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Which other phases share this obstruction is open.","posed_since":"2017","precise":"Ringel and Kovrizhin (2017) and Golan, Smith and Ringel (2020) showed that chiral topological phases for which exp(2 pi i c/24) (c the chiral central charge) differs from every topological spin cannot be realized by any local sign-free Hamiltonian. Determine whether non-chiral topological orders, gapless phases (e.g. non-Fermi liquids or Fermi surfaces with nontrivial Luttinger volume) or symmetry-enriched phases carry intrinsic sign problems, and give a complete criterion. An answer is a classification with proofs.","problem_ref":null,"references":"","settled_by":"A theorem extending the chiral criterion to all gapped phases, or explicit sign-free local models for the remaining phases.","status_note":"The chiral case covered by the topological-spin criterion was settled by 2020; non-chiral and gapless cases remain open.","title":"Which phases of matter carry an intrinsic sign 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