Non-hermitian topology as a unifying framework for the Andreev versus Majorana states controversy

Abstract Zero-energy Andreev levels in hybrid semiconductor-superconductor nanowires mimic all expected Majorana phenomenology, including $$2{e}^{2}/h$$ 2 e 2 ∕ h conductance quantisation, even where band topology predicts trivial phases. This surprising fact has been used to challenge the interpretation of various transport experiments in terms of Majorana zero modes. Here we show that the Andreev versus Majorana controversy is clarified when framed in the language of non-Hermitian topology, the natural description for quantum systems open to the environment. This change of paradigm allows one to understand topological transitions and the emergence of zero modes in more general systems than can be described by band topology. This is achieved by studying exceptional point bifurcations in the complex spectrum of the system’s non-Hermitian Hamiltonian. Within this broader topological classification, Majoranas from both conventional band topology and a large subset of Andreev levels at zero energy are in fact topologically equivalent, which explains why they cannot be distinguished.

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Paths of unitary access to exceptional points

Journal of Physics Conference Series ◽

10.1088/1742-6596/2038/1/012026 ◽

2021 ◽

Vol 2038 (1) ◽

pp. 012026

Author(s):

Miloslav Znojil

Keyword(s):

Hilbert Space ◽

Phase Transitions ◽

Quantum Phase Transitions ◽

Quantum Systems ◽

Point Of View ◽

Explicit Description ◽

Direct Access ◽

Exceptional Point ◽

Exceptional Points ◽

Innovative Idea

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

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THE DIFFEOMORPHISM GROUP APPROACH TO NONLINEAR QUANTUM SYSTEMS

International Journal of Modern Physics B ◽

10.1142/s0217979292000931 ◽

1992 ◽

Vol 06 (11n12) ◽

pp. 1905-1916 ◽

Cited By ~ 27

Author(s):

GERALD A. GOLDIN

Keyword(s):

Nonlinear Effects ◽

Quantum Systems ◽

Incompressible Fluids ◽

Diffeomorphism Groups ◽

Group Approach ◽

Quantum Vortex ◽

Nonlinear Quantum ◽

New Applications ◽

Fundamental Physical ◽

Natural Description

Unitary representations of diffeomorphism groups predict some unusual possibilities in quantum theory, including non-standard statistics and certain nonlinear effects. Many of the fundamental physical properties of “anyons” were first derived from their study by R. Menikoff, D.H. Sharp, and the author. This paper surveys new applications in two other domains: first (with Menikoff and Sharp) some surprising conclusions about the nature of quantum vortex configurations in ideal, incompressible fluids; second (with H.-D. Doebner) a natural description of dissipative quantum mechanics by means of a nonlinear Schrödinger equation different from the sort usually studied. Our equation follows from including a diffusion current in the equation of continuity.

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The Past, Present and Possible Future for Systems

International Journal of Systems and Society ◽

10.4018/ijss.2017010101 ◽

2017 ◽

Vol 4 (1) ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Gianfranco Minati

Keyword(s):

Informal Learning ◽

Social Systems ◽

Quantum Systems ◽

Economic Crises ◽

The Past ◽

Complex Interactions ◽

Industrial Age ◽

Emergent Systems ◽

General Systems ◽

Post Industrial

In this article, the author briefly summarise the characteristics of the science of complexity or post- Bertalanffy General Systems. The author discusses the shift from considering systems as acquiring properties due to their explicit or supposed design, to self-organised, emergent systems. Characteristics, approaches to modelling and interventions to change vary in nature with the post-Bertalanffy Systemics. While new suitable models and approaches are under study in sciences, such as physics, chemistry, biology, mathematics, engineering, and neurosciences, the author detects significant backwardness when dealing with the complexity of social systems and related problems that are developing in the post-industrial age. These problems include economic crises, security, defence, privacy, managing prisons, and supporting development. Such social problems are inadequately faced by using classical Bertalanffy's systemic concepts or by simply transposing models and changing the meaning of variables. This inadequacy is based on the underestimation of the peculiarities of Human Systems that consist of complex interactions that allow coherence and are also cognitive, informal, learning, evolutionary, ecological and non-governable Luhmannian subsystems. The non-cultural or low-cultural accessibility of the approaches considered by the science of complexity contribute to this inadequacy. Finally, the author presents some comments on how the science of systems may further evolve by considering new types of systems and systemic properties such as systemic fields and quantum systems. He speculates about some possible future understanding of human social systems.

