Anomaly Indicators for Time-Reversal Symmetric Topological Orders

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Topological Phases of Matter

Background:

• Topological orders with time-reversal symmetry can be anomalous.
• These anomalous orders cannot exist in 2D systems and require 3D bulk realization.
• Symmetry-protected topological phases are crucial for understanding these phenomena.

Purpose of the Study:

• To propose new quantitative measures, termed anomaly indicators, for detecting anomalous time-reversal symmetric topological orders.
• To provide tools for classifying topological orders based on their dimensional requirements.
• To investigate the relationship between anyon properties and the existence of anomalies.

Main Methods:

• Formulating anomaly indicators using quantum dimensions, topological spins, and time-reversal properties of anyons.
• Developing indicators specific to bosonic systems (η₂) and fermionic systems in the DIII class (ηf).
• Relating these indicators to known classification schemes and anomalies.

Main Results:

• Introduced two novel anomaly indicators, η₂ and ηf.
• Expressed these indicators in terms of fundamental anyon characteristics.
• Conjectured the ability of η₂ (with η₁) to detect Z₂ anomalies in bosonic systems.
• Conjectured the ability of ηf to detect Z₁₆ anomalies in fermionic systems.

Conclusions:

• The proposed anomaly indicators provide a powerful method for identifying anomalous time-reversal symmetric topological orders.
• These indicators offer a pathway to experimentally or theoretically verify the nature of topological phases.
• The findings contribute to a deeper understanding of topological phase classification and the role of symmetry.