Rotation Symmetry-Protected Topological Phases of Fermions

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.Comment: are welcom

Fermion Condensation and Gapped Domain Walls in Topological Orders

We propose the concept of fermion condensation in bosonic topological orders in two spatial dimensions. Fermion condensation can be realized as gapped domain walls between bosonic and fermionic topological orders, which are thought of as a real-space phase transitions from bosonic to fermionic topological orders. This generalizes the previous idea of understanding boson condensation as gapped domain walls between bosonic topological orders. We show that generic fermion condensation obeys a Hierarchy Principle by which it can be decomposed into a boson condensation followed by a minimal fermion condensation, which involves a single self-fermion that is its own anti-particle and has unit quantum dimension. We then develop the rules of minimal fermion condensation, which together with the known rules of boson condensation, provides a full set of rules of fermion condensation. Our studies point to an exact mapping between the Hilbert spaces of a bosonic topological order and a fermionic topological order that share a gapped domain wall.Comment: 20 pages, 2-colum