Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems

TL;DR

This paper explores the relationship between Uhlmann and Wilczek-Zee phases in degenerate quantum systems, revealing their differing behaviors at zero temperature and near Dirac points, with implications for quantum topology.

Contribution

It provides an exact analysis of the Uhlmann and WZ phases in a four-level degenerate system, highlighting their distinct behaviors and conditions for equivalence.

Findings

Uhlmann phase may differ from scalar WZ phase at zero temperature due to Dirac point obstructions.

Uhlmann phase detects singularities at Dirac points, unlike WZ phase.

WZ connection vanishes along a zero-field axis, contrasting with Uhlmann phase behavior.

Abstract

The Wilczek-Zee (WZ) holonomy arises in degenerate states while the Uhlmann holonomy characterizes finite-temperature topology. We investigate possible relationships between the Uhlmann phase and the scalar WZ phase, which reflects the Uhlmann and WZ holonomy respectively, in an exemplary four-level model with two doubly degenerate subspaces. Through exact solutions, we contrast the behavior of the Uhlmann and WZ connections and their associated phases. In the zero-temperature limit, the Uhlmann phase may or may not agree with the scalar WZ phase of the degenerate ground states due to obstructions from the Hamiltonian manifested as Dirac points. This is in stark contrast to non-degenerate systems where the correspondence between the Uhlmann and Berry phases in general holds. Our analyses further show that for the example studied here, the Uhlmann phase catches the singular behavior at the Dirac points while the WZ connection and scalar WZ phase vanish along a zero-field axis. We also briefly discuss possible experimental implications.