Lecture 5: Connection with the Properties of a Crystal. Born–Oppenheimer Approximation. Edge States and Role of Topology

A semiconductor nanowire with strong spin-orbit coupling in proximity to a superconductor is predicted to display Majorana edge states emerging under a properly oriented magnetic field. The experimental investigation of these exotic states requires assessing the one-dimensional (1D) character of the nanowire and understanding the superconducting proximity effect in the presence of a magnetic field. Here, we explore the quasi-ballistic 1D transport regime of an InAs nanowire with Ta contacts. Fine-tuned by means of local gates, the observed plateaus of approximately quantized conductance hide the presence of a localized electron, giving rise to a lurking Coulomb blockade effect and Kondo physics. When Ta becomes superconducting, this local charge causes an unusual, reentrant magnetic field dependence of the supercurrent, which we ascribe to a 0 - π transition. Our results underline the relevant role of unintentional charge localization in the few-channel regime where helical subbands and Majorana quasi-particles are expected to arise.

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Bridging the Physics and Chemistry of Graphene(s): From Hückel’s Aromaticity to Dirac’s Cones and Topological Insulators

10.26434/chemrxiv.10299062 ◽

2020 ◽

Author(s):

Aristides Zdetsis

Keyword(s):

Shell Structure ◽

Resonance Structure ◽

Edge States ◽

Many Body ◽

Opposite Parity ◽

Many Body Theory ◽

Conduction Bands ◽

Polycyclic Aromatic ◽

Body Theory

By bridging graphene and benzene through a well-defined sequence of polycyclic aromatic hydrocarbons and their inherent shell structure, it is shown that graphene is actually a coherent arrangement of interwoven benzene molecules, coordinated by aromaticity, shell structure, and topology, all interrelated and microscopically realized through dynamical flipping of the atomic pz-orbitals, playing the role of pseudospin or “qubits”. This renders graphene resonance structure, “resonating” between two complementary aromaticity patterns, involving 2k, k→∞ kekulé type of resonances resulting in “robust electronic coherence”, with dual “molecular-crystalline” nature, and two valence-conduction bands of opposite parity, driven by inversion symmetry competition, which is essentially a “molecule-versus-crystal” competition, in accord with topological-insulator and many-body theory. The “average picture” converges to the usual band structure with two aromatic π-electrons per ring, and the fingerprints of inversion-competition at the D3h-symmetric Dirac points, which for rectangular nanographene(s) appear as gapless topological edge states without real spin-polarization, contrary to opposite claims. <br>

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Bridging the Physics and Chemistry of Graphene(s): From Hückel’s Aromaticity to Dirac’s Cones and Topological Insulators

10.26434/chemrxiv.10299062.v1 ◽

2019 ◽

Author(s):

Aristides Zdetsis

Keyword(s):

Shell Structure ◽

Resonance Structure ◽

Edge States ◽

Many Body ◽

Opposite Parity ◽

Many Body Theory ◽

Conduction Bands ◽

Polycyclic Aromatic ◽

Body Theory

By bridging graphene and benzene through a well-defined sequence of polycyclic aromatic hydrocarbons and their inherent shell structure, it is shown that graphene is actually a coherent arrangement of interwoven benzene molecules, coordinated by aromaticity, shell structure, and topology, all interrelated and microscopically realized through dynamical flipping of the atomic pz-orbitals, playing the role of pseudospin or “qubits”. This renders graphene resonance structure, “resonating” between two complementary aromaticity patterns, involving 2k, k→∞ kekulé type of resonances resulting in “robust electronic coherence”, with dual “molecular-crystalline” nature, and two valence-conduction bands of opposite parity, driven by inversion symmetry competition, which is essentially a “molecule-versus-crystal” competition, in accord with topological-insulator and many-body theory. The “average picture” converges to the usual band structure with two aromatic π-electrons per ring, and the fingerprints of inversion-competition at the D3h-symmetric Dirac points, which for rectangular nanographene(s) appear as gapless topological edge states without real spin-polarization, contrary to opposite claims. <br>

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From Chern–Simons to Tomonaga–Luttinger

International Journal of Modern Physics A ◽

10.1142/s0217751x18500136 ◽

2018 ◽

Vol 33 (02) ◽

pp. 1850013 ◽

Cited By ~ 4

Author(s):

Nicola Maggiore

Keyword(s):

