Real-space recipes for general topological crystalline states

Zhida Song,Yang Qi,Chen Fang

doi:10.1038/s41467-020-17685-5

Abstract

Topological crystalline states (TCSs) are short-range entangled states jointly protected by onsite and crystalline symmetries. Here we present a unified scheme for constructing all TCSs, bosonic and fermionic, free and interacting, from real-space building blocks and connectors. Building blocks are lower-dimensional topological states protected by onsite symmetries alone, and connectors are glues that complete the open edges shared by two or multiple building blocks. The resulted assemblies are selected against two physical criteria we call the no-open-edge condition and the bubble equivalence. The scheme is then applied to obtaining the full classification of bosonic TCSs protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our construction scheme can give the complete set of TCSs for bosons and fermions, and prove the boson case analytically using a spectral-sequence expansion.

Highlights

Topological crystalline states (TCSs) are short-range entangled states jointly protected by onsite and crystalline symmetries
SPT protected by onsite symmetries have been studied for years, and we know that bosonic SPT is classified by group cohomology of the symmetry group[1,2,3,4,10], and SPT of free fermions is classified by the K theory in the “tenfold way”[15,16]
In contrast to SPT protected by onsite symmetries are crystalline symmetry-protected topological states, or topological crystalline states (TCS)

Summary

Introduction

Topological crystalline states (TCSs) are short-range entangled states jointly protected by onsite and crystalline symmetries. Building blocks are lower-dimensional topological states protected by onsite symmetries alone, and connectors are glues that complete the open edges shared by two or multiple building blocks. For three-dimensional TCS, one considers p = 3, 2, 1, 0 blocks When all open edges are completed, that is, when the “no-open-edge condition” is met, we obtain assemblies that are (i) symmetric under onsite and spatial symmetries and (ii) gapped This does not mean that the crystal is topologically nontrivial, as we require that it cannot be deformed into a product state. Torsors are not SPT (but may be understood as fractions of SPT), and their topological properties should be separately considered

Methods

Results

Conclusion