Unification of bosonic and fermionic theories of spin liquids on the kagome lattice

Recent numerical studies have provided strong evidence for a gapped $Z_2$ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either bose (Schwinger bosons) or fermi (Abrikosov fermions) statstics, which are then treated within the mean-field theory. A longstanding question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all 8 $Z_2$ spin liquid phases in Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct. The key idea is that for $Z_2$ spin liquid states that admit a SBMF description on kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for kagome Heisenberg model, Sachdev's $Q_{1}=Q_{2}$ SBMF state and Lu-Ran-Lee's $Z_2[0,\pi]\beta$ Abrikosov fermion state, are found to describe the same symmetric spin liquid phase. We expect these results to aid in a complete specification of the numerically observed spin liquid phase. We also discuss a set of $Z_2$ spin liquid phases in fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states, that do not admit a SBMF description.