Theory of quantum kagome ice and vison zero modes

We derive an effective $Z_2$ gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)] in Monte Carlo studies of the $S=\frac{1}{2}$ kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a $Z_2$ spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely, that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin susceptibility, and a characteristic $(N_{\mathrm{dis}}-1)ln2$ contribution to the entropy, where $N_{\mathrm{dis}}$ is the number of disclinations. Only the $Z_2$ Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary $Z_2$ symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the $Z_2$ spin liquid, and phase transitions out of the $Z_2$ spin liquid to nearby ordered phases.

Figure 1

Schematic zero-temperature phase diagram of the XYZh model, based on the quantum Monte Carlo results of Ref. [33], showing quantum kagome ice (QKI), ferromagnetic (FM), and valence bond solid (VBS) states. Only $h>0,J_a>0$ is shown, as the phase diagram is symmetric under $h\to -h$ and $J_a$ changes sign under unitary $\pi /2$ spin rotation. At small $J_a/J_z$, the system can be mapped to a honeycomb lattice quantum dimer model where we believe VBS order is the most likely possibility [34], although VBS order was not observed in Ref. [33]; this point is discussed further in the text. The phase transition from QKI to FM was found to be first order.

Figure 2

(a) The sites of the kagome lattice, where spins of the XYZh model reside, are identified with nearest-neighbor links of the honeycomb lattice. Honeycomb sites, which correspond to kagome triangles, naturally divide into A and B sublattices, shown as open and closed circles, due to the bipartite nature of the honeycomb lattice. The generators of the $p6m$ space group are shown, with $P_d$ and $P_y$ reflections (dashed lines) and $T_1$ and $T_2$ translations (thick arrows). (b), (c) Illustrate the hopping processes in the $\mathrm{U}\left(1\right)$ gauge theory that correspond to $J_{\perp }$ and $J_a$ terms, respectively.

Figure 3

Illustration of how $J_a$ and $J_{\perp }$ terms can combine to give nearest-neighbor charge-two hopping. Three different $J_a$ coordinated hopping processes are shown. Two of these are shown with red dashed arrows, one with green solid arrows. Acting in succession with the two red/dashed processes gives a charge-one hopping from ${\mathsf{r}}^{\prime \prime }$ to $\mathsf{r}$, the same process as the $J_{\perp }$ term. Combining this with the green/solid process then gives a charge-two hopping from ${\mathsf{r}}^{\prime \prime }$ to ${\mathsf{r}}^{\prime }$.

Figure 4

$\pi$ disclination at a hexagon center of the kagome lattice, with the dual honeycomb lattice also shown. The disclination is a defect where the shaded region is cut out, and sites of the remaining lattice that are related by a $\pi$ rotation about the disclination center are identified. Equivalently, rather than cut out the shaded region, we can simply identify all sites related by a $\pi$ rotation, such as the kagome sites $k$ and $k^{\prime }$. Similarly, the honeycomb site $a$ is identified with $a^{\prime }$. The hexagonal honeycomb plaquette at the disclination center becomes a triangular plaquette with sides $ab,bc,c{a}^{\prime }\simeq ca$.