Freed and Hopkins developed an ansatz for classifying interacting SPT phases using invertible field theories with a natural Free to Interacting (FTI) map from free fermion phases. This ansatz has been generalized to include crystalline phases and a crystalline equivalence principle (CEP). However, motivated by failure of the CEP for weak free fermions and the FTI, here we generalize the original Freed and Hopkins ansatz to a fully equivariant version for symmorphic crystallographic symmetries and show there is a natural equivariant FTI map from symmorphic crystalline weak free fermions. We further discuss why this equivariant ansatz is both mathematically and physically more natural than the spatial symmetry extension by Freed and Hopkins and why full equivariance should hold over the CEP.
The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.
It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups
G, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the
nth twisted group cohomology of
G with
Z coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if
g reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the
E8 state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in
d spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of
E8 states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for
G. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.
The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space-group SPT phases were also proposed for interacting bosonic systems. In this Rapid Communication, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point-group SPT phases for two-dimensional (2D) interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these different classes of point-group SPT phases in 2D correlated superconductors is addressed.
Freed-Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.
We present a framework for investigating the effects of interactions on crystalline symmetry-protected topological (SPT) phases. Within this framework, one can establish a direct connection between the equivalence classes of free-fermion systems and their corresponding interacting classes. A central component of this framework is the Atiyah-Hirzebruch spectral sequence, which provides a systematic way to represent crystalline SPT phases as SPT phases with internal symmetries on subspaces. We demonstrate the application of this approach through examples in various dimensions: 1d systems with U(1) and reflection symmetry, 2d systems with U(1) and C_{n} rotation symmetry, and 3d systems with U(1) and inversion symmetry.