Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter

Area of Science:

• Condensed Matter Physics
• Materials Science
• Quantum Mechanics

Background:

• Local topological markers are crucial for classifying topological phases in materials lacking translation invariance.
• The Chern marker is effective in even dimensions, but analogous local markers for odd dimensions are lacking.
• Characterizing topological phases in disordered or amorphous materials presents unique challenges.

Purpose of the Study:

• To develop general analytic expressions for local topological markers in odd dimensions.
• To introduce practical tools for classifying free-fermion topological states protected by local symmetries.
• To demonstrate the application of these markers in characterizing amorphous topological materials.

Main Methods:

• Introduction of a one-parameter family of single-particle density matrices (P_ϑ).
• Interpretation of the parameter ϑ as an additional dimension to calculate the Chern marker.
• Development of analytic expressions for the Chiral marker (Z) and Chern-Simons marker (Z_2).

Main Results:

• General analytic expressions for local topological markers in odd dimensions are provided.
• A Chiral marker, equivalent to the chiral winding number in translationally invariant systems, is introduced.
• A Chern-Simons marker, characterizing nonchiral phases in odd dimensions, is presented.
• The markers successfully classified amorphous topological superconductors (Z) and topological insulators (Z_2) in 3D.

Conclusions:

• The developed local markers provide a powerful framework for characterizing topological phases in odd dimensions.
• These markers are applicable to systems lacking translation invariance, such as amorphous materials.
• The findings offer new avenues for exploring and classifying novel topological states of matter.