Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1805.04753 · Physical review. B./Physical review. B

Thermal Uhlmann-Chern number from the Uhlmann connection for extracting topological properties of mixed states

Yan He, Hao Guo, Chih-Chun Chien

Published Jun 22, 2018• 19 citations
arXivHTMLPDF

Abstract

The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so-called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well-defined global section. This property implies that the Uhlmann connection is topologically trivial and, as a consequence, the corresponding Chern character vanishes. We propose a modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the nontopological thermal Uhlmann Chern number, which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.

DOI
10.1103/physrevb.97.235141

Related papers

Thermal Uhlmann-Chern number: Bridging pure and mixed states

Xin Wang, Xu-Yang Hou, Yan He +1 more

Topological properties of quantum systems at finite temperatures, described by mixed states, pose significant challenges due to the triviality of the Uhlmann bundle. By inserting the density matrix into the Chern character, we introduce the thermal Uhlmann-Chern number, a generalization of the Chern number that reduces to the pure-state value in the zero-temperature limit and vanishes at infinite temperature, providing a framework to study the temperature-dependent evolution of topological features in mixed states. We provide, for the first time, a rigorous mathematical proof that the first- and higher-order Uhlmann-Chern numbers converge to the corresponding Chern numbers in the zero-temperature limit, differing only by a factor of $1/D$ for $D$-fold degenerate ground states. We demonstrate the utility of this framework through applications to a two-level system, the coherent state model, the 2D Haldane model, and a four-band model, highlighting the temperature-dependent behavior of topological invariants. Our results establish a robust bridge between the topological properties of pure and mixed states, offering new insights into finite-temperature topological phases.

arXiv:2506.18022•Physical review. B./Physical review. B•Nov 8, 2025•2 citations

Chern number and Berry curvature for Gaussian mixed states of fermions

Lukas Wawer, Michael Fleischhauer

We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) to two-dimensional band structures. In contrast to the geometric Uhlmann phase for density matrices, the EGP leads to a proper Chern number for Gaussian, finite-temperature, or nonequilibrium steady states. The Chern number can be expressed as an integral of the ground-state Berry curvature of a fictitious lattice Hamiltonian, constructed from single-particle correlations. For the Chern number to be nonzero the fictitious Hamiltonian has to break time-reversal symmetry.

arXiv:2104.12115•Physical review. B./Physical review. B•Sep 15, 2021•18 citations

Mixed-State Topology in Non-Hermitian Systems

Shou-Bang Yang, Pei-Rong Han, Wen Ning +3 more

Non-Hermitian (NH) systems, owing to the existence of exceptional point (or ring and surface), exhibit exotic topological features which are inaccessible in Hermitian systems. While current studies on NH topology has primarily focused on pure states at zero temperature, the topological properties of mixed states remain largely unexplored. In this work, we investigate the mixed-state topology in two-dimensional NH systems using the Uhlmann phase and the thermal Uhlmann-Chern number, both structured via the Uhlmann connection at specific temperatures, revealing distinct topological characteristics compared to those of pure states. Furthermore, we extend our analysis to mixed states in three-dimensional Abelian and four-dimensional non-Abelian NH systems, confirming the existence of the higher-order mixed-state topology. Our study establishes a conceptual and practical pathway for exploring topological phenomena in the mixed-state regime of NH physics.

quant-pharXiv:2602.10831•Feb 12, 2026

Comparison of finite-temperature topological indicators based on Uhlmann connection

Ye Zhang, Aixin Pi, Yan He +1 more

Two indicators of finite-temperature topological properties based on the Uhlmann connection, one generalizing the Wilson loop to the Uhlmann-Wilson loop and the other generalizing the Berry phase to the Uhlmann phase, are constructed explicitly for a time-reversal invariant topological insulators with a ${Z}_{2}$ index. While the phases of the eigenvalues of the Wilson loop reflect the ${Z}_{2}$ index of the model at zero temperature, it is found that the signature from the Uhlmann-Wilson loop gradually fades away as temperature increases. On the other hand, the Berry phase exhibits quantization due to the underlying holonomy group. The Uhlmann phase retains the quantization at finite temperatures and serves as an indicator of topological properties. A phase diagram showing where jumps of the Uhlmann phase can be found is presented. By modifying the model to allow higher winding numbers, finite-temperature topological regimes sandwiched between trivial regimes at high and low temperatures may emerge.

arXiv:2108.05016•Physical review. B./Physical review. B•Oct 21, 2021•14 citations

Theory of the Uhlmann Phase in Quasi-Hermitian Quantum Systems

Xu-Yang Hou, Xin Wang, Hao Guo

Geometric phases play a fundamental role in understanding the geometric structure of quantum states, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.

quant-pharXiv:2603.01908•Physical review. B./Physical review. B•Mar 3, 2026•0 citations

