The tenfold way redux: Fermionic systems with $N$-body interactions

TL;DR

This paper extends the tenfold way classification of fermionic systems to include arbitrary N-body interactions, identifying symmetry constraints and providing a group cohomological perspective to aid topological classification.

Contribution

It systematically characterizes fermionic systems with N-body interactions using symmetry analysis and group cohomology, leading to canonical symmetry representations and Hamiltonian structures.

Findings

Identifies four types of symmetries in fermionic systems.

Derives canonical symmetry representations for each of the ten classes.

Shows the structure of N-body Hamiltonians forms an affine subspace in certain classes.

Abstract

We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary $N$-body interactions. We identify four types of symmetries that such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary $N$-body interactions in each of the ten classes. We show that the space of $N$-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address open questions on the topological classification of interacting fermionic systems.