{"paper":{"title":"Symmetry Protected Topological Hopf Insulator and its Generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.str-el","authors_text":"Cenke Xu, Chunxiao Liu, Farzan Vafa","submitted_at":"2016-12-15T02:02:47Z","abstract_excerpt":"We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry $\\mathcal{C}^\\prime$, a generalized particle-hole symmetry that gives the Hopf insulator a $\\mathbb{Z}_2$ classification. The $4d$ analogue of the Hopf insulator with symmetry $\\mathcal{C}^\\prime$ has the same $\\mathbb{Z}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as \"Chern-insulator$ \\rtimes S^1$\" and \"Chern-insulator$\\rtimes T^2$\" respectively. We also discuss the relati"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04905","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}