Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions

Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of $G$-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional $H^1(G, \mathbb{Z}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group $G\times\mathbb{Z}_2^f$ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.Comment: references added; published versio

Study on the Restrictive Factors of Wushu Sanda Movement in Weinan Middle School of Shan xi Province

This paper analyzes the development of Wushu Sanda movement in middle schools in Weinan City, Shan xi Province through the research methods of literature, questionnaire, logic analysis and expert interview, clears out the constraints of Wushu Sanda movement in middle schools, and puts forward corresponding solutions. The results show that: the effective time can not be well mastered; The site is inadequate or unsuitable; May cause fighting or disorder; Lack of martial arts education; Lack of teachers; Prone to sports injury; The lack of interest is the restricting factor that restricts the development of Wushu Sanda movement among young people. The author suggests that we should classify teaching according to the purpose of learning; Optimize the allocation and use of space; Changing the perception of parents and students; Strengthen military morality education; Recruit excellent Sanda teachers; Improve teaching quality and optimize service; Injury prevention and rehabilitation treatment are the effective countermeasures to solve the problem of Wushu Sanda movement among young people