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:2402.15964 · Chinese Physics Letters

Probing the Topology of Fermionic Gaussian Mixed States with U(1) symmetry by Full Counting Statistics

Liang Mao, Hui Zhai, Fan Yang

Published Apr 21, 2025• 8 citations
arXivHTMLPDF

Abstract

Abstract Topological band theory has been studied for free fermions for decades, and one of the most profound physical results is the bulk-boundary correspondence. Recently a focus in topological physics is extending topological classification to mixed states. Here, we focus on Gaussian mixed states for which the modular Hamiltonians of the density matrix are quadratic free fermion models with U(1) symmetry and can be classified by topological invariants. The bulk-boundary correspondence is then manifested as stable gapless modes of the modular Hamiltonian and degenerate spectrum of the density matrix. In this article, we show that these gapless modes can be detected by the full counting statistics, mathematically described by a function introduced as F(θ). A divergent derivative at θ = π can be used to probe the gapless modes in the modular Hamiltonian. Based on this, a topological indicator, whose quantization to unity senses topologically nontrivial mixed states, is introduced. We present the physical intuition of these results and also demonstrate these results with concrete models in both one- and two-dimensions. Our results pave the way for revealing the physical significance of topology in mixed states.

DOI
10.1088/0256-307x/42/6/067401

Related papers

Bulk-Boundary Correspondence and Singularity-Filling in Long-Range Free-Fermion Chains

Nick G. Jones, Ryan Thorngren, Ruben Verresen

The bulk-boundary correspondence relates topologically protected edge modes to bulk topological invariants and is well understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent α, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with α>1, linking together the quantized winding invariant, bulk topological string-order parameters, and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalize these results by (i) identifying a family of BDI chains with α<1 where our results still hold and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when α-1 exceeds the dynamical critical exponent.

arXiv:2211.15690•Physical Review Letters•Jun 16, 2023•21 citations

Measures of Chirality in Mixed-State Topological Phases

Shijun Sun, Bader Aldossari, Rasmit Devkota +1 more

What does it mean for a mixed-state topological phase to be chiral? Mathematically, chirality can be sharply characterized through the symmetry algebra of the mixed state. Physically, however, the question is far more subtle. In pure states, chirality in topological phase is tied to a web of familiar diagnostics, involving bulk-boundary correspondence, a gapless entanglement spectrum, a nontrivial modular commutator, and a quantized thermal Hall response. We show that none of these diagnostics remain reliable in mixed states. Instead, for decohered topological phases with a known error-free parent state, we propose two relative-entropy-based measures that can diagnose chirality, with one of them further extracting the chiral central charge. As a concrete example, we use these measures to analyze the $em^2$-decohered $\mathbb{Z}_3$ toric code, identifying the chirality transition to be in the random-bond Potts class for the particle-hole symmetric channel, and the polarized random bond Potts class for asymmetric channels. Our results emphasize how mixed-state topology demands intrinsically new diagnostics beyond direct analogues of pure-state probes.

quant-pharXiv:2606.26235•Jun 26, 2026•0 citations

Symmetry-preserving quadratic Lindbladian and dissipation driven topological transitions in Gaussian states

Liang Mao, Fan Yang, Hui Zhai

Abstract The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U (1) symmetry and the DIII class without U (1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.

arXiv:2301.04345•Reports on Progress in Physics•Jun 20, 2024•21 citations

Symmetry and topology of monitored quantum dynamics

Zhenyu Xiao, Kohei Kawabata

The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective non-Hermitian dynamical generators, thereby establishing the tenfold classification for symmetry and topology of monitored free fermions. Our classification elucidates the role of topology in measurement-induced phase transitions and identifies potential topological terms in the corresponding nonlinear sigma models. Furthermore, we establish the bulk-boundary correspondence in monitored quantum dynamics: nontrivial topology in spacetime manifests itself as topologically nontrivial steady states and gapless boundary states in Lyapunov spectra, such as Lyapunov zero modes and chiral edge modes, leading to the topologically protected slowdown of dynamical purification.

arXiv:2412.06133•Physical review. B./Physical review. B•Mar 12, 2026•2 citations

Absence of topology in Gaussian mixed states of bosons

Christopher D. Mink, Michael Fleischhauer, Razmik G. Unanyan

In a recent paper [Bardyn et al., Phys. Rev. X 8, 011035 (2018)], it was shown that the generalization of many-body polarization to mixed states can be used to construct a topological invariant that is also applicable to finite-temperature and nonequilibrium Gaussian states of lattice fermions. The many-body polarization defines an ensemble geometric phase that is identical to the Zak phase of a fictitious Hamiltonian, whose symmetries determine the topological classification. Here we show that in the case of Gaussian states of bosons, the corresponding topological invariant is always trivial. This also applies to finite-temperature states of bosons in lattices with a topologically nontrivial band structure. As a consequence, there is no quantized topological charge pumping for translationally invariant bulk states of noninteracting bosons.

arXiv:1901.04287•Physical review. B./Physical review. B•Jul 25, 2019•10 citations

Z2 topological invariants for mixed states of fermions in time-reversal invariant band structures

