Free and Interacting Short-Range Entangled Phases of Fermions: Beyond the Ten-Fold Way

TL;DR

This paper extends the classification of free fermionic phases beyond the ten-fold way by incorporating arbitrary on-site symmetries, analyzing their stability under interactions, and identifying phases that are intrinsically interacting.

Contribution

It provides a generalized classification framework for free fermionic phases with arbitrary symmetries and studies their stability and realizability in the presence of interactions.

Findings

Classification extended to arbitrary symmetries and dimensions.

Identification of free phases destabilized by interactions.

Discovery of intrinsically interacting fermionic phases.

Abstract

We extend the periodic table of phases of free fermions in the ten-fold way symmetry classes to a classification of free fermionic phases protected by an arbitrary on-site unitary symmetry $\hat G$ in an arbitrary dimension. The classification is described as a function of the real representation theory of $\hat G$ and the data of the original periodic table. We also systematically study in low dimensions the relationship between the free invariants and the invariants of short-range entangled interacting phases of fermions. Namely we determine whether a given symmetry protected phase of free fermions is destabilized by sufficiently strong interactions or it remains stable even in the presence of interactions. We also determine which interacting fermionic phases cannot be realized by free fermions. Examples of both destabilized free phases and intrinsically interacting phases are common in all dimensions.