Interacting fermionic crystalline topological phases for all 3D space groups
In plain words
Crystal symmetries such as rotations and reflections protect extra topological phases, and for interacting electrons in three dimensions the complete list for all 230 crystal symmetry groups is not known.
Precise statement
For interacting fermions with U(1) charge conservation, spin-orbit coupling and time reversal, classify 3D symmetry-protected topological phases for each of the 230 space groups, for example by real-space block-state (topological crystal) constructions, and determine which free-fermion topological crystalline insulators stay distinct with interactions. An answer is a table of classification groups per space group with explicit generators.
What would settle it
A complete block-state or equivalent computation for all 230 space groups, cross-checked against free-fermion symmetry-indicator results.
Status in the literature
Unverified note
Bosonic crystalline phases and 2D fermionic wallpaper-group cases were classified by about 2020; to this survey's knowledge the full 3D interacting fermionic table is not complete as of 2026.