Figure 1

(Color online) Distorted triangular lattice with nearest-neighbor Heisenberg exchanges $J^{\prime }$ (on all horizontal bonds) and $J$ (on all other bonds), representing the geometry of systems examined in this paper.Reuse & Permissions

Figure 2

(Color online) Global phase diagram for the case with $N_v=1$ and a particular model of the spinons (model BIII); a similar phase diagram appeared in Ref. 8. The Néel antiferromagnet is found in a many materials, and has ordering wave vector $\mathbf{Q}=2\pi \left(1,0\right)$. The geometry of the VBS state coincides with that found (Refs. 20, 21) in ${\text{EtMe}}_3\text{P}{\left[\text{Pd}{\left(\text{dmit}\right)}_2\right]}_2$. The spiral antiferromagnet shown in the figure has an ordering wave vector $\mathbf{Q}=2\pi \left(1-ϵ,0\right)$ with $ϵ=1/6$, small, as expected for $J^{\prime }<J$. The experimentally realized spiral state in ${\text{Cs}}_2{\text{CuCl}}_4$ has $J^{\prime }>J$, and consequently a larger value of $ϵ\approx 1/2$. The $Z_2$ spin liquid in this phase diagram is similar to that proposed in Ref. 32 to explain observations (Refs. 26, 27, 28, 29) in $\kappa \text{−}{\left(\text{ET}\right)}_2{\text{Cu}}_2{\left(\text{CN}\right)}_3$. The transitions have been discussed previously: (i) the ${\text{CP}}^1$ field theory for the Néel-VBS transition (Ref. 3), (ii) the O(4) field theory for the spiral-$Z_2$ spin-liquid transition (Refs. 54, 55, 56), (iii) the mean-field theory for the spiral-Néel transition (Ref. 8), and (iv) the O(2) field theory for the VBS-$Z_2$ spin-liquid transition (Refs. 7, 16). All these field theories are contained in our theory in Eq. (1.1), which also describes the multicritical point M.Reuse & Permissions

Figure 3

Schematic phase diagram of the doubled CS theory in Eq. (1.1) for $k=2$. All nonzero order parameters which characterize the broken symmetry in each phase are shown.Reuse & Permissions

Figure 4

(Color online) The dual Ising model describing the vison dynamics in the dual honeycomb lattice. All the vertical bonds have hopping amplitude $w^{\prime }$; all the other bonds have hopping amplitude $w$. Note that a small $J^{\prime }/J$ implies a large $w^{\prime }/w$, and vice versa. Besides the lattice anisotropy, there is a $\pi$ flux through every hexagon, which frustrates the vison kinetics. This flux is implemented by changing the sign of the dotted $w$ bonds.Reuse & Permissions

Figure 5

Phase diagram of model AI with $g>0$. The stripe order is illustrated in Fig. 7, while the spiral antiferromagnet is as in Fig. 2.Reuse & Permissions

Figure 6

Phase diagram of model AI with $g<0$. The square lattice VBS state is as illustrated in Fig. 2.Reuse & Permissions

Figure 7

(Color online) Illustration of the $Z_2$ stripe order. This can be viewed as a bond density wave, with alternating signs on successive rows of bonds.Reuse & Permissions

Figure 8

Phase diagram of model AII with $g<0$. The phase diagram of model AII with $g>0$ is the same as that for model AI with $g>0$.Reuse & Permissions

Figure 9

Phase diagram of model AIII.Reuse & Permissions

Figure 10

(Color online) Schematic of the orientation of the orthogonal vectors ${\vec{n}}_1$, ${\vec{n}}_2$, and ${\vec{n}}_3$ around a vortex in the SO(3) GSM of the spiral antiferromagnet. One of the vectors has a constant orientation, while the other two precess by an angle of $2\pi$.Reuse & Permissions

Figure 11

(Color online) The remnant spin order pattern after proliferation of one single flavor of group 2 and 3 vortices in Eq. (6.1). (a) Proliferation of the first flavor in group 2; the spin pattern is up-down-down, and the GSM is $S^2\times Z_3$. Notice that the moment of the up-spin site is twice as much as the down-spin site, so the total magnetization is zero. (b) Proliferation of first flavor in group 3; the spin pattern is up-down-zero with GSM $S^2\times Z_3$.Reuse & Permissions