Uhlmann number in translational invariant systems.

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory

Background:

• Topological invariants like the Chern number characterize quantum systems.
• Understanding topological properties at finite temperatures is crucial for realistic systems.

Purpose of the Study:

• To define and utilize the Uhlmann number for characterizing the topology of 2D fermionic systems at finite temperatures.
• To investigate topological phase transitions in paradigmatic systems using the Uhlmann number.

Main Methods:

• Definition of the Uhlmann number as an extension of the Chern number.
• Application of linear response theory to connect geometric quantities with physical observables.
• Derivation of a finite-temperature generalization of the Thouless-Kohmoto-Nightingale-den Nijs formula.

Main Results:

• The Uhlmann number successfully describes the topology of 2D fermionic systems at finite temperatures.
• The mean Uhlmann curvature and Uhlmann number are linked to dynamical susceptibility and conductivity, respectively.
• A generalized Thouless-Kohmoto-Nightingale-den Nijs formula valid at non-zero temperatures is derived.

Conclusions:

• The Uhlmann number provides a powerful tool for analyzing topological properties of quantum systems at finite temperatures.
• The established links between geometric and physical quantities offer new avenues for experimental verification.