Real-space recipes for general topological crystalline states

TL;DR

This paper introduces a comprehensive real-space construction scheme for classifying all topological crystalline states, both bosonic and fermionic, in various dimensions, including interacting cases, using building blocks and connectors.

Contribution

It provides a unified real-space framework for constructing and classifying all topological crystalline states, extending beyond non-interacting limits and including interactions.

Findings

Complete classification of bosonic topological crystalline states in 2D and 3D.

Analytical proof of the boson classification using spectral sequence expansion.

Scheme applicable to all wallpaper and space groups, covering free and interacting states.

Abstract

Topological crystalline states are short-range entangled states jointly protected by onsite and crystalline symmetries. While the non-interacting limit of these states, e.g., the topological crystalline insulators, have been intensively studied in band theory and have been experimentally discovered, the classification and diagnosis of their strongly interacting counterparts are relatively less well understood. Here we present a unified scheme for constructing all topological crystalline states, bosonic and fermionic, free and interacting, from real-space "building blocks" and "connectors". Building blocks are finite-size pieces of lower dimensional topological states protected by onsite symmetries alone, and connectors are "glue" that complete the open edges shared by two or multiple pieces of building blocks. The resulted assemblies are selected against two physical criteria we call the "no-open-edge condition" and the "bubble equivalence", which, respectively, ensure that each selected assembly is gapped in the bulk and cannot be deformed to a product state. The scheme is then applied to obtaining the full classification of bosonic topological crystalline states 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 real-space recipes give the complete set of topological crystalline states for bosons and fermions, and prove the boson case analytically using a spectral sequence expansion of group cohomology.