Rotation Symmetry-Protected Topological Phases of Fermions

TL;DR

This paper classifies interacting fermionic SPT phases with rotation and Abelian symmetries, revealing new interaction-enabled phases and clarifying the crystalline-internal symmetry correspondence.

Contribution

It demonstrates the crystalline-internal symmetry correspondence requires a modified symmetry group structure and uncovers new intrinsically fermionic SPT phases enabled by interactions.

Findings

Explicit block-state constructions support the symmetry correspondence.

New intrinsically fermionic SPT phases found in 2D and 3D with four-fold rotation.

Lieb-Schultz-Mattis-type theorems for Majorana fermionic SPTs established.

Abstract

We study classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the $\mathbb{Z}_2$ fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D with four-fold rotation. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPTs are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.