Phenomenological $Z_2$ lattice gauge theory of the spin-liquid state of the kagome Heisenberg antiferromagnet

We construct a phenomenological $Z_2$ lattice gauge theory to describe the spin-liquid state of the $S=\frac{1}{2}$ kagome Heisenberg antiferromagnet in the sector with zero total spin. The model is a natural generalization of the Misguich-Serban-Pasquier Hamiltonian with added interactions that are strongly constrained by lattice symmetries. We are able to reproduce qualitatively many of the characteristic features observed in recent numerical studies. We also make connections to the previous works along similar lines.

Figure 1

(a) The kagome Heisenberg antiferromagnet. $S=\frac{1}{2}$ spins are located on the lattice sites, and the antiferromagnetic Heisenberg interactions are between nearest-neighboring spins. (b) The rules of the arrow representation of the dimer coverings. (c) An example of the arrow representation. (d) Mapping the arrow patterns to Ising variables. Shaded (unshaded) triangles are A (B) sites of the honeycomb.

Figure 2

(a) The flux $F^z$ is defined as the product of ${\sigma }_{ij}^z$ around the hexagon. (b) The magnetic flux threading any hexagon is 1 in the ground state. (c) A vison state contains a $-1$ flux threading through a plaquette (shaded hexagon). (d) The effect of $F_{\alpha }^z$ on the electric fluxes is to reverse all the arrows around the hexagonal plaquette $\alpha$, or equivalently flipping the dimers in a David Star.

Figure 3

(a) The links highlighted in red (thick solid) are a pair of nearest neighbors, and the links highlighted in blue (thick dashed) are another pair. (b) Similar to (a), the links highlighted in the same format are second neighbors. (c) A pair of third neighbor links are highlighted in red (thick solid). (d) The two quasidegenerate dimer coverings in the “diamond” resonance observed by DMRG calculations. (e) The arrow representation for the diamond resonance.

Figure 4

(a) The dual Ising variables are defined on the hexagonal plaquettes, and they form a triangular lattice (blue dashed lines). Latin (greek) letters are labels of the sites in the honeycomb (triangular) lattice. (b) The constraints on the phase factors ${\lambda }_{\alpha \beta }$. The product of the ${\lambda }_{\alpha \beta }$ around the shaded (unshaded) triangles is $-1$ (1). A possible arrangements for ${\lambda }_{\alpha \beta }$ is that ${\lambda }_{\alpha \beta }=1$ on black solid links and $-1$ on red dashed links. (c) The four sublattices of the triangular lattice: $A$, $B$, $C$, and $D$. Sites in the same sublattice are connected by the third-neighbor interaction. (d) The exchange interaction between the spins in $A$ sublattice. The interaction is $K$ on black solid links and $-K$ on red dashed links. The primitive vectors ${\mathit{\delta }}_{1,2}$ are also shown.

Figure 5

(a) An up triangle on kagome. ${\sigma }_{01}^x$, ${\sigma }_{02}^x$, and ${\sigma }_{03}^x$ are $Z_2$ electric flux operators on the links of the honeycomb lattice. $d_{12}$, $d_{13}$, and $d_{23}$ are the dimer density modulation on the links of kagome. (b) A triangle occupied by a dimer. (c) An empty triangle.

Figure 6

(a) The YC8 cylinder. The periodic boundary condition is imposed along the vertical direction. Sites with the same number are identical. (b) The labeling system employed in the main text. (c) The corresponding $Z_2$ gauge theory model on the honeycomb lattice. Arrows are the primitive vectors ${\mathbf{e}}_u$ and ${\mathbf{e}}_v$ of the dual triangular lattices. The coordinates of the dual triangular sites are shown. (d) The dual model for YC8 cylinder in the even sector. The phase factor $\lambda =1(-1)$ on black solid (red dashed) links. (e) The same dual model in the odd sector. (f) The exchange interaction between dual spins in A sublattice in the even sector. The interaction is $K$ on black solid links and $-K$ on red dashed links. (g) The exchange interaction between dual spins in A sublattice in the odd sector.

Figure 7

Ground-state energy per site $\epsilon$ of YC$4m$ cylinder in even $X=1$ (blue crosses) and odd $X=-1$ sectors (red open circles). The solid and dashed lines correspond to $K/\Delta =0.12$ and $0.15$, respectively.

Figure 8

(a) A YC6 cylinder and the predicted valence-bond density modulation pattern. Lattice sites with the same numeral label are identical. (b) The corresponding $Z_2$ gauge theory. (c) The dual Ising model. $A$ and $B$ are sublattice labels. $\lambda =1(-1)$ on black solid (red dashed) links. (d) The interaction between $A$ dual spins. The interaction is $K(-K)$ on black solid (red dashed) links.

Figure 9

Dimer density modulation $\left|d\right|$ in the ground states of YC$4m+2$ cylinders as a function of the cylinder index $m$.

Figure 10

(a) Valence-bond modulation pattern on the YC9-2 cylinder calculated from the phenomenological theory. The spin-spin correlation is enhanced on the solid blue bonds and weakened on the dashed red bonds, respectively. Sites labeled with the same number are identical. (b) The corresponding gauge theory model and the dual Ising model. Arrows indicate the nonvanishing expectation value of ${\sigma }_{ij}^x$ in the ground state. Boundary bonds labeled with the same number are identical. Thick green bonds show the contour with respect to which the total electric flux operator $X$ is defined. $A$ and $B$ stand for two independent sublattice sites of the dual lattice. The phase factor $\lambda =1(-1)$ on black solid (red dashed) bonds. Here the even sector $X=1$ is shown.

Figure 11

(a) A singlet or dimer is pinned on a bond with enhanced Heisenberg exchange interaction. (b) A pinned dimer is amount to frozen arrows in the arrow representation. (c) The dimer density modulation pattern on YC8 cylinder calculated from the phenomenological theory. We choose $\Delta =1$, $K=0.15$, and $V=0.2$. The dimer density is increased on blue solid bonds and decreased on red dashed bonds. The thickness is proportional to the magnitude of modulation.

Figure 12

(a) The open boundary of YC$4m$ cylinders employed by Yan et al.[10] The singlets are pinned at the boundary. (b) The arrow representation for the frozen dimer covering shown in (a). The numbers $\pm 1$ are the $Z_2$ electric fluxes on the dangling bonds. (c) Another open boundary condition of YC$4m$ cylinders and the pinned singlets. (d) The arrow representation and the $Z_2$ electric fluxes corresponding to the frozen dimer covering shown in (c). (e) The electric flux in the bulk is screened by a spinon.