${\mathcal{A}}_2$ $\left({\mathcal{B}}_2\right)$ and vice versa.

Figure 5

(Color online) The phase diagram with zero magnetic field. When $D>J_z$ and $\mid J_{xy}\mid$ is small, the system is in the nondegenerate gapped photon phase; when $0<D<J_z$ and with small $\mid J_{xy}\mid$, the ground state breaks both translational and time-reversal symmetry, resulting in the sixfold degenerate plaquette order. When $J_{xy}$ is negative and large, basically the system is in superfluid order which spontaneously breaks the global spin $S^z$ conservation; when $J_{xy}$ is positive and large, $\sqrt{3}\times \sqrt{3}$ order is supposed to be favored; and when $D$ is negative and large, the system has nematic order, with nonzero expectation of ${\left(S^{+}\right)}^2$.

Figure 6

(Color online) The vortex and vortex core configuration in the sixfold plaquette order. Every vortex is surrounded by six phase domains and six domain walls; the vortex core is exactly a (0,0,0) defect. In this figure, $\mathrm{B}+$ denotes resonating (1,0,1,0,1,0) plaquette on sublattice B. The six domains around this vortex core are (count counterclockwise) $A+$ with $\Phi \sim {\Phi }_0$ and $\vec{\phi }={\vec{\phi }}_0+(1∕3,2∕3)$, $C-$ with $\Phi \sim \exp (i\pi ∕3){\Phi }_0$ and $\vec{\phi }={\vec{\phi }}_0+(1∕3,0)$, $B+$ with $\Phi \sim \exp (i2\pi ∕3){\Phi }_0$ and $\vec{\phi }={\vec{\phi }}_0+(2∕3,0)$, $A-$ with $\Phi \sim \exp \left(i\pi \right){\Phi }_0$ and $\vec{\phi }={\vec{\phi }}_0+(2∕3,1∕3)$, $C+$ with $\Phi \sim \exp (i4\pi ∕3){\Phi }_0$ and $\vec{\phi }={\vec{\phi }}_0+(0,1∕3)$, and $B-$ with $\Phi \sim \exp (i5\pi ∕3){\Phi }_0$ and $\vec{\phi }={\vec{\phi }}_0+(0,2∕3)$. The arrows show the circulation of the phase of $\Phi$. Also, this vortex is a bound state of one vortex of ${\phi }_1$ and one vortex of ${\phi }_2$. The spins are defined on the links of the honeycomb lattice, which are the sites of the kagome lattice. Links with + and − are occupied by spin states $S^z=+1$ and $-1$. The red dashed lines denote the domain walls.

Figure 7

(Color online) The vortex and vortex core configuration of ${\phi }_1$. Every vortex is surrounded by three phase domains and three domain walls; the vortex core is exactly a $(1,1,-1)$ defect. The three domains around this vortex core are (count counterclockwise) $B+$ with $\vec{\phi }={\vec{\phi }}_0+(2∕3,0)$, $B-$ with $\vec{\phi }={\vec{\phi }}_0+(0,2∕3)$, and $A+$ with $\vec{\phi }={\vec{\phi }}_0+(1∕3,0)$. The arrows show the circulation of ${\phi }_1$.

Figure 8

(Color online) Defects which violate the three-color constraint hop on one of the two triangular sublattices (red and green) of the honeycomb lattice.

Figure 9

(Color online) The $\sqrt{3}\times \sqrt{3}$ order of $\theta$ obtained from the ordered pattern of ${\theta }_{1A}$ and ${\theta }_{1B}$. The black arrow corresponds to angle $\theta$, and the pattern of $\theta$ can be obtained from its adjacent red arrow $\left({\theta }_{1A}\right)$ and green arrow $\left({\theta }_{1B}\right)$.

Figure 10

(Color online) $q=0$ order of $\theta$ obtained from ordered pattern of ${\theta }_{1A}$ and ${\theta }_{1B}$.

Figure 11

(Color online) The distribution of $S^z=1$ and $S^z=-1$ sites on the kagome lattice is equivalent to the distribution of dimers of dimer model on the dual honeycomb lattice. The ring-exchange term (49) plays the same role as the dimer resonating term in the quantum dimer model.

Figure 12

(Color online) The phase diagram in small longitudinal magnetic field $h$; PL stands for the plaquette phase. The difference between this phase diagram and the one without magnetic field in Fig. 5 is that between the plaquette phase and the superfluid phase, there is a supersolid phase. In the negative and large $D$ case, small $J_{xy}$ is going to generate a gapped crystalline order.

Figure 13

The dimer model on the kagome lattice. Since this is a spin-1 system, every site connects to two dimers. The dimer flipping term flips configuration ${\mathcal{C}}_1$ to ${\mathcal{C}}_2$ and vice versa.

Figure 14

(Color online) A typical state in the Haldane phase. In the bulk, every site is shared by two dimers. However, at each edge, there is one residual unpaired spin-$1∕2$ variable.

Figure 15

(Color online) The edge of AKLT state on the square lattice. There is one extra spin-$1∕2$ degree of freedom on each site of the edge; therefore, the edge state is effectively a spin-$1∕2$ spin chain, which is either gapless or breaks translational symmetry. The AKLT state is qualitatively different from the state with $S^z=0$ everywhere.

Figure 16

(Color online) The edge of the kagome lattice. Every unit cell at the edge has even number of spin-$1∕2$ quantities. Presumably, the edge state is gapped and featureless.