Symmetry enriched U(1) topological orders for dipole-octupole doublets on a pyrochlore lattice

Symmetry plays a fundamental role in our understanding of both conventional symmetry breaking phases and the more exotic quantum and topological phases of matter. We explore the experimental signatures of symmetry enriched U(1) quantum spin liquids (QSLs) on the pyrochlore lattice. We point out that the Ce local moment of the newly discovered pyrochlore QSL candidate ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$, is a dipole-octupole doublet. The generic model for these unusual doublets supports two distinct symmetry enriched U(1) QSL ground states in the corresponding quantum spin ice regimes. These two U(1) QSLs are dubbed dipolar U(1) QSL and octupolar U(1) QSL. While the dipolar U(1) QSL has been discussed in many contexts, the octupolar U(1) QSL is rather unique. Based on the symmetry properties of the dipole-octupole doublets, we predict the peculiar physical properties of the octupolar U(1) QSL, elucidating the unique spectroscopic properties in the external magnetic fields. We further predict the Anderson-Higgs transition from the octupolar U(1) QSL driven by the external magnetic fields. We identify the experimental relevance with the candidate material ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and other dipole-octupole doublet systems.

Figure 1

The electron configuration and the ${\mathrm{D}}_{\text{3d}}$ crystal electric field (CEF) splitting of the ${\mathrm{Ce}}^{3+}$ ion in ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$. The CEF ground state wave functions are combinations of $J^z=\pm 3/2$ states [14], thus the CEF ground state is a DO doublet. $\mathrm{\Delta }$ is the CEF gap and was fitted to be $\mathrm{\Delta }=50\pm 5\mathrm{meV}$ [14].

Figure 2

Phase diagrams for magnetic fields along (a) $\left[111\right]$, (b) $\left[001\right]$, and (c) $\left[110\right]$ directions. Outside the QSL phases are the induced magnetic ordered phase via the spinon condensation. For $h=0$, the spinons are condensed at ${\mathbf{k}}_c=(0,0,0)$, and we choose the local moments to order in the local $\widehat{z}$ direction. In (a), a large magnetic field near the vertical axis drives the spinon condensation at ${\mathbf{k}}_c=\pi (1,1,1)$, and the resulting order is depicted in the figure. This order smoothly connects to the order on the horizontal axis. The cases in (b) and (c) are similar, except that in (b) the field on the vertical axis drives the condensation at ${\mathbf{k}}_c=2\pi (0,0,1)$, while in (c) ${\mathbf{k}}_c=\pi (1,1,0)$ near the vertical axis. We set the diamond lattice constant to unity.

Figure 3

Lower excitation edges of the spinon continuum in the dynamic spin structure factor under (a) zero magnetic field, and field along (b) $\left[111\right]$, (c) $\left[001\right]$, and (d) $\left[110\right]$ directions. In the figure, we set $J_{\pm }=0.1J_y$. The inset of (a) is the Brillouin zone [33].