Real-space recipes for general topological crystalline states.

Area of Science:

• Condensed Matter Physics
• Quantum Materials
• Symmetry-Protected Topological States

Background:

• Topological crystalline states (TCSs) are quantum states protected by both onsite and crystalline symmetries.
• Existing methods for classifying TCSs can be complex and fragmented.

Purpose of the Study:

• To present a unified, real-space construction scheme for all bosonic and fermionic TCSs.
• To provide a complete classification of TCSs based on symmetry groups and dimensionality.

Main Methods:

• Constructing TCSs from lower-dimensional topological building blocks and symmetry-preserving connectors.
• Applying selection criteria: no-open-edge condition and bubble equivalence.
• Utilizing spectral-sequence expansion for analytical proof in the bosonic case.

Main Results:

• A comprehensive scheme for constructing and classifying TCSs, applicable to both free and interacting systems.
• Full classification of bosonic TCSs for 2D wallpaper and 3D space groups.
• Demonstrated the scheme's ability to encompass all known TCSs.

Conclusions:

• The presented unified scheme offers a powerful and systematic approach to understanding topological crystalline states.
• This work provides a complete classification of bosonic TCSs and a framework for fermionic TCSs.
• The real-space construction method simplifies the identification and characterization of complex topological phases.