Generic quantum spin ice

We consider possible exotic ground states of quantum spin ice as realized in rare earth pyrochlores. Prior work [Savary and Balents, Phys. Rev. Lett. 108, 037202 (2012).] introduced a gauge mean-field theory (gMFT) to treat spin or pseudospin Hamiltonians for such systems, reformulated as a problem of bosonic spinons coupled to a $U\left(1\right)$ gauge field. We extend gMFT to treat the most general nearest-neighbor exchange Hamiltonian, which contains a further exchange interaction. This term leads to interactions between spinons and requires a significant extension of gMFT, which we provide. As an application, we focus especially on the non-Kramers materials Pr${}_{2}T{M}_2$O${}_7$ ($TM=$ Sn, Zr, Hf, and Ir), for which the additional term is especially important, but for which an Ising-planar exchange coupling discussed previously is forbidden by time-reversal symmetry. In this case, when the planar $\mathit{XY}$ exchange is unfrustrated, we perform a full analysis and find three quantum ground states: a $U\left(1\right)$ quantum spin liquid (QSL), an antiferroquadrupolar ordered state and a noncoplanar ferroquadrupolar ordered one. We also consider the case of frustrated $\mathit{XY}$ exchange, and find that it favors a $\pi$-flux QSL, with an emergent line degeneracy of low-energy spinon excitations. This feature greatly enhances the stability of the QSL with respect to classical ordering.

Figure 1

(a) Mapping onto $U\left(1\right)$ gauge theory: spinons are defined on diamond lattice sites (black sphere) and gauge fields live on their links (black solid line). $J_{\pm }$ ($J_{\pm \pm }$) shows quadratic (quartic) spinon hopping process in Eq. (11). (b) ${\gamma }_{1,2,3}$ [defined in Eq. (2)] depending on the bond directions: bonds with double solid lines have the matrix ${\gamma }_1$, bonds with single solid lines have ${\gamma }_2$ and bonds with dotted lines have ${\gamma }_3$.

Figure 2

Mean-field Ansatz for gauge fields that preserve the symmetry of Hamiltonian Eq. (11): (a) zero-flux state, and (b) $\pi$-flux state. In (b), the thick links have ${\overline{A}}_{\mathbf{r},{\mathbf{r}}^{\prime }}=\pi$ and other links have ${\overline{A}}_{\mathbf{r},{\mathbf{r}}^{\prime }}=0$. Thick solid (dashed) links are aligned in upper (lower) hexagonal plane perpendicular to $\langle 111\rangle$ direction in a diamond lattice.

Figure 3

Phase diagram of two dimensionless parameters $J_{\pm }/J_{zz}$ vs $J_{\pm \pm }/J_{zz}$. Four distinct phases exist: classical spin ice (at the origin), $U\left(1\right)$ QSL, AFQ, and FQ. (more details in the main context)

Figure 4

(a) AFQ: an $f$-electron charge distribution under the antiferro-quadrupolar order induced by a spinon condensation $\langle \Phi \rangle \ne 0$ at $\mathbf{k}=\left(000\right)$ wave vector. Here, we have taken the Pr${}^{3+}$ case with the local ground-state non-Kramers doublet $\alpha |4\sigma \rangle -\beta \sigma |\sigma \rangle +\gamma |-2\sigma \rangle$ with $\alpha \approx 0.970$, $\beta \approx 0.075,$ and $\gamma \approx 0.230$, where $|J_z\rangle$ is an eigenstate of the $z$ component of the total angular momentum in the local frame.[9] (b) Noncoplanar FQ: an $f$-electron charge distribution under the noncoplanar ferroquadrupolar order induced by a spinon condensation $\langle \Phi \rangle \ne 0$ at $\mathbf{k}=2\pi \left(100\right)$ for the Pr${}^{3+}$ case. (c) Antiferromagnet: an $\mathit{XY}$ antiferromagnetic ordering induced by a spinon condensation $\langle \Phi \rangle \ne 0$ at $\mathbf{k}=\left(000\right)$ wave vector in the case of half-integer spins. (d) Noncoplanar ferromagnet: an $\mathit{XY}$ noncoplanar ferromagnetic ordering induced by a spinon condensation $\langle \Phi \rangle \ne 0$ at $\mathbf{k}=2\pi \left(100\right)$ in the case of half-integer spins.