Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems

Viyuela, Oscar; Rivas, Ana María Rivas; Martín-Delgado, M. A.

doi:10.1103/physrevlett.112.130401

Cited by 156 publications

(257 citation statements)

References 42 publications

(80 reference statements)

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“…We call it the "quark number holonomy." The quark number holonomy seems to be a similar quantity to the Uhlmann phase [4,5]. The quark number holonomy defined in Eq.…”

Section: Quark Number Holonomymentioning

confidence: 92%

“…Motivated by this progress, it has been suggested that the confinement-deconfinement transition can be described by using the analogy of the topological order and then that the free-energy degeneracy plays a crucial role [3]. The idea of topological order is extended to finite temperature in terms of the Uhlmann phase [4,5]. The Uhlmann phase is an extension of the Berry phase to mixed quantum states.…”

Section: A Topological Ordermentioning

confidence: 99%

“…The Uhlmann phase is an extension of the Berry phase to mixed quantum states. The Uhlmann phase can describe the topological order at finite T in the one-dimensional fermion systems such as the topological insulator and the superconductor [5]. The Uhlmann phase is defined by using the amplitude for the density matrix where amplitudes form the Hilbert space.…”

Section: A Topological Ordermentioning

confidence: 99%

See 2 more Smart Citations

Quark number holonomy and confinement-deconfinement transition

Kashiwa

Ohnishi

2016

Phys. Rev. D

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We propose a new quantity which describes the confinement-deconfinement transition based on topological properties of QCD. The quantity which we call the quark number holonomy is defined as the integral of the quark number susceptibility along the closed loop of θ where θ is the dimensionless imaginary chemical potential. The expected behavior of the quark number holonomy at finite temperature is discussed, and its asymptotic behaviors are shown.

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“…We call it the "quark number holonomy." The quark number holonomy seems to be a similar quantity to the Uhlmann phase [4,5]. The quark number holonomy defined in Eq.…”

Section: Quark Number Holonomymentioning

confidence: 92%

Section: A Topological Ordermentioning

confidence: 99%

Section: A Topological Ordermentioning

confidence: 99%

See 1 more Smart Citation

Quark number holonomy and confinement-deconfinement transition

Kashiwa

Ohnishi

2016

Phys. Rev. D

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“…However, in a recent work [25] we have shown that it is possible to characterize a thermal topological phase for topological insulators and superconductors in one-dimensional systems. This thermal topological phase, classified by Uhlmann holonomies [26,27], is separated from a trivial phase by a critical finite temperature, at which the topological phase abruptly disappears.…”

mentioning

confidence: 99%

Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

2014

Self Cite

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We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number nU. With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature T is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two nontrivial phases in a model with high Chern numbers. At small temperatures we recover the standard topological phases as the Uhlmann number approaches to the Chern number. Although these studies provide a successful picture for quantum systems in pure states, typically the ground state, very little is known about the fate of those topological phases of matter when the system is in a mixed quantum state represented by a density matrix. In fact, the correct understanding of this situation becomes particularly relevant in order to address unavoidable thermal effects on topological phases and nonequilibrium dynamics under dissipation [19].Over the last few years, there have been some results establishing the absence of stable topological phases subject to thermal effects, like for TOs with spins [20][21][22], for SP-TOs with spins [23], or SP-TOs with fermions under certain conditions [24]. However, in a recent work [25] we have shown that it is possible to characterize a thermal topological phase for topological insulators and superconductors in one-dimensional systems. This thermal topological phase, classified by Uhlmann holonomies [26,27], is separated from a trivial phase by a critical finite temperature, at which the topological phase abruptly disappears. This result paves the way towards the characterization of SP-TOs with fermions in thermal states, or more general density matrices.In this paper, we have achieved this goal for twodimensional fermion systems, either insulating or superconducting. This extension from one to two-dimensional systems is nontrivial in the sense that a direct generalization of the topological invariants for pure states (Chern numbers) to density matrices via the Uhlmann approach leads to trivial results. However, we have succeeded in circumventing this problem and introducing a suitable notion of topological Uhlmann numbers. These are gauge invariant and observable quantities which allows for a classification of topological phases of density matrices in two-dimensional (2D) quantum systems. Specifically, we

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“…The quark number holonomy seems to be a similar quantity with the Uhlmann phase [5,6]. The Uhlmann phase can describe the topological order at finite T at least in the one-dimensional fermion systems such as the topological insulator and the superconductor [6]. The Uhlmann phase is the extension of the Berry phase.…”

Section: Introductionmentioning

confidence: 89%

Topological Nature of Deconfinement Transition in QCD

Kashiwa

2018

Proceedings of the Workshop on Quarks and Compact Stars 2017 (QCS2017)

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We propose a new conjecture to determine the confinement-deconfinement transition by using the imaginary chemical potential. The imaginary chemical potential can be treated as the AharonovBohm phase and then an analogy of the topological order suggests that the Roberge-Weiss endpoint would define the temperature of the confinement-deconfinement transition. Based on the topological nature, we can construct a new quantum order-parameter which describes the confinementdeconfinement transition. This quantity is defined as the integral of the quark number susceptibility along the closed loop of θ = 0 ∼ 2π where θ is the dimensionless imaginary chemical potential. Expected behavior of the quantity at finite temperature is discussed and its asymptotic behaviors are drawn.

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Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems

Cited by 156 publications

References 42 publications

Quark number holonomy and confinement-deconfinement transition

Quark number holonomy and confinement-deconfinement transition

Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

Topological Nature of Deconfinement Transition in QCD

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