Complete Crystalline Topological Invariants from Partial Rotations in (2+1)D Invertible Fermionic States and...

Area of Science:

• Condensed Matter Physics
• Topological Phases of Matter
• Crystalline Symmetry

Background:

• Topological phases of matter possess invariants protected by crystalline symmetry.
• Extracting these invariants from microscopic calculations has been challenging.

Purpose of the Study:

• To develop a general method for extracting many-body topological invariants from microscopic calculations.
• To provide a complete characterization of topological states in (2+1)D invertible fermionic systems.

Main Methods:

• Extraction of many-body invariants {Θ_{o}^{±}} using partial rotations in (2+1)D fermionic states.
• Application of the method in the presence of magnetic fields and non-zero Chern numbers.
• Numerical computations on the square lattice Hofstadter model.

Main Results:

• A set of many-body invariants {Θ_{o}^{±}} was successfully extracted.
• These invariants, along with Chern number (C), chiral central charge (c_{-}), and filling (ν), fully characterize the topological state.
• Invariants were obtained from a single bulk ground state, without additional defects.
• Numerical results align with conformal and topological field theory calculations.

Conclusions:

• The developed method provides a robust way to identify and characterize topological phases protected by crystalline symmetries.
• The findings extend the understanding of topological phases, particularly in systems with magnetic fields.
• The results offer new insights into Hofstadter's butterfly spectrum and topological quantum chemistry.