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In the case when $\\epsilon_r$ is real, to the first order in $\\epsilon_r$, the band structures show a phase transition from an exact phase with real quasienergies to a broken phase with complex quasienergies inside a region of wave vector space, the so-called k-gap."},"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":"1804.07095","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.optics","submitted_at":"2018-04-19T11:38:35Z","cross_cats_sorted":[],"title_canon_sha256":"b8b8ce7213a907398c2123a86e4ca775d870257e1e4ada6c032b0fc62db31640","abstract_canon_sha256":"6c17172b26c343539d0c46247cefed4b3a085b8617779e8d0d7cf9955e908ad8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:00.114200Z","signature_b64":"mwv5nXZedAOmgtcQjqbFacwFCEeGaUzw8SB3bOmypCcqAJKHPS4ko5NzG9sxqd8h6/apr3FjbnvQrh0qKhXjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5fb2adb78d4507e53ecc6db927ffdf34553a7a9edcc162e0501de13c836ba4c","last_reissued_at":"2026-05-18T00:07:00.113578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:00.113578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Photonic Floquet media with a complex time-periodic permittivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"physics.optics","authors_text":"C. 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