Classification of Topological Insulators and Superconductors: the ``Ten-Fold Way''

Jin Ming Yao

Andreas W W Ludwig

After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.

Tudor D. Stanescu

The “ten-fold way” underlying the classification of nonin- teracting topological insulators and superconductors maps the infrastructure of the “civilized” part of the topological world. With interactions, we step into the wild. The main challenges that stem from the presence of interactions can be organized into three categories. The first set of problems is generated by the lack of a general classification of interacting topological phases (i.e., the interacting analog of the “ten-fold way”), which could be used as the basis for a systematic understanding of these quantum states. Moreover, it is not even clear what type of mathematical structure should underlie such a classification. Nonetheless, significant progress has been made in the past few years involving partial classifications of certain types of interacting topological states based on tensor categories [ 512 ], group cohomology theory [ 98 , 226 ], cobordism [ 276 ], and other “scary mathematical beasts.”

Xiao-Liang Qi

Zhong Wang

Tudor D. Stanescu

T he “ten-fold way” underlying the classification of non-interacting topological insulators and superconductors maps the infras- tructure of the “civilized” part of the topological world. With interactions, we step into the wild. The main challenges that stem from the presence of interactions can be organized into three categories. The first set of problems is generated by the lack of a general classification of interacting topological phases (i.e., the interacting analog of the “ten-fold way”), which could be used as the basis for a systematic understanding of these quantum states. Moreover, it is not even clear what type of mathematical structure should underlie such a classification. Nonetheless, significant progress has been made in the past few years involving partial classifications of certain types of interacting topological states based on tensor categories [429], group cohomology theory [82, 200], cobordism [246], and other “scary mathematical beasts.”