对称保护拓扑相是指有能隙的短程纠缠态，这些态中没有发生任何对称破缺。拓扑绝缘体/拓扑超导体是无相互作用的对称保护拓扑相的例子。 本项目旨在研究有相互作用的对称保护拓扑相,包括有相互作用的拓扑绝缘体和拓扑超导体。 通过群上同调理论，在相互作用玻色系统中，我们构造了许多对称保护拓扑相。 通过推广的群超上同调论，在相互作用费米子系统中，我们可以构造自由费米子系统中不存在的拓扑绝缘体或拓扑超导体。 我们将尝试推广群上同调论来对所有的对称保护拓扑相进行分类。同时，我们将研究对称保护的无能隙的边界激发及其对外场的响应。我们将构造简单的格点模型来实现这些相，从而指导实验学家去寻找具有这些对称保护拓扑相的相关材料。

Symmetry protected topological(SPT) states are gapped short-range entangled states that do not break any symmetry of the Hamiltonian. Topological insulators/superconductors are special cases of SPT phases for free fermion systems. This porject will study all interacting symmetry protected topological phase, including interacting topological insulators and topological superconductors. We have constructed interacting bosonic SPT phases throuth group cohomology theory and interacting fermionic SPT phases through super-cohomology theory. We will try to generalize the group cohomology theory and find a systematic method to classify all the SPT phases. Further, we will study the boundary excitations of the SPT phases and their respose to external probes. Then, we will construct simple lattice models to realized these phases. With these models, we can identify materials that host the SPT phases.