Fermionic Symmetry Protected Topological Phases and Cobordisms

TL;DR

The paper tests a cobordism-based framework for classifying fermionic SPT phases with ${\mathbb Z}_2$ symmetry (time-reversal or internal) by constructing long-distance effective actions and matching to known results in space-time dimensions $d \le 4$. It develops a systematic scheme using Spin/Pin bordism with torsion, showing that all $d \le 3$ phases align with free-fermion classifications and that higher dimensions host inherently interacting SPTs, with richer structures such as eta- and Arf-invariants governing the topological actions. A unified description across symmetry types is provided via twisted cobordism groups and decorated domain-wall constructions, including Smith isomorphism relations and domain-wall decorations that generate lower-dimensional SPTs. The work highlights both the power and limitations of bordism-based classifications, predicting nontrivial interacting phases in higher dimensions and pointing to nonlocal features of fermionic SPT effective actions that warrant further study.

Abstract

It has been proposed recently that interacting Symmetry Protected Topological (SPT) phases can be classified using cobordism theory. We test this proposal in the case of fermionic SPT phases with Z/2 symmetry, where Z/2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimension less than or equal to 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases.