Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems

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.

17 pages, 2 figures, submitted

References in corpus (20)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Field Theory of Time-Reversal Invariant Insulators
• On the $Z_2$ classification of Quantum Spin Hall Models
• An experimental observation of geometric phases for mixed states using NMR interferometry
• Abelian and Non-Abelian Quantum Geometric Tensor
• Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
• Topological indices for open and thermal systems via Uhlmann's phase
• Measurement of geometric phase for mixed states using single photon interferometry
• Symmetry-protected Topological Phases at Finite Temperature
• Observation of nonadditive mixed state phases with polarized neutrons
• Ubiquity of zeros of Loschmidt amplitude for mixed states in different physical processes and their implications
• Topological Uhlmann phase transitions for a spin-j particle in a magnetic field
• Finite-temperature topological phase transitions of spin-$j$ systems in Uhlmann processes: General formalism and experimental protocols
• Local geometry and quantum geometric tensor of mixed states
• Non-Abelian quantum geometric tensor in degenerate topological semimetals
• Uhlmann phase of coherent states and the Uhlmann-Berry correspondence
• Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases
• Uhlmann quench and geometric dynamic quantum phase transition of mixed states
• Holonomic implementation of CNOT gate on topological Majorana qubits