Effect of strong interactions on the classification of fermionic phases

Determine how the established classification of zero-temperature phases of non-interacting fermionic systems changes when arbitrarily strong interactions are allowed, with the goal of identifying the interaction-stable classification of phases (including invertible phases) beyond the free-fermion framework.

Background

The periodic table of topological insulators and superconductors provides a comprehensive classification of non-interacting fermionic phases at zero temperature. However, introducing interactions can collapse or refine this classification, and understanding the interaction-stable classification is a central problem in condensed matter theory.

This paper focuses on invertible (short-range entangled) phases in two dimensions and defines indices intended to capture interaction-robust invariants. Resolving how interactions alter the full classification would contextualize these indices within a complete interacting-phase framework.

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)

Extending the results to quantum many-body systems remains an open problem. Interactions or bosonic degrees of freedom may permit odd values excluded by our results.

— 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