Time reversal symmetry breaking chiral spin liquids: Projective symmetry group approach of bosonic mean-field theories

Projective symmetry groups are the mathematical tools which make it possible to list and classify mean-field spin liquids (SLs) based on a parton construction. The seminal work of Wen [Phys. Rev. B 65, 165113 (2002)] and its subsequent extension to bosons by Wang and Vishwanath [Phys. Rev. B 74, 174423 (2006)] concerned the so-called symmetric SLs; i.e., states that break neither lattice symmetries nor time reversal invariance. Here we generalize this tool to chiral (time reversal symmetry breaking) SLs described in a Schwinger boson mean-field approach and illustrate it on the triangular lattice, which can harbor nine different weakly symmetric SLs (two symmetric SLs and seven chiral SLs) with nearest neighbor bond operators only. Results for other lattices (square and kagome) are given in the Appendixes. Application of this new approach has recently led to the discovery of two chiral ground states on the kagome lattice [Messio et al., Phys. Rev. Lett. 108, 207204 (2012); Fåk et al., Phys. Rev. Lett. 109, 037208 (2012)]. The signature of a time reversal symmetry breaking SL is the presence in the ground state of nontrivial fluxes of loop operators that break some lattice point group symmetries. The physical significance of these gauge invariant quantities is discussed both in the classical limit and in the quantum SL and their expressions in terms of spin observables are given.

Figure 1

Tetrahedral order on the triangular lattice.

Figure 2

Generators of the lattice symmetries $\mathcal{X}$ on the triangular and square lattices. ${\mathcal{V}}_i$ is a translation, $\sigma$ is a reflection, and ${\mathcal{R}}_i$ is a rotation of order $i$.

Figure 3

Ansätze respecting the ${\mathcal{X}}_e$ symmetries on the triangular lattice. All arrows carry ${\mathcal{B}}_{ij}$ parameters of modulus $B_1$ and of argument ${\varphi }_{B_1}$ and ${\mathcal{A}}_{ij}$ parameters of modulus $A_1$ and of argument 0 on red arrows (choice of the gauge), $2k\pi /3$ on blue ones, and $4k\pi /3$ on green ones. On dashed arrows ${\mathcal{A}}_{ij}$ and ${\mathcal{B}}_{ij}$ take an extra $p_1\pi$ phase.

Figure 4

Ansätze respecting the ${\mathcal{X}}_e$ symmetries on the kagome lattice up to a gauge transformation. Blue arrows carry ${\mathcal{B}}_{ij}$ parameters of modulus $B_1$ and of argument ${\varphi }_{B_1}$ and ${\mathcal{A}}_{ij}$ parameters of modulus $A_1$ and of argument 0. Red arrows are for moduli $B_1^{\prime }$ and $A_1^{\prime }$ and arguments ${\varphi }_{B_1^{\prime }}$ and ${\varphi }_{A_1^{\prime }}$. On dashed arrows ${\mathcal{A}}_{ij}$ and ${\mathcal{B}}_{ij}$ take an extra $p_1\pi$ phase.

Figure 5

Triangular lattice with 12 sites (solid circles) and periodicity ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$, respecting all the lattice symmetries of Fig. 2. Three nonlocal loops ${\ell }_1$, ${\ell }_2$, and ${\ell }_3$ are drawn in dashed arrows. The green dotted lines cross the bonds where the MF parameters are multiplied by $-1$.

Figure 6

The hexagon is the Brillouin zone of the triangular lattice, the rectangle, this of the ansatz. The wave vectors of the 12-site lattice with PBCs are the blue ones, whereas they are the green ones for PBC in the ${\mathcal{L}}_1$ direction and APBC in the ${\mathcal{L}}_2$ direction. The background intensity is the value of the spinon energy (dark for minima).