Symmetry

Research

23 pages, 3223 KiB

Open AccessArticle

Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps

by Nir Hacker and Boris A. Malomed

Symmetry 2021, 13(3), 372; https://doi.org/10.3390/sym13030372 - 25 Feb 2021

Cited by 6 | Viewed by 3642

We consider a two-component linearly coupled system with the intrinsic cubic nonlinearity and the harmonic-oscillator (HO) confining potential. The system models binary settings in BEC and optics. In the symmetric system, with the HO trap acting in both components, we consider Josephson oscillations [...] Read more.

We consider a two-component linearly coupled system with the intrinsic cubic nonlinearity and the harmonic-oscillator (HO) confining potential. The system models binary settings in BEC and optics. In the symmetric system, with the HO trap acting in both components, we consider Josephson oscillations (JO) initiated by an input in the form of the HO’s ground state (GS) or dipole mode (DM), placed in one component. With the increase of the strength of the self-focusing nonlinearity, spontaneous symmetry breaking (SSB) between the components takes place in the dynamical JO state. Under still stronger nonlinearity, the regular JO initiated by the GS input carries over into a chaotic dynamical state. For the DM input, the chaotization happens at smaller powers than for the GS, which is followed by SSB at a slightly stronger nonlinearity. In the system with the defocusing nonlinearity, SSB does not take place, and dynamical chaos occurs in a small area of the parameter space. In the asymmetric half-trapped system, with the HO potential applied to a single component, we first focus on the spectrum of confined binary modes in the linearized system. The spectrum is found analytically in the limits of weak and strong inter-component coupling, and numerically in the general case. Under the action of the coupling, the existence region of the confined modes shrinks for GSs and expands for DMs. In the full nonlinear system, the existence region for confined modes is identified in the numerical form. They are constructed too by means of the Thomas–Fermi approximation, in the case of the defocusing nonlinearity. Lastly, particular (non-generic) exact analytical solutions for confined modes, including vortices, in one- and two-dimensional asymmetric linearized systems are found. They represent bound states in the continuum. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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16 pages, 632 KiB

Open AccessArticle

Kramers Degeneracy and Spin Inversion in a Lateral Quantum Dot

by Konstantin Pichugin, Antonio Puente and Rashid Nazmitdinov

Symmetry 2020, 12(12), 2043; https://doi.org/10.3390/sym12122043 - 10 Dec 2020

Cited by 1 | Viewed by 2491

Abstract

We show that the axial symmetry of the Bychkov–Rashba interaction can be exploited to produce electron spin-flip in a circular quantum dot, without lifting the time reversal symmetry. In order to elucidate this effect, we consider ballistic electron transmission through a two-dimensional circular [...] Read more.

We show that the axial symmetry of the Bychkov–Rashba interaction can be exploited to produce electron spin-flip in a circular quantum dot, without lifting the time reversal symmetry. In order to elucidate this effect, we consider ballistic electron transmission through a two-dimensional circular billiard coupled to two one-dimensional electrodes. Using the tight-binding approximation, we derive the scattering matrix and the effective Hamiltonian for the considered system. Within this approach, we found the conditions for the optimal realization of this effect in the transport properties of the quantum dot. Numerical analysis of the system, extended to the case of two-dimensional electrodes, confirms our findings. The relatively strong quantization of the quantum dot can make this effect robust against the temperature effects. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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13 pages, 546 KiB

Open AccessFeature PaperArticle

On Symmetry Properties of The Corrugated Graphene System

by Mihal Pudlak, Jan Smotlacha and Rashid Nazmitdinov

Symmetry 2020, 12(4), 533; https://doi.org/10.3390/sym12040533 - 03 Apr 2020

Cited by 3 | Viewed by 2110

Abstract

The properties of the ballistic electron transport through a corrugated graphene system are analysed from the symmetry point of view. The corrugated system is modelled by a curved surface (an arc of a circle) connected from both sides to flat sheets. The spin–orbit [...] Read more.

The properties of the ballistic electron transport through a corrugated graphene system are analysed from the symmetry point of view. The corrugated system is modelled by a curved surface (an arc of a circle) connected from both sides to flat sheets. The spin–orbit couplings, induced by the curvature, give rise to equivalence between the transmission (reflection) probabilities of the transmitted (reflected) electrons with the opposite spin polarisation, incoming from opposite system sides. We find two integrals of motion that explain the chiral electron transport in the considered system. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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24 pages, 2559 KiB

Open AccessArticle

Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases

by Thudiyangal Mithun, Aleksandra Maluckov, Kenichi Kasamatsu, Boris A. Malomed and Avinash Khare

Symmetry 2020, 12(1), 174; https://doi.org/10.3390/sym12010174 - 18 Jan 2020

Cited by 51 | Viewed by 4044

Abstract

Quantum droplets are ultradilute liquid states that emerge from the competitive interplay of two Hamiltonian terms, the mean-field energy and beyond-mean-field correction, in a weakly interacting binary Bose gas. We relate the formation of droplets in symmetric and asymmetric two-component one-dimensional boson systems [...] Read more.

