© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2506.18022 · Physical review. B./Physical review. B

Thermal Uhlmann-Chern number: Bridging pure and mixed states

Xin Wang, Xu-Yang Hou, Yan He, Hao Guo

Published Nov 8, 2025• 2 citations

Abstract

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.

DOI: 10.1103/prq8-c9ns

Related papers

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

Yan He, Hao Guo, Chih-Chun Chien

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.

arXiv:1805.04753•Physical review. B./Physical review. B•Jun 22, 2018•19 citations

Uhlmann Measure in Topological Insulators and Superconductors at Finite Temperature

Oscar Viyuela, ́Angel Rivas, Miguel Angel Martin-Delgado

I will introduce the Uhlmann geometric phase as a tool to characterise density matrices of 1D and 2D topological insulators and superconductors. We achieve this goal by constructing new topological invariants called Topological Uhlmann numbers. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. New effects appear such as the existence of critical temperatures, novel thermal-topological transitions in models with high Chern numbers, breakdown of the usual bulk-edge correspondence, … Moreover, as the Uhlmann phase is an observable itself, we analyse potential measurement schemes that could be applicable to current experimental setups like cold atoms in optical lattices.

Bulletin of the American Physical Society•Mar 5, 2015•0 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

2D Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

Oscar Viyuela, ́Angel Rivas, Miguel Angel Martin-Delgado

We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in 2D fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number ${\rm n}_{\rm U}$. With it, we study thermal topological phases in several 2D models of topological insulators and superconductors, computing phase diagrams where the temperature $T$ is on equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two non-trivial phases in a model with high Chern numbers. At small temperature we recover the standard topological phases as the Uhlmann number approaches to the Chern number.

•May 23, 2014•0 citations

Topological Indices for Open and Thermal Systems Via Uhlmann’s Phase

Zhoushen Huang, Daniel P. Arovas

Two-dimensional topological phases are characterized by Thouless-Kohmoto-Nightingale--den Nijs integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmann's parallel transport for density matrices [Rep. Math. Phys. 24, 229 (1986)], an integer classification of topological phases can be defined for a finite generalized temperature $T$ or dephasing Lindbladian. This scheme reduces to the familiar Thouless-Kohmoto-Nightingale--den Nijs classification for $T<{T}_{\mathrm{c},1}$, becomes trivial for $T>{T}_{\mathrm{c},2}$, and exhibits a ``gapless'' intermediate regime where topological indices are not well defined. We demonstrate these ideas in detail, applying them to Haldane's honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multiband Chern insulators.

arXiv:1405.4567•Physical Review Letters•Aug 13, 2014•102 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

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.

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.

I will introduce the Uhlmann geometric phase as a tool to characterise density matrices of 1D and 2D topological insulators and superconductors. We achieve this goal by constructing new topological invariants called Topological Uhlmann numbers. Since this phase is formulated for general mixed quantum states, it provides a way to extend topological properties to finite temperature situations. New effects appear such as the existence of critical temperatures, novel thermal-topological transitions in models with high Chern numbers, breakdown of the usual bulk-edge correspondence, … Moreover, as the Uhlmann phase is an observable itself, we analyse potential measurement schemes that could be applicable to current experimental setups like cold atoms in optical lattices.

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.

We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in 2D fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number

{\rm n}_{\rm U}

. With it, we study thermal topological phases in several 2D models of topological insulators and superconductors, computing phase diagrams where the temperature

T

is on equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two non-trivial phases in a model with high Chern numbers. At small temperature we recover the standard topological phases as the Uhlmann number approaches to the Chern number.

Two-dimensional topological phases are characterized by Thouless-Kohmoto-Nightingale--den Nijs integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmann's parallel transport for density matrices [Rep. Math. Phys. 24, 229 (1986)], an integer classification of topological phases can be defined for a finite generalized temperature

T

or dephasing Lindbladian. This scheme reduces to the familiar Thouless-Kohmoto-Nightingale--den Nijs classification for

T<{T}_{\mathrm{c},1}

, becomes trivial for

T>{T}_{\mathrm{c},2}

, and exhibits a ``gapless'' intermediate regime where topological indices are not well defined. We demonstrate these ideas in detail, applying them to Haldane's honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multiband Chern insulators.

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.