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Kitaev’s Chain Model and Majorana Zero Modes

Physics and High Technology ◽

10.3938/phit.29.018 ◽

2020 ◽

Vol 29 (6) ◽

pp. 3-7

Author(s):

Sangjun JEON ◽

Kwang-Yong CHOI

Keyword(s):

Conceptual Development ◽

Fault Tolerant ◽

Condensed Matter ◽

Majorana Fermion ◽

Quantum Computers ◽

Chain Model ◽

Zero Energy ◽

Experimental Realization ◽

Zero Modes ◽

History Of

A Majorana Fermion (MF), defined as a particle that is its own anti-particle, can be engineered in the form of quasi-particles appearing in condensed matter systems. Majorana zero modes (MZMs), topologically protected zero-energy states, can exist at the topological phase boundary of one- or two-dimensional systems. These MZMs follow non-Abelian statistics and can be used as a building block for fault-tolerant quantum computers. Here, we introduce the conceptual development of MZMs based on Kitaev’s chain model and give a brief history of the experimental realization of MZMs.

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Observation of a Majorana zero mode in a topologically protected edge channel

Science ◽

10.1126/science.aax1444 ◽

2019 ◽

Vol 364 (6447) ◽

pp. 1255-1259 ◽

Cited By ~ 35

Author(s):

Berthold Jäck ◽

Yonglong Xie ◽

Jian Li ◽

Sangjun Jeon ◽

B. Andrei Bernevig ◽

...

Keyword(s):

Scanning Tunneling Microscopy ◽

Zero Mode ◽

Scanning Tunneling ◽

Zero Energy ◽

Tunneling Microscopy ◽

Model Calculations ◽

Iron Clusters ◽

Edge Channel ◽

Zero Modes ◽

Gap States

Superconducting proximity pairing in helical edge modes, such as those of topological insulators, is predicted to provide a unique platform for realizing Majorana zero modes (MZMs). We used scanning tunneling microscopy measurements to probe the influence of proximity-induced superconductivity and magnetism on the helical hinge states of bismuth(111) films grown on a superconducting niobium substrate and decorated with magnetic iron clusters. Consistent with model calculations, our measurements revealed the emergence of a localized MZM at the interface between the superconducting helical edge channel and the iron clusters, with a strong magnetization component along the edge. Our experiments also resolve the MZM’s spin signature, which distinguishes it from trivial in-gap states that may accidentally occur at zero energy in a superconductor.

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General systems, classical systems, quantum systems and fuzzy systems: an introductory survey

International Journal of General Systems ◽

10.1080/03081079.2010.505028 ◽

2011 ◽

Vol 40 (1) ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Rudolf Seising

Keyword(s):

Fuzzy Systems ◽

Quantum Systems ◽

Classical Systems ◽

General Systems

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Bulk–edge correspondence and stability of multiple edge states of a $\mathcal{PT}$-symmetric non-Hermitian system by using non-unitary quantum walks

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptaa034 ◽

2020 ◽

Vol 2020 (12) ◽

Cited By ~ 1

Author(s):

Makio Kawasaki ◽

Ken Mochizuki ◽

Norio Kawakami ◽

Hideaki Obuse

Keyword(s):

Symmetry Breaking ◽

Open Quantum Systems ◽

Quantum Walk ◽

Quantum Systems ◽

Quantum Walks ◽

Multiple Edge ◽

Exceptional Point ◽

Topological Phase ◽

Edge States ◽

The Stability

Abstract Topological phases and the associated multiple edge states are studied for parity and time-reversal ($\mathcal{PT}$)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.

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Double-frequency Aharonov-Bohm effect and non-Abelian braiding properties of Jackiw-Rebbi zero-mode

National Science Review ◽

10.1093/nsr/nwz189 ◽

2019 ◽

Vol 7 (3) ◽

pp. 572-578 ◽

Cited By ~ 2

Author(s):

Yijia Wu ◽

Haiwen Liu ◽

Jie Liu ◽

Hua Jiang ◽

X C Xie

Keyword(s):

Zero Mode ◽

Fusion Rule ◽

Quantum Spin ◽

Zero Energy ◽

Double Frequency ◽

Zero Modes ◽

Charge Fractionalization ◽

Bohm Effect ◽

Aharonov Bohm ◽

Special Case

Abstract Ever since its first proposal in 1976, Jackiw-Rebbi zero-mode has been drawing extensive attention for its charming properties including charge fractionalization, topologically protected zero-energy and possible non-Abelian statistics. We investigate these properties through the Jackiw-Rebbi zero-modes in quantum spin Hall insulators. Though charge fractionalization is not manifested, Jackiw-Rebbi zero-mode's zero-energy nature leads to a double-frequency Aharonov-Bohm effect, implying that it can be viewed as a special case of Majorana zero-mode without particle-hole symmetry. Such relation is strengthened for Jackiw-Rebbi zero-modes also exhibiting non-Abelian properties in the absence of superconductivity. Furthermore, in the condition that the degeneracy of Jackiw-Rebbi zero-modes is lifted, we demonstrate a novel non-Abelian braiding with continuously tunable fusion rule, which is a generalization of Majorana zero-modes’ braiding properties.

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