Measuring Mixed-State Topological Invariant in Open Photonic Quantum Walk

Qin-Qin Wang, Xiao-Ye Xu, Yong-Jian Han +2 more

Pure-state manifestations of geometric phase are well established and have found applications across essentially all branches of physics, yet their generalization to mixed-state regimes remains largely unexplored experimentally. The Uhlmann geometric phase offers a natural extension of pure-state paradigms and can exhibit a topological character. However, observation of this invariant is impeded by the incompatibility between Uhlmann parallel transport and Hamiltonian dynamics, as well as the difficulty of preparing topologically nontrivial mixed states. To address this challenge, we report an experimentally accessible protocol for directly measuring the mixed-state topological invariant. By engineering controlled nonunitary dynamics in a photonic quantum walk, we prepare topologically nontrivial mixed states from a trivial initial state. Furthermore, by machine-learning the full density matrix in momentum space, we directly extract the quantized geometric phase of the nontrivial mixed states. These results highlight a geometric phase framework that naturally extends to open quantum systems both in and out of thermal equilibrium.

quant-pharXiv:2512.24857•Jan 1, 2026•0 citations
The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so-called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well-defined global section. This property implies that the Uhlmann connection is topologically trivial and, as a consequence, the corresponding Chern character vanishes. We propose a modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the nontopological thermal Uhlmann Chern number, which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.
Topological properties of quantum systems at finite temperatures, described by mixed states, pose significant challenges due to the triviality of the Uhlmann bundle. By inserting the density matrix into the Chern character, we introduce the thermal Uhlmann-Chern number, a generalization of the Chern number that reduces to the pure-state value in the zero-temperature limit and vanishes at infinite temperature, providing a framework to study the temperature-dependent evolution of topological features in mixed states. We provide, for the first time, a rigorous mathematical proof that the first- and higher-order Uhlmann-Chern numbers converge to the corresponding Chern numbers in the zero-temperature limit, differing only by a factor of 1/D1/D1/D for DDD-fold degenerate ground states. We demonstrate the utility of this framework through applications to a two-level system, the coherent state model, the 2D Haldane model, and a four-band model, highlighting the temperature-dependent behavior of topological invariants. Our results establish a robust bridge between the topological properties of pure and mixed states, offering new insights into finite-temperature topological phases.
We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) to two-dimensional band structures. In contrast to the geometric Uhlmann phase for density matrices, the EGP leads to a proper Chern number for Gaussian, finite-temperature, or nonequilibrium steady states. The Chern number can be expressed as an integral of the ground-state Berry curvature of a fictitious lattice Hamiltonian, constructed from single-particle correlations. For the Chern number to be nonzero the fictitious Hamiltonian has to break time-reversal symmetry.
Non-Hermitian (NH) systems, owing to the existence of exceptional point (or ring and surface), exhibit exotic topological features which are inaccessible in Hermitian systems. While current studies on NH topology has primarily focused on pure states at zero temperature, the topological properties of mixed states remain largely unexplored. In this work, we investigate the mixed-state topology in two-dimensional NH systems using the Uhlmann phase and the thermal Uhlmann-Chern number, both structured via the Uhlmann connection at specific temperatures, revealing distinct topological characteristics compared to those of pure states. Furthermore, we extend our analysis to mixed states in three-dimensional Abelian and four-dimensional non-Abelian NH systems, confirming the existence of the higher-order mixed-state topology. Our study establishes a conceptual and practical pathway for exploring topological phenomena in the mixed-state regime of NH physics.
Two indicators of finite-temperature topological properties based on the Uhlmann connection, one generalizing the Wilson loop to the Uhlmann-Wilson loop and the other generalizing the Berry phase to the Uhlmann phase, are constructed explicitly for a time-reversal invariant topological insulators with a Z2{Z}_{2}Z2​ index. While the phases of the eigenvalues of the Wilson loop reflect the Z2{Z}_{2}Z2​ index of the model at zero temperature, it is found that the signature from the Uhlmann-Wilson loop gradually fades away as temperature increases. On the other hand, the Berry phase exhibits quantization due to the underlying holonomy group. The Uhlmann phase retains the quantization at finite temperatures and serves as an indicator of topological properties. A phase diagram showing where jumps of the Uhlmann phase can be found is presented. By modifying the model to allow higher winding numbers, finite-temperature topological regimes sandwiched between trivial regimes at high and low temperatures may emerge.
Geometric phases play a fundamental role in understanding the geometric structure of quantum states, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.
Pure-state manifestations of geometric phase are well established and have found applications across essentially all branches of physics, yet their generalization to mixed-state regimes remains largely unexplored experimentally. The Uhlmann geometric phase offers a natural extension of pure-state paradigms and can exhibit a topological character. However, observation of this invariant is impeded by the incompatibility between Uhlmann parallel transport and Hamiltonian dynamics, as well as the difficulty of preparing topologically nontrivial mixed states. To address this challenge, we report an experimentally accessible protocol for directly measuring the mixed-state topological invariant. By engineering controlled nonunitary dynamics in a photonic quantum walk, we prepare topologically nontrivial mixed states from a trivial initial state. Furthermore, by machine-learning the full density matrix in momentum space, we directly extract the quantized geometric phase of the nontrivial mixed states. These results highlight a geometric phase framework that naturally extends to open quantum systems both in and out of thermal equilibrium.