Vison states and confinement transitions of ${\mathbb{Z}}_2$ spin liquids on the kagome lattice

We present a projective symmetry group (PSG) analysis of the spinless excitations of ${\mathbb{Z}}_2$ spin liquids on the kagome lattice. In the simplest case, vortices carrying ${\mathbb{Z}}_2$ magnetic flux (“visons”) are shown to transform under the 48 element group $\mathit{GL}(2,{\mathbb{Z}}_3)$. Alternative exchange couplings can also lead to a second case with visons transforming under 288-element group $\mathit{GL}(2,{\mathbb{Z}}_3)\times {\mathrm{D}}_3$. We study the quantum phase transition in which visons condense into confining states with valence bond solid order. The critical field theories and confining states are classified using the vison PSGs.

Figure 1

Visualization of two different valence bond solid states as dimer coverings of the kagome lattice. Each dimer represents a frustrated bond in the ordered phase of the corresponding Ising model on the dual dice lattice. The left pattern, representing the VBS $1_F$ phase (see text), is a hard-core dimer covering with a 12-site unit cell that maximizes the number of perfectly flippable diamonds (highlighted in gray). Note, however, that the choice of diamonds is not unique because the dimer pattern is symmetric with respect to $\pi /3$ rotations around the hexagons marked by a black dot. The pattern on the right represents the VBS $1_A$ phase (see text), has a 36-site unit cell, and maximizes the number of perfectly flippable hexagons (highlighted in gray).

Figure 2

A vison on the kagome lattice. The center of the vison is marked by the X. We have $\text{sgn}\left(Q_{ij}^v\right)=-\text{sgn}\left(Q_{ij}\right)$ only on the links marked by the wavy lines.

Figure 3

A vison on the triangular lattice (a similar solution is expected on the kagome lattice). The center of the vison is marked by the X. The wavy line is the “branch-cut” where we have $\text{sgn}\left(Q_{ij}^v\right)=-\text{sgn}\left(Q_{ij}\right)$ only on the links crossed by the line. Plotted is the minimization result of $E\left[\left\{Q_{ij}\right\}\right]$ with $\alpha =1,\beta =-2,K=-0.5$. Minimization is done with the cluster embedded in a vison-free lattice with all nearest-neighbor links $Q_{ij}$. The numbers are $(Q_{ij}-Q_{ij}^v)$ and the thickness of the links are proportional to ${(Q_{ij}^v-Q_{ij})}^{1/2}$. $K<0$ gives rise to the zero flux state while $K>0$ favors the $\pi$ flux state where the unit cell is doubled. The zero and $\pi$ flux states without the vison have been studied previously.[11, 31]

Figure 4

Periodic motion of a vison around a closed loop $\mathcal{C}$ on the triangular lattice. Here $\mathcal{C}$ encloses the single site marked by the filled circle. The wavy lines indicate $\text{sgn}\left(Q_{ij}^v\right)=-\text{sgn}\left(Q_{ij}\right)$, as in Fig. 2. The bottom state is gauge equivalent to the top state, after the gauge transformation $b_{i\alpha }\to -b_{i\alpha }$ only for the site $i$ marked by the filled circle.

Figure 5

(Left) Unit cell of the dice lattice consisting of the three independent sites labeled $A$, $B$, and $C$. The $A$ sites are six-coordinated, whereas the $B$ and $C$ sites are three-coordinated. The basis vectors $\mathbf{u}=(3/2,\sqrt{3}/2)$ and $\mathbf{v}=(3/2,-\sqrt{3}/2)$ are indicated by dashed arrows. (Right) Gauge choice for the fully frustrated Ising model on the dice lattice with a 12-site unit cell. Red thick bonds are frustrated, i.e., $\text{sgn}\left(J_{ij}\right)=-1$.

Figure 6

Gauge transformations associated with translations by $\mathbf{u}$ and reflections about the $x$ axis. Shown is the unit cell of the fully frustrated dice model after a lattice translation by $\mathbf{u}$ (left) and after a reflection about the $x$ axis (right). The corresponding gauge-transformations consist of spin flips on the sites marked by blue points, which restore the original gauge pattern as in Fig. 5.

Figure 7

Additional next-nearest-neighbor interactions between three-coordinated sites on the dice lattice (shown as dashed blue lines) that are allowed by the PSG. For illustrative clarity not all additional bonds in the unit cell are shown. All other bonds can be obtained from the ones that are shown via translations by the basis vectors $\mathbf{u}$ and/or $\mathbf{v}$.

Figure 8

Phase diagram of Eq. (3.14) as a function of the two couplings $a$ and $b$ (here we have set $u=1$). The VBS $1_F$ phase is not reflection symmetric, whereas the VBS $2_F$ phase is reflection symmetric. In the VBS $2_F$ phase there is a crossover (indicated by the dashed line) from a phase where one of the ${\psi }_n$’s is zero (left of the dashed line) to a phase where three of the ${\psi }_n$’s are zero (right of the dashed line).

Figure 9

Bond pattern in the VBS $1_F$ phase. Plotted is the gauge-invariant bond-strength $J_{ij}{\phi }_i{\phi }_j$ for nearest-neighbor bonds on the dice lattice at $a=b=1$, which has been assigned to each respective kagome bond. Black dashed lines indicate satisfied bonds ($-J_{ij}{\phi }_i{\phi }_j<0$); red solid lines are frustrated bonds ($-J_{ij}{\phi }_i{\phi }_j>0$). The thickness of the bonds is proportional to the bond strength.

Figure 10

Bond patterns in the VBS $2_F$ phase. Plotted is the gauge invariant bond-strength $J_{ij}{\phi }_i{\phi }_j$ for nearest-neighbor bonds on the dice lattice, shown on the corresponding kagome bonds. Black dashed lines represent satisfied bonds ($-J_{ij}{\phi }_i{\phi }_j<0$). In this phase there are no frustrated bonds at all. The thickness of the bonds is proportional to the bond strength.

Figure 11

Phase diagram obtained from the GL functional (3.25) as a function of the couplings $a_2$ and $a_4$. The other parameters are fixed at $u=1$, $a_1=1/2$, $a_3=a_2$, $a_5=1/20$, and $a_6=-1/25$. The different phases are described in the text.

Figure 12

Bond patterns in the VBS $1_A$, $2_A$, $3_A$, and $4_A$ phase (clockwise, starting from the upper left). Again we plot the gauge-invariant bond-strength $J_{ij}{\phi }_i{\phi }_j$ for nearest-neighbor bonds on the dice lattice, mapped to the corresponding kagome bonds. Black dashed lines represent satisfied bonds ($-J_{ij}{\phi }_i{\phi }_j<0$), red lines are frustrated bonds. The thickness of the lines is proportional to the bond strength.

Figure 13

Confining phase on the triangular lattice. Plotted is the gauge invariant bond-strength $J_{ij}{\phi }_i{\phi }_j$ for nearest-neighbor bonds on the frustrated honeycomb lattice, shown on the corresponding triangular lattice bonds. Black lines represent satisfied bonds ($-J_{ij}{\phi }_i{\phi }_j<0$). There are no frustrated bonds in the confining phase. The thickness of the bonds is proportional to the bond-strength. The two different patterns arise due to a crossover when the sign of the O(4)-breaking term in the GL-functional changes.