Lukas Wawer, Michael Fleischhauer

The topological classification of fermion systems in mixed states is a long-standing quest. For Gaussian states, reminiscent of noninteracting unitary fermions, some progress has been made. While the topological quantization of certain observables such as the Hall conductivity is lost for mixed states, directly observable many-body correlators exist which preserve the quantized nature and naturally connect to known topological invariants in the ground state. For systems that break time-reversal (TR) symmetry, the ensemble geometric phase was identified as such an observable which can be used to define a Chern number in $(1+1)$ and two dimensions. Here we propose a corresponding ${\mathbb{Z}}_{2}$ topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known ${\mathbb{Z}}_{2}$ invariants for the ground state of the so-called fictitious Hamiltonian, which for thermal states is just the ground state of the system Hamiltonian itself. We illustrate our findings for finite-temperature states of a paradigmatic ${\mathbb{Z}}_{2}$ topological insulator, the Kane-Mele model.

arXiv:2109.01487•Physical review. B./Physical review. B•Dec 21, 2021•8 citations
Abstract Topological band theory has been studied for free fermions for decades, and one of the most profound physical results is the bulk-boundary correspondence. Recently a focus in topological physics is extending topological classification to mixed states. Here, we focus on Gaussian mixed states for which the modular Hamiltonians of the density matrix are quadratic free fermion models with U(1) symmetry and can be classified by topological invariants. The bulk-boundary correspondence is then manifested as stable gapless modes of the modular Hamiltonian and degenerate spectrum of the density matrix. In this article, we show that these gapless modes can be detected by the full counting statistics, mathematically described by a function introduced as F(θ). A divergent derivative at θ = π can be used to probe the gapless modes in the modular Hamiltonian. Based on this, a topological indicator, whose quantization to unity senses topologically nontrivial mixed states, is introduced. We present the physical intuition of these results and also demonstrate these results with concrete models in both one- and two-dimensions. Our results pave the way for revealing the physical significance of topology in mixed states.
The bulk-boundary correspondence relates topologically protected edge modes to bulk topological invariants and is well understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent α, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with α>1, linking together the quantized winding invariant, bulk topological string-order parameters, and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalize these results by (i) identifying a family of BDI chains with α<1 where our results still hold and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when α-1 exceeds the dynamical critical exponent.
What does it mean for a mixed-state topological phase to be chiral? Mathematically, chirality can be sharply characterized through the symmetry algebra of the mixed state. Physically, however, the question is far more subtle. In pure states, chirality in topological phase is tied to a web of familiar diagnostics, involving bulk-boundary correspondence, a gapless entanglement spectrum, a nontrivial modular commutator, and a quantized thermal Hall response. We show that none of these diagnostics remain reliable in mixed states. Instead, for decohered topological phases with a known error-free parent state, we propose two relative-entropy-based measures that can diagnose chirality, with one of them further extracting the chiral central charge. As a concrete example, we use these measures to analyze the em2em^2em2-decohered Z3\mathbb{Z}_3Z3​ toric code, identifying the chirality transition to be in the random-bond Potts class for the particle-hole symmetric channel, and the polarized random bond Potts class for asymmetric channels. Our results emphasize how mixed-state topology demands intrinsically new diagnostics beyond direct analogues of pure-state probes.
Abstract The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U (1) symmetry and the DIII class without U (1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.
The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective non-Hermitian dynamical generators, thereby establishing the tenfold classification for symmetry and topology of monitored free fermions. Our classification elucidates the role of topology in measurement-induced phase transitions and identifies potential topological terms in the corresponding nonlinear sigma models. Furthermore, we establish the bulk-boundary correspondence in monitored quantum dynamics: nontrivial topology in spacetime manifests itself as topologically nontrivial steady states and gapless boundary states in Lyapunov spectra, such as Lyapunov zero modes and chiral edge modes, leading to the topologically protected slowdown of dynamical purification.
In a recent paper [Bardyn et al., Phys. Rev. X 8, 011035 (2018)], it was shown that the generalization of many-body polarization to mixed states can be used to construct a topological invariant that is also applicable to finite-temperature and nonequilibrium Gaussian states of lattice fermions. The many-body polarization defines an ensemble geometric phase that is identical to the Zak phase of a fictitious Hamiltonian, whose symmetries determine the topological classification. Here we show that in the case of Gaussian states of bosons, the corresponding topological invariant is always trivial. This also applies to finite-temperature states of bosons in lattices with a topologically nontrivial band structure. As a consequence, there is no quantized topological charge pumping for translationally invariant bulk states of noninteracting bosons.
The topological classification of fermion systems in mixed states is a long-standing quest. For Gaussian states, reminiscent of noninteracting unitary fermions, some progress has been made. While the topological quantization of certain observables such as the Hall conductivity is lost for mixed states, directly observable many-body correlators exist which preserve the quantized nature and naturally connect to known topological invariants in the ground state. For systems that break time-reversal (TR) symmetry, the ensemble geometric phase was identified as such an observable which can be used to define a Chern number in (1+1)(1+1)(1+1) and two dimensions. Here we propose a corresponding Z2{\mathbb{Z}}_{2}Z2​ topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known Z2{\mathbb{Z}}_{2}Z2​ invariants for the ground state of the so-called fictitious Hamiltonian, which for thermal states is just the ground state of the system Hamiltonian itself. We illustrate our findings for finite-temperature states of a paradigmatic Z2{\mathbb{Z}}_{2}Z2​ topological insulator, the Kane-Mele model.