We study a family of materials of 5d or 4f ions on pyrochlore lattice. Considering both spin-orbit coupling and crystal field splitting, a type of Kramers doublets transform according to two distinct 1-dimensional irreducible representation under the $D_{3d}$ site symmetry can be realized. We dub these doublets dipolar-octupolar(DO) doublets because the pseudo-spin operator formed by those doublets transform in a non-trivial way under symmetry operation. Such model has the potential to realize not only topological insulator but also quantum spin ices which are long ranged entangled quantum states in 3+1D at different limit [1].
After our theoretical prediction, recent experiments suggests novel phenomena in the Cerium based pyrochlores [2][3][4]
[1] Yi-Ping Huang, Gang Chen, and Michael Hermele, Phys. Rev. Lett. 112, 167203 (2014)
[3] Quantum Spin Ice Dynamics in the Dipole-Octupole Pyrochlore Magnet Ce2Zr2O7
[4] A quantum liquid of magnetic octupoles on the pyrochlore lattice
Recently, significant advances in the theoretical understanding of the topological structure and the space group symmetry shed new lights on the possibility to detect global topological structure using symmetry defects. Several theoretical models demonstrate such behavior. However, very little attention has been paid to their physical consequences, and they have not been discussed in a simple, physically relevant model. In our work, we point out such idea provides striking numerical signals that can be measured using large scale unbiased quantum Monte Carlo simulation [1]. We find the topological excitation transforms non-trivially under particular space group symmetry and global symmetry which leads to non-trivial local degeneracy, vison zero mode, that indicates the topological structure of the quantum states [2].
[1] A two-dimensional spin liquid in quantum kagome ice
[2] Yi-Ping Huang and Michael Hermele, Phys. Rev. B 95, 075130 (2017)
Topological phases protected by the geometrical symmetries of crystal lattices turn out to be surprisingly simple. They can be built from simpler lower-dimensional states, arranged in a crystalline pattern. We study the classification of symmetry-protected topological (SPT) phases with crystalline symmetry (cSPT phases). Focusing on bosonic cSPT phases in two and three dimensions, we introduce a simple family of cSPT states, where the system is comprised of decoupled lower-dimensional building blocks that are themselves SPT states. We introduce a procedure to classify these block states, which surprisingly reproduces a classification of cSPT phases recently obtained by Thorngren and Else (arXiv:1612.00846) using very different methods, for all wallpaper and space groups. The explicit constructions underlying our results clarify the physical properties of the phases classified by Thorngren and Else, and expose additional structure in the classification. Moreover, the states we classify can be completely characterized by point-group SPT (pgSPT) invariants and related weak pgSPT invariants that we introduce. In many cases, the weak invariants can be visualized in terms of translation-symmetric stacking of lower-dimensional pgSPT states. We apply our classification to propose a Lieb-Shultz-Mattis–type constraint for two-dimensional spin systems with only crystalline symmetry, and establish this constraint by a dimensional reduction argument. Finally, the surprising matching with the Thorngren-Else classification leads us to conjecture that all SPT phases protected only by crystalline symmetry can be built from lower-dimensional blocks of invertible topological states. We argue that this conjecture holds if we make a certain physically reasonable but unproven assumption.
There is an old saying : 醉翁之意不在酒，在乎山水之間也。 In describe the following scenario. During the hiking, tasting the good wine in the mountain is not the goal. Instead the true purpose is to savour the beautiful view of the mountains.
[1] Sheng-Jie Huang, Hao Song, Yi-Ping Huang, and Michael Hermele, Phys. Rev. B 96, 205106 (2017)