Interaction effects on the classification of non-interacting fermionic phases

Determine how the established zero-temperature classification of phases of non-interacting fermionic systems changes when arbitrarily strong interactions are allowed, with particular attention to short-range entangled (invertible) phases where a complete interacting classification may be tractable.

Background

The paper begins by recalling that the classification of phases of non-interacting fermionic systems at zero temperature is well understood, referencing prior work on the periodic table of topological insulators and superconductors. However, once interactions are introduced, the free-fermion classification may collapse or refine, and a general interacting classification is largely unknown.

The authors note that while the fully general problem appears intractable, progress may be possible for short-range entangled (invertible) phases, motivating their construction of an index intended to serve as an invariant for such phases without assuming additional symmetries.

References

An important open problem is to determine how this classification changes when we allow arbitrarily strong interaction.

— An index for invertible phases of two-dimensional quantum spin systems  (2410.02059 - Sopenko, 2024) in Section 1 (Introduction)

For a system with onsite charge conservation, the question is whether such an index would require a four-dimensional quantum Hall response corresponding to an odd value of the free-fermion invariant ${\rm ch}_2$. Constructing the index and realizing an odd response are both open problems.

— Symmetry-enforced topological parity: revisiting $2\mathbb{Z}$ classifications beyond cellwise symmetry actions  (2609.37106 - Shiozaki, 29 Sep 2026) in Section 12, Conclusions and outlook