Projective symmetry group classification of chiral ${\mathbb{Z}}_2$ spin liquids on the pyrochlore lattice: Application to the spin-$\frac{1}{2}$ XXZ Heisenberg model

We give a complete classification of fully symmetric as well as chiral ${\mathbb{Z}}_2$ quantum spin liquids on the pyrochlore lattice using a projective symmetry group analysis of Schwinger boson mean-field states. We find 50 independent Ansätze, including the 12 fully symmetric nearest-neighbor ${\mathbb{Z}}_2$ spin liquids that have been classified by Liu et al. [Phys. Rev. B 100, 075125 (2019)]. For each class, we specify the most general symmetry-allowed mean-field Hamiltonian. Additionally, we test the properties of a subset of the spin-liquid Ansätze by solving the mean-field equations for the spin-$1/2$ XXZ model near the antiferromagnetic Heisenberg point. We find four chiral spin liquids that break the screw symmetry of the lattice modulo time-reversal symmetry. These states have a different symmetry from the previously studied monopole flux state, and their unique characteristic is a $\frac{\pi }{3}$ flux enclosed by every rhombus of the lattice.

Figure 1

(a) A unit cell of the pyrochlore lattice. Nearest-neighbor sites are connected by bonds. Blue bonds (main tetrahedron) are within a unit cell, and red bonds (inverse tetrahedron) are between neighboring unit cells. (b) The local spin basis of the inverse tetrahedron (A1). (c) The enlarged unit cell consisting of eight tetrahedra.

Figure 2

The normalized spin-spin correlation in local spin basis ${\mathcal{S}}^{z,z}\left(\mathbf{q}\right)$ plotted along the $[h,h,l]$ plane. The spin-liquid states are labeled by the fluxes of the $\widehat{A}$ operators on the hexagonal, bow tie, and rhombus loops: $({\varphi }_{⎔},{\varphi }_{\bowtie },{\varphi }_{\diamond })=(p_1,p_1+n_1,1+\frac{2k}{3})\pi$. The dashed and solid white lines in (a) indicate the momentum cuts presented in Fig. 3.

Figure 3

Cut through the pinch points of the spin-spin correlation ${\mathcal{S}}^{z,z}\left(\mathbf{q}\right)$ in the local spin basis at (i) $\mathbf{q}=(0,0,4\pi )$ as indicated by the solid line in Fig. 2 and (ii) $\mathbf{q}=(-2\pi ,-2\pi ,+2\pi )$ as indicated by the dashed line in Fig. 2. For better comparability, the plots are normalized such that the pinch points have a magnitude of 1.

Figure 4

The normalized total neutron scattering amplitude ${\mathcal{S}}_{\text{TOT}}\left(\mathbf{q}\right)$ plotted along the $[h,h,l]$ plane. The spin-liquid states are labeled by the fluxes of the $\widehat{A}$ operators on the hexagonal, bow tie, and rhombus loops: $({\varphi }_{⎔},{\varphi }_{\bowtie },{\varphi }_{\diamond })=(p_1,p_1+n_1,1+\frac{2k}{3})\pi$.

Figure 5

The normalized spin-flip channel of the neutron scattering amplitude ${\mathcal{S}}_{\text{SF}}\left(\mathbf{q}\right)$ plotted along the $[h,h,l]$ plane. The spin-liquid states are labeled by the fluxes of the $\widehat{A}$ operators on the hexagonal, bow tie, and rhombus loops: $({\varphi }_{⎔},{\varphi }_{\bowtie },{\varphi }_{\diamond })=(p_1,p_1+n_1,1+\frac{2k}{3})\pi$.

Figure 6

The normalized no-spin-flip channel of the neutron scattering amplitude ${\mathcal{S}}_{\text{NSF}}\left(\mathbf{q}\right)$ plotted along the $[h,h,l]$ plane. The spin-liquid states are labeled by the fluxes of the $\widehat{A}$ operators on the hexagonal, bow tie, and rhombus loops: $({\varphi }_{⎔},{\varphi }_{\bowtie },{\varphi }_{\diamond })=(p_1,p_1+n_1,1+\frac{2k}{3})\pi$.

Figure 7

Transformations of the triangle, rhombus and bow-tie loops under $I$ and $\mathrm{\Sigma }$ symmetry. $T_0$ labels the tetrahedra of the (0,0,0) unit cell, $T_3^{-1}$ labels the tetrahedra of the $(0,0,-1)$ unit cell.

Figure 8

The 16-site unit cell of the $n_1=1$ Ansätze. The expectation values of the bond operators are defined by the $u_{\mu \nu }^t$ matrices given in Eq. (F6). The direction $\mu \stackrel{}{\to }\nu$ is indicated by the arrowheads. On each dashed bond, the $u_{\mu \nu }^t$ matrices have to be multiplied by an extra phase factor of $exp\left(i\pi n_1\right)$. For $n_1=0$, the mean-field Ansatz is fully described by the four-site unit cell without any dashed lines. The dependence of the $u_{\mu \nu }^t$ matrices on the mean fields is described in Eq. (17).