Quantum droplets are ultradilute liquid states that emerge from the competitive interplay of two Hamiltonian terms, the mean-field energy and beyond-mean-field correction, in a weakly interacting binary Bose gas. We relate the formation of droplets in symmetric and asymmetric two-component one-dimensional boson systems to the modulational instability of a spatially uniform state driven by the beyond-mean-field term. Asymmetry between the components may be caused by their unequal populations or unequal intra-component interaction strengths. Stability of both symmetric and asymmetric droplets is investigated. Robustness of the symmetric solutions against symmetry-breaking perturbations is confirmed. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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31 pages, 1860 KiB

Open AccessArticle

Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

by Eugen Mircea Anitas

Symmetry 2020, 12(1), 65; https://doi.org/10.3390/sym12010065 - 01 Jan 2020

Cited by 17 | Viewed by 5339

Abstract

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been [...] Read more.

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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9 pages, 241 KiB

Open AccessArticle

Metal–Insulator Transition in Three-Dimensional Semiconductors

by Klaus Ziegler

Symmetry 2019, 11(11), 1345; https://doi.org/10.3390/sym11111345 - 01 Nov 2019

Cited by 1 | Viewed by 1850

Abstract

We use a random gap model to describe a metal–insulator transition in three-dimensional semiconductors due to doping, and find a conventional phase transition, where the effective scattering rate is the order parameter. Spontaneous symmetry breaking results in metallic behavior, whereas the insulating regime [...] Read more.

We use a random gap model to describe a metal–insulator transition in three-dimensional semiconductors due to doping, and find a conventional phase transition, where the effective scattering rate is the order parameter. Spontaneous symmetry breaking results in metallic behavior, whereas the insulating regime is characterized by the absence of spontaneous symmetry breaking. The transition is continuous for the average conductivity with critical exponent equal to 1. Away from the critical point, the exponent is roughly 0.6, which may explain experimental observations of a crossover of the exponent from 1 to 0.5 by going away from the critical point. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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22 pages, 1944 KiB

Open AccessArticle

Analysis of a Trapped Bose–Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance

by Ofir E. Alon

Symmetry 2019, 11(11), 1344; https://doi.org/10.3390/sym11111344 - 01 Nov 2019

Cited by 16 | Viewed by 3466

Abstract

We analyze, analytically and numerically, the position, momentum, and in particular the angular-momentum variance of a Bose–Einstein condensate (BEC) trapped in a two-dimensional anisotropic trap for static and dynamic scenarios. Explicitly, we study the ground state of the anisotropic harmonic-interaction model in two [...] Read more.

We analyze, analytically and numerically, the position, momentum, and in particular the angular-momentum variance of a Bose–Einstein condensate (BEC) trapped in a two-dimensional anisotropic trap for static and dynamic scenarios. Explicitly, we study the ground state of the anisotropic harmonic-interaction model in two spatial dimensions analytically and the out-of-equilibrium dynamics of repulsive bosons in tilted two-dimensional annuli numerically accurately by using the multiconfigurational time-dependent Hartree for bosons method. The differences between the variances at the mean-field level, which are attributed to the shape of the BEC, and the variances at the many-body level, which incorporate depletion, are used to characterize position, momentum, and angular-momentum correlations in the BEC for finite systems and at the limit of an infinite number of particles where the bosons are

100 %

condensed. Finally, we also explore inter-connections between the variances. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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52 pages, 789 KiB

Open AccessFeature PaperArticle

Symmetry Breaking in Stochastic Dynamics and Turbulence

by Michal Hnatič, Juha Honkonen and Tomáš Lučivjanský

Symmetry 2019, 11(10), 1193; https://doi.org/10.3390/sym11101193 - 23 Sep 2019

Cited by 4 | Viewed by 2780

Abstract

Symmetries play paramount roles in dynamics of physical systems. All theories of quantum physics and microworld including the fundamental Standard Model are constructed on the basis of symmetry principles. In classical physics, the importance and weight of these principles are the same as [...] Read more.

Symmetries play paