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A hopfion is a confined topological soliton of $\\mathbf S(\\mathbf r)$ characterized by an {\\it emergent} magnetic field $B_\\gamma(\\mathbf r) \\equiv \\epsilon_{\\alpha\\beta\\gamma} \\,\\mathbf S\\cdot(\\nabla_\\alpha \\mathbf S\\times \\nabla_\\beta \\mathbf S)/4 \\neq 0$ with vanishing average value $\\langle \\mathbf B(\\mathbf r)\\rangle = 0$. We evaluate the scattering amplitude in the opposite limits of large and small hopfion radius $R$ using t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2104.09584","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.mes-hall","submitted_at":"2021-04-19T19:34:13Z","cross_cats_sorted":[],"title_canon_sha256":"546a721ab9d0dcb056e7389dbca67911c7b6250ab5b6bae979f0e7b4f95df034","abstract_canon_sha256":"29f057fa000576dca83aa98a596a14325a047acb3638a791476d0d904b47dfb6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:04:50.927233Z","signature_b64":"fUGYooiIucJFfNnNPIl7DDsJzNurnYd9G/U/lWKwiQApRC6Viu0lyWLknOIvUwuVcsZCZD8xABEDOX0alJ4fBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35f7450a4bf6450cb385cc76aedd9ddb9f9945c2d2d2f236a901f28324b0b04b","last_reissued_at":"2026-07-05T03:04:50.926882Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:04:50.926882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Electronic scattering off a magnetic hopfion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.mes-hall","authors_text":"Domenico Andreoli, Jiadong Zang, Sergey S. Pershoguba","submitted_at":"2021-04-19T19:34:13Z","abstract_excerpt":"We study scattering of itinerant electrons off a magnetic hopfion in a three-dimensional metallic magnet described by a magnetization vector $\\mathbf S(\\mathbf r)$. A hopfion is a confined topological soliton of $\\mathbf S(\\mathbf r)$ characterized by an {\\it emergent} magnetic field $B_\\gamma(\\mathbf r) \\equiv \\epsilon_{\\alpha\\beta\\gamma} \\,\\mathbf S\\cdot(\\nabla_\\alpha \\mathbf S\\times \\nabla_\\beta \\mathbf S)/4 \\neq 0$ with vanishing average value $\\langle \\mathbf B(\\mathbf r)\\rangle = 0$. 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