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) statistics, which are then treated within mean-field theory. A long-standing question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all $8Z_2$ spin liquid phases in the 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 the kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for the kagome Heisenberg model, Sachdev's $Q_1=Q_2$ SBMF state and the Lu-Ran-Lee $Z_2[0,\pi ]\beta$ 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 the fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states that do not admit a SBMF description.

Figure 1

Crystal symmetries of kagome lattice with 4 generators $\{T_{1,2},R_{\pi /3},R_y\}$. Translations $(T_1,T_2)$ along the direction 1 and 2 are drawn as the directed arrow. $R_{\pi /3}$ stands for ${60}^{\circ }$ rotation about a hexagon center. The mirror reflection 1 is denoted by $R_y$, while reflection 2 corresponds to $R_{\pi /3}{R}_y$.

Figure 2

Nontrivial twist factor ${\mathrm{\Omega }}_{f,v}^b(I,I)=-1$ associated with inversion square operation ${\left(R_{\pi /3}\right)}^6=I^2=\mathit{e}$, as discussed in Sec. 2c1. The symmetry operator associated with ${\left(R_{\pi /3}\right)}^6=I^2$ is illustrated by the solid (dashed) closed hexagon string with red (blue) color for fermionic spinons $f$ (vison $v)$. The phase factor acquired by a bosonic spinon $b=f\times v$ in this process is ${\omega }_b(I,I)=-{\omega }_f(I,I)·{\omega }_v(I,I)$, where the extra $-1$ sign comes from the crossing of the fermion string (black solid line) and vison string (blue dashed line).

Figure 3

Nontrivial twist factor ${\mathrm{\Omega }}_{f,v}^b(R,R)=-1$ associated with reflection square operation $R^2=\mathit{e}$, as discussed in Sec. 2c2. Reflection axis of $R$ is denoted by the dotted green line. The symmetry operator associated with reflection $R$ is illustrated by the solid (dashed) closed hexagon string $O_f\left(O_v\right)$ with red (blue) color for fermionic spinons $f_{1,2}$ (vison $v_{1,2})$, in the bottom of the figure. The phase factor acquired by each bosonic spinon $b_i=f_i\times v_i,i=1,2$ in the process of $R^2=\mathit{e}$ is ${\omega }_b(R,R)=-{\omega }_f(R,R)·{\omega }_v(R,R)$, where the extra $-1=S_f{O}_v{S}_f^{-1}{O}_v^{-1}$ sign comes from the crossing of open fermionic spinon string $S_f$ (black solid line) and closed vison string $O_v$ (blue dashed line). Note that the pair of fermionic spinons $f_{1,2}$ (and visons $v_{1,2})$ are related by translation $T_a$ (parallel to reflection axis), to guarantee that they share the same reflection quantum number.