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8060 KiB

Open AccessArticle

Massless Majorana-Like Charged Carriers in Two-Dimensional Semimetals

by Halina Grushevskaya and George Krylov

Symmetry 2016, 8(7), 60; https://doi.org/10.3390/sym8070060 - 08 Jul 2016

Cited by 10 | Viewed by 4566

The band structure of strongly correlated two-dimensional (2D) semimetal systems is found to be significantly affected by the spin-orbit coupling (SOC), resulting in SOC-induced Fermi surfaces. Dirac, Weyl and Majorana representations are used for the description of different semimetals, though the band structures [...] Read more.

The band structure of strongly correlated two-dimensional (2D) semimetal systems is found to be significantly affected by the spin-orbit coupling (SOC), resulting in SOC-induced Fermi surfaces. Dirac, Weyl and Majorana representations are used for the description of different semimetals, though the band structures of all these systems are very similar. We develop a theoretical approach to the band theory of two-dimensional semimetals within the Dirac–Hartree–Fock self-consistent field approximation. It reveals partially breaking symmetry of the Dirac cone affected by quasi-relativistic exchange interactions for 2D crystals with hexagonal symmetry. Fermi velocity becomes an operator within this approach, and elementary excitations have been calculated in the tight-binding approximation when taking into account the exchange interaction of

π (p_{z})

-electron with its three nearest

π (p_{z})

-electrons. These excitations are described by the massless Majorana equation instead of the Dirac one. The squared equation for this field is of the Klein–Gordon–Fock type. Such a feature of the band structure of 2D semimetals as the appearance of four pairs of nodes is shown to be described naturally within the developed formalism. Numerical simulation of band structure has been performed for the proposed 2D-model of graphene and a monolayer of Pb atoms. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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1588 KiB

Open AccessArticle

Entangled Harmonic Oscillators and Space-Time Entanglement

by Sibel Başkal, Young S. Kim and Marilyn E. Noz

Symmetry 2016, 8(7), 55; https://doi.org/10.3390/sym8070055 - 28 Jun 2016

Cited by 9 | Viewed by 6102

Abstract

The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator [...] Read more.

The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the

O (3, 2)

de Sitter group containing two

O (3, 1)

Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the

S p (4)

group, which serves as the basic language for two-mode squeezed states. Since the

S p (4)

symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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266 KiB

Open AccessArticle

Coherent States of Harmonic and Reversed Harmonic Oscillator

by Alexander Rauh

Symmetry 2016, 8(6), 46; https://doi.org/10.3390/sym8060046 - 13 Jun 2016

Cited by 3 | Viewed by 4327

Abstract

A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart [...] Read more.

A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart from the standard coherent states, a further class of solutions is derived with a time-dependent width parameter. The width of the corresponding probability density fluctuates, or "breathes" periodically with the oscillator frequency. In the case of the RO, one also obtains normalized wave packets which, however, show diffusion through exponential broadening with time. At the initial time, the integration constants give rise to complete sets of coherent states in the three cases considered. The results are applicable to the quantum mechanics of the Kepler-Coulomb problem when transformed to the model of a four-dimensional harmonic oscillator with a constraint. In the classical limit, as was shown recently, the wave packets of the RO basis generate the hyperbolic Kepler orbits, and, by means of analytic continuation, the elliptic orbits are also obtained quantum mechanically. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

1353 KiB

Open AccessArticle

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

by Gabriel Amador, Kiara Colon, Nathalie Luna, Gerardo Mercado, Enrique Pereira and Erwin Suazo

Symmetry 2016, 8(6), 38; https://doi.org/10.3390/sym8060038 - 28 May 2016

Cited by 9 | Viewed by 4830

Abstract

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them [...] Read more.

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark- and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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870 KiB

Open AccessArticle

Higher Order Nonclassicality from Nonlinear Coherent States for Models with Quadratic Spectrum

by Anaelle Hertz, Sanjib Dey, Véronique Hussin and Hichem Eleuch

Symmetry 2016, 8(5), 36; https://doi.org/10.3390/sym8050036 - 19 May 2016

Cited by 5 | Viewed by 4564

Abstract

Harmonic oscillator coherent states are well known to be the analogue of classical states. On the other hand, nonlinear and generalised coherent states may possess nonclassical properties. In this article, we study the nonclassical behaviour of nonlinear coherent states for generalised classes of [...] Read more.

