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Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems

Xin Wang, Hao Guo, Chih-Chun Chien

TL;DR

The work investigates whether the finite-temperature Uhlmann phase for a degenerate four-level system reduces to the scalar Wilczek-Zee phase of its degenerate ground subspace as $T\to0$. By deriving explicit Uhlmann and Wilczek-Zee connections for a two-doubly-degenerate model and two concrete examples (a simple sphere-like path and a 4D tight-binding model), the authors show that the zero-temperature limit of the Uhlmann phase need not reproduce the scalar WZ phase, due to obstructions from Dirac points and symmetry-induced zero-field axes. In one example the two phases agree in the $T\to0$ limit, while in another they diverge, with the Uhlmann phase capturing Dirac-point singularities even as the scalar WZ phase vanishes along a zero-field axis. The results highlight nuanced, case-dependent links between mixed-state and pure-state topologies at finite temperature, with implications for finite-temperature topology, holonomic quantum computation, and experimental simulations of Uhlmann phases.

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.

Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems

TL;DR

The work investigates whether the finite-temperature Uhlmann phase for a degenerate four-level system reduces to the scalar Wilczek-Zee phase of its degenerate ground subspace as . By deriving explicit Uhlmann and Wilczek-Zee connections for a two-doubly-degenerate model and two concrete examples (a simple sphere-like path and a 4D tight-binding model), the authors show that the zero-temperature limit of the Uhlmann phase need not reproduce the scalar WZ phase, due to obstructions from Dirac points and symmetry-induced zero-field axes. In one example the two phases agree in the limit, while in another they diverge, with the Uhlmann phase capturing Dirac-point singularities even as the scalar WZ phase vanishes along a zero-field axis. The results highlight nuanced, case-dependent links between mixed-state and pure-state topologies at finite temperature, with implications for finite-temperature topology, holonomic quantum computation, and experimental simulations of Uhlmann phases.

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.

Paper Structure

This paper contains 24 sections, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Four-level model with two doubly degenerate spaces
  3. The Hamiltonian
  4. Uhlmann holonomy
  5. Wilczek-Zee holonomy
  6. Scalar Wilczek-Zee phase
  7. Examples
  8. A simple illustration
  9. 4D tight-binding model
  10. Correspondence between the Uhlmann and scalar WZ phases
  11. Complications from degeneracy
  12. Correspondence between Uhlmann and WZ phases is not guaranteed
  13. Obstructions to Phase Relations
  14. Zero-field axis: Geometric Origins and Physical Implications
  15. Singular behavior of the Uhlmann connection and phase at Dirac points
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Uhlmann phase $\theta_{\text{U}}$ as a function of temperature $T$ for the simple case. A $\pi$-jump occurs at $T\approx0.75R$.
  • Figure 2: (Top panel) Uhlmann phase $\theta_{\text{U}}$ as a function of temperature $T$ on a semi-logarithmic scale for $m = -3$. A $\pi$-jump occurs at $T \approx 0.75R_0$, signaling a temperature-induced topological phase transition. (Bottom panel) Uhlmann-phase diagram of the 4D tight-binding model. The numerical values of $T_c$ are marked by the blue circles. The background and dome-shape regions correspond to the trivial ($\theta_{\text{U}} = 0$) and topological ($\theta_{\text{U}} = \pi$) phases, respectively. The scalar WZ phase at $T=0$, however, vanishes after the system traverses the path $C$ regardless of $m$. Here $R_0=R(m=-3)$.