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

摘要

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); Fak 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.