{"work":{"id":"d1714641-e909-476f-8215-c1fc8a7fc3d3","openalex_id":null,"doi":null,"arxiv_id":"1106.4772","raw_key":null,"title":"Symmetry protected topological orders and the group cohomology of their symmetry group","authors":null,"authors_text":"X","year":2011,"venue":"cond-mat.str-el","abstract":"Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\\Psi}, H^{1+d}[G_{\\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\\Psi} the symmetry group of the ground states.","external_url":"https://arxiv.org/abs/1106.4772","cited_by_count":null,"metadata_source":"pith","metadata_fetched_at":"2026-05-24T00:35:56.212958+00:00","pith_arxiv_id":"1106.4772","created_at":"2026-05-11T12:41:04.010378+00:00","updated_at":"2026-05-24T00:35:56.212958+00:00","title_quality_ok":true,"display_title":"Chen, Z.-C","render_title":"Chen, Z.-C"},"hub":{"state":{"work_id":"d1714641-e909-476f-8215-c1fc8a7fc3d3","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":13,"external_cited_by_count":null,"distinct_field_count":3,"first_pith_cited_at":"2014-12-16T20:10:29+00:00","last_pith_cited_at":"2026-05-15T18:00:00+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-05-26T03:36:16.777858+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"background","n":8}],"polarity_counts":[{"context_polarity":"background","n":8}],"runs":{},"summary":{},"graph":{},"authors":[]}}