Degrees Of Freedom ◽

Quantum Hall ◽

Three Dimensional ◽

Two Dimensional ◽

Edge States ◽

Fractional Quantum ◽

Fractional Quantum Hall ◽

Chern Simons ◽

Weyl Fermion

A single-sided boundary is introduced in the three-dimensional Chern–Simons model. It is shown that only one boundary condition for the gauge fields is possible, which plays the twofold role of chirality condition and bosonization rule for the two-dimensional Weyl fermion describing the degrees of freedom of the edge states of the Fractional Quantum Hall Effect. The symmetry on the boundary is derived, which determines the effective two-dimensional action, whose equation of motion coincides with the continuity equation of the Tomonaga–Luttinger theory. The role of Lorentz symmetry and of discrete symmetries on the boundary is also discussed.

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THEORETICAL ASPECTS OF QUANTUM HALL EFFECT AND TWO-DIMENSIONAL CFT

International Journal of Modern Physics B ◽

10.1142/s0217979292001110 ◽

1992 ◽

Vol 06 (11n12) ◽

pp. 2217-2239 ◽

Cited By ~ 1

Author(s):

G. CRISTOFANO ◽

G. MAIELLA ◽

R. MUSTO ◽

F. NICODEMI

Keyword(s):

Quantum Hall ◽

Coulomb Gas ◽

Topological Order ◽

Two Dimensional ◽

Edge States ◽

Conformal Field ◽

Highest Weight ◽

Finite Set ◽

Magnetic Translation

A review of the QHE is presented where the emphasis is placed on the role of the magnetic translation group and of the related topological properties. After a presentation of general experimental and theoretical features we briefly summarize known results of two-dimensional Conformal Field Theory relevant for the QHE. Then we show how to evaluate groundstate wave functions on the plane and on the torus by the use of CFT techniques. In the latter case it is shown how for filling ν=1/m a consistent description is achieved by means of a finite set of Coulomb Gas Vertex Operators. They describe (fractional) charged particles with the associated quantized magnetic flux (“anyons"). Furthermore, from these vertices and relative highest weight states, one does find that the g.s.w.f. for the torus should be m-fold degenerate showing, also, the role of a new kind of long-range topological order recently advocated. Then we show that the presence of m sectors of edge states for a cylinder is strictly related to such a degeneracy, and their relation with a Kac-Moody algebra is discussed.

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Conductance of quantum spin Hall edge states from first principles: The critical role of magnetic impurities and inter-edge scattering

Physical Review B ◽

10.1103/physrevb.101.155404 ◽

2020 ◽

Vol 101 (15) ◽

Cited By ~ 4

Author(s):

Luca Vannucci ◽

Thomas Olsen ◽

Kristian S. Thygesen

Keyword(s):

First Principles ◽

Critical Role ◽

Quantum Spin ◽

Magnetic Impurities ◽

Edge States ◽

Edge Scattering ◽

Quantum Spin Hall

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EDGE STATES AND TRAJECTORIES IN QUANTUM DOTS: PROBING THE QUANTUM-CLASSICAL TRANSITION

International Journal of Modern Physics B ◽

10.1142/s0217979207042744 ◽

2007 ◽

Vol 21 (08n09) ◽

pp. 1278-1287

Author(s):

D. K. FERRY ◽

R. AKIS ◽

J. P. BIRD

Keyword(s):

Quantum Dot ◽

Quantum Hall Effect ◽

Quantum Hall ◽

Oscillatory Behavior ◽

Wave Functions ◽

Edge States ◽

Open Quantum ◽

Pointer States ◽

Classical Transition

Edge states have been a backbone of our understanding of the experimental basis of the quantum Hall effect for quite some time. Interestingly, this comprises a quantum system with well defined currents and particle trajectories. The role of trajectories in quantum mechanics has been a problematic question of interpretation for quite some time, and the open quantum dot is a natural system in which to probe this question. Contrary to early speculation, a set of well defined quantum states survives in the open quantum dot. These states are the pointer states and provide a transition into the classical states that can be found in these structures. These states provide resonances, which are observable as oscillatory behavior in the magnetoconductance of the dots. But, they have well defined current directions within the dots. Consequently, one expects trajectories to be a property of these states as well. As one crosses from the low to the high field regime, quite steady trajectories and consequent wave functions can easily be identified and examined. In this talk, we review the current understanding and the support for the decoherence theory.

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