Spin structure factors of chiral quantum spin liquids on the kagome lattice

We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\mathit{k}\to -\mathit{k}$ symmetry as well as the sixfold rotation symmetry around the $\mathrm{\Gamma }$ point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the cuboc1 state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.

Figure 1

The six-site unit cell of the general chiral SBMFT Ansatz as discussed in Refs. [38, 37]. The bond operators $\langle {\widehat{\mathcal{A}}}_{lj}\rangle =\left|{\widehat{\mathcal{A}}}_{lj}\right|e^{\text{i}{\theta }_{\mathcal{A}}}$ and $\langle {\widehat{\mathcal{B}}}_{lj}\rangle =\left|{\widehat{\mathcal{B}}}_{lj}\right|e^{\text{i}{\theta }_{\mathcal{B}}}$ between two neighboring sites $l$ and $j$ are such that at every bond one has $|{\widehat{\mathcal{A}}}_{lj}|=\mathcal{A}$ and $|{\widehat{\mathcal{B}}}_{lj}|=\mathcal{B}$. On purple (dark) bonds ${\theta }_{\mathcal{A}}=0$ and ${\theta }_{\mathcal{B}}={\varphi }_{\mathcal{B}}$, while on orange (bright) bonds the phases are ${\theta }_{\mathcal{A}}={\varphi }_{{\mathcal{A}}^{\prime }}+\phi$ and ${\theta }_{\mathcal{B}}={\varphi }_{{\mathcal{B}}^{\prime }}+\phi$ with $\phi =0$ on undashed bonds and $\phi =p_1\pi$ on dashed bonds, where $p_1\in \{0,1\}$ depending on the Ansatz. Finally, the red arrows indicate the real-space vectors ${\mathit{e}}_1=a(1/2,\sqrt{3}/2)$, ${\mathit{e}}_2=a(1/2,-\sqrt{3}/2)$, and ${\mathit{e}}_3=a(-1,0)$, with $a$ the spacing between two neighboring sites, and $k_j=\mathit{k}·{\mathit{e}}_j$.

Figure 2

The normalized dynamic structure factor along the $\mathrm{\Gamma }\text{−}M\text{−}K\text{−}\mathrm{\Gamma }$ high-symmetry lines for the nonchiral SBMFT Ansätze $q=0$ (left) and $\sqrt{3}\times \sqrt{3}$ (right) in the gapped spin liquid phase at $\mathcal{S}=0.2$.

Figure 3

The normalized dynamic structure factor for the chiral cuboc1 Ansatz along the $\mathrm{\Gamma }\text{−}M\text{−}K\text{−}\mathrm{\Gamma }$ high-symmetry lines (left) and in the $\mathit{k}$ plane at $\omega =0.45J$ (right) in the gapped spin liquid phase at $\mathcal{S}=0.2$. The white hexagon in the right panel marks the extended Brillouin zone. Note that the dynamic structure factor at fixed frequency (right) is not symmetric under inversion and only has a threefold rotation symmetry due to time-reversal symmetry breaking (see main text).

Figure 4

The normalized dynamic structure factor for the cuboc2 ansatz along the $\mathrm{\Gamma }$-M-K-$\mathrm{\Gamma }$ high-symmetry lines (left) and in the $\mathit{k}$-plane at $\omega =0.45J$ (right) in the gapped spin liquid phase at $\mathcal{S}=0.2$. Note that the saddle point values of the mean-field parameters for the cuboc2 ansatz preserve time-reversal symmetry at $\mathcal{S}=0.2$, consequently this state is not chiral.

Figure 5

The normalized dynamic structure factor for the octahedral ansatz along the $\mathrm{\Gamma }$-M-K-$\mathrm{\Gamma }$ high-symmetry lines (left) and in the $\mathit{k}$-plane at $\omega =0.48J$ (right) in the gapped spin liquid phase at $\mathcal{S}=0.2$. Note that the saddle point is non-chiral, as in the case of the cuboc2 ansatz.

Figure 6

Normalized static spin structure factors for the cuboc1, cuboc2, and octahedral Ansätze in the gapped spin liquid phase with spin $\mathcal{S}=0.2$. Note that the static structure factor of the chiral cuboc1 state does not show signs of time-reversal symmetry breaking (see main text).