Classification of three-dimensional gapped phases including fractons
In plain words
In three dimensions some gapped phases have excitations that cannot move freely (fractons), and no complete list of 3D gapped phases that includes them exists.
Precise statement
Classify gapped phases of 3D local bosonic and fermionic lattice Hamiltonians up to adiabatic deformation and stacking with decoupled 2D gapped layers (foliation equivalence), including fracton phases whose ground-state degeneracy grows with system size. Existing classifications of 3D topological orders (Lan, Kong, Wen 2018; Lan, Wen 2019) assume no fracton excitations. An answer is a set of invariants that distinguish all phases, including a criterion separating foliated from non-foliated fracton orders.
What would settle it
A set of invariants for 3D gapped lattice phases shown to distinguish all known fracton models and to be complete within a stated framework.
Status in the literature
Unverified note
3D bosonic topological orders with bosonic (Lan, Kong, Wen 2018) and fermionic (Lan, Wen 2019) pointlike excitations were classified for non-fracton orders; a classification including fracton orders is incomplete as of 2026.