Which phases of matter carry an intrinsic sign problem
In plain words
Many phases of matter with one-way edge currents provably cannot be simulated sign-free by any model built from local terms. Which other phases share this obstruction is open.
Precise statement
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.
What would settle it
A theorem extending the chiral criterion to all gapped phases, or explicit sign-free local models for the remaining phases.
Status in the literature
The chiral case covered by the topological-spin criterion was settled by 2020; non-chiral and gapless cases remain open.