Quantum Spin Ices and Topological Phases from Dipolar-Octupolar Doublets on the Pyrochlore Lattice

We consider a class of $d$- and $f$-electron systems in which dipolar-octupolar Kramers doublets arise on the sites of the pyrochlore lattice. For such doublets, two components of the pseudospin transform like a magnetic dipole, while the other transforms like a component of the magnetic octupole tensor. Based on a symmetry analysis, we construct and study models of dipolar-octupolar doublets in itinerant and localized limits. In both limits, the resulting models are of surprisingly simple form. In the itinerant limit, we find topological insulating behavior. In the localized limit, the most general nearest-neighbor spin model is the $XYZ$ model. We show that this $XYZ$ model exhibits two distinct quantum spin ice (QSI) phases, that we dub dipolar QSI, and octupolar QSI. We conclude with a discussion of potential relevance to real material systems.

Figure 1

(a) The evolution of $d$ electron states under cubic crystal field, SOC, and trigonal distortion. (b) The energies for the three local doublets under different trigonal distortions. Compression (elongation) along the $C_3$ axis corresponds to ${\mathrm{\Delta }}_3>0$ (${\mathrm{\Delta }}_3<0$).

Figure 2

Phase diagram of the tight-binding model wtih first- and second-neighbor hopping, as a function of (${\tilde{w}}_0$, ${\tilde{w}}_x$, ${\tilde{w}}_z$), setting ${\tilde{t}}_{nn}^3=1$. Very small fourth-neighbor hopping is included to remove unstable band touchings at the $W$ point. Metallic ($M$) and strong topological insulator (TI) phases are found. The phase diagram is symmetric under ${\tilde{w}}_x\to -{\tilde{w}}_x$ and ${\tilde{w}}_0\to -{\tilde{w}}_0$.

Figure 3

Left: Unit cube in (${\tilde{J}}_x$, ${\tilde{J}}_y$, ${\tilde{J}}_z$) parameter space of the $XYZ$ model. Shaded regions were analyzed via gMFT. Right: gMFT phase diagram on the ${\tilde{J}}_z=1$ surface of the cube, where dQSI, AIAO, and AFO phases are found. Within gMFT, the phase transition is 1st order (2nd order) at the dashed (solid) boundary. The dotted line is the $XXZ$ line. We did not apply gMFT for ${\tilde{J}}_x+{\tilde{J}}_y\ge 0$. There, the exchange is frustrated, and QSI is likely to be more stable than for ${\tilde{J}}_x+{\tilde{J}}_y<0$ [22]. The phase diagram on the other surfaces of the cube can be obtained by relabeling parameters, with the nature of phases changing according to the anisotropic character of ${\tilde{\tau }}_{\mathit{r}}^{\mu }$. dQSI occurs on the ${\tilde{J}}_z=1$ and ${\tilde{J}}_x=1$ faces, while oQSI occurs on the ${\tilde{J}}_y=1$ face.