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It is known that above the lower critical dimension $d_{\\mathrm{lc}}=4$ the infinite models undergo a paramagnetic-ferromagnetic transition for $N>N_c$ ($N_c=2.835$ for random field and $N_c=9.441$ for random anisotropy). For $N<N_c$ and $d<d_{\\mathrm{lc}}$ there exists a quasi-long-range ordered phase with zero order parameter and a power-law decay of spin correlations. Using functional renormalization group we derive the surface s"},"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":"1205.1338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2012-05-07T10:46:56Z","cross_cats_sorted":[],"title_canon_sha256":"973b5a42e0e933185fa30bda15d0fc8986c384b0fc9c36fd50c5bdcad39a210e","abstract_canon_sha256":"c3b486324ee8a2fcd292368988528dca1e71fb8cfa6ee99f5cdeb929adffc802"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:16.632612Z","signature_b64":"gzhwVylXd7cx6wkV/nq7HaDjjwm4nnM8kvOpA8LYW9i8pJwvQie8CWI8qxMQ12xTbLgn+0qdc35OibJc7c6dBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ee38264d0eb762c538a9f0f3c7353f3c792c54ae4f65515fa81c5f5d8153a42","last_reissued_at":"2026-05-18T03:46:16.631930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:16.631930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random field and random anisotropy O(N) spin systems with a free surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.dis-nn","authors_text":"Andrei A. Fedorenko","submitted_at":"2012-05-07T10:46:56Z","abstract_excerpt":"We study the surface scaling behavior of a semi-infinite $d$-dimensional O(N) spin system in the presence of quenched random field and random anisotropy disorders. It is known that above the lower critical dimension $d_{\\mathrm{lc}}=4$ the infinite models undergo a paramagnetic-ferromagnetic transition for $N>N_c$ ($N_c=2.835$ for random field and $N_c=9.441$ for random anisotropy). For $N<N_c$ and $d<d_{\\mathrm{lc}}$ there exists a quasi-long-range ordered phase with zero order parameter and a power-law decay of spin correlations. 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