Harmonic oscillator coherent states are well known to be the analogue of classical states. On the other hand, nonlinear and generalised coherent states may possess nonclassical properties. In this article, we study the nonclassical behaviour of nonlinear coherent states for generalised classes of models corresponding to the generalised ladder operators. A comparative analysis among them indicates that the models with quadratic spectrum are more nonclassical than the others. Our central result is further underpinned by the comparison of the degree of nonclassicality of squeezed states of the corresponding models. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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238 KiB

Open AccessArticle

Local Dynamics in an Infinite Harmonic Chain

by M. Howard Lee

Symmetry 2016, 8(4), 22; https://doi.org/10.3390/sym8040022 - 15 Apr 2016

Cited by 11 | Viewed by 3441

Abstract

By the method of recurrence relations, the time evolution in a local variable in a harmonic chain is obtained. In particular, the autocorrelation function is obtained analytically. Using this result, a number of important dynamical quantities are obtained, including the memory function of [...] Read more.

By the method of recurrence relations, the time evolution in a local variable in a harmonic chain is obtained. In particular, the autocorrelation function is obtained analytically. Using this result, a number of important dynamical quantities are obtained, including the memory function of the generalized Langevin equation. Also studied are the ergodicity and chaos in a local dynamical variable. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

386 KiB

Open AccessArticle

Analytical Solutions of Temporal Evolution of Populations in Optically-Pumped Atoms with Circularly Polarized Light

by Heung-Ryoul Noh

Symmetry 2016, 8(3), 17; https://doi.org/10.3390/sym8030017 - 19 Mar 2016

Cited by 3 | Viewed by 3670

Abstract

We present an analytical calculation of temporal evolution of populations for optically pumped atoms under the influence of weak, circularly polarized light. The differential equations for the populations of magnetic sublevels in the excited state, derived from rate equations, are expressed in the [...] Read more.

We present an analytical calculation of temporal evolution of populations for optically pumped atoms under the influence of weak, circularly polarized light. The differential equations for the populations of magnetic sublevels in the excited state, derived from rate equations, are expressed in the form of inhomogeneous second-order differential equations with constant coefficients. We present a general method of analytically solving these differential equations, and obtain explicit analytical forms of the populations of the ground state at the lowest order in the saturation parameter. The obtained populations can be used to calculate lineshapes in various laser spectroscopies, considering transit time relaxation. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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364 KiB

Open AccessEssay

Old Game, New Rules: Rethinking the Form of Physics

by Christian Baumgarten

Symmetry 2016, 8(5), 30; https://doi.org/10.3390/sym8050030 - 06 May 2016

Cited by 3 | Viewed by 4402

Abstract

We investigate the modeling capabilities of sets of coupled classical harmonic oscillators (CHO) in the form of a modeling game. The application of the simple but restrictive rules of the game lead to conditions for an isomorphism between Lie-algebras and real Clifford algebras. [...] Read more.

We investigate the modeling capabilities of sets of coupled classical harmonic oscillators (CHO) in the form of a modeling game. The application of the simple but restrictive rules of the game lead to conditions for an isomorphism between Lie-algebras and real Clifford algebras. We show that the correlations between two coupled classical oscillators find their natural description in the Dirac algebra and allow to model aspects of special relativity, inertial motion, electromagnetism and quantum phenomena including spin in one go. The algebraic properties of Hamiltonian motion of low-dimensional systems can generally be related to certain types of interactions and hence to the dimensionality of emergent space-times. We describe the intrinsic connection between phase space volumes of a 2-dimensional oscillator and the Dirac algebra. In this version of a phase space interpretation of quantum mechanics the (components of the) spinor wavefunction in momentum space are abstract canonical coordinates, and the integrals over the squared wave function represents second moments in phase space. The wave function in ordinary space-time can be obtained via Fourier transformation. Within this modeling game, 3+1-dimensional space-time is interpreted as a structural property of electromagnetic interaction. A generalization selects a series of Clifford algebras of specific dimensions with similar properties, specifically also 10- and 26-dimensional real Clifford algebras. Full article

(This article belongs to the Special Issue Harmonic Oscillators In Modern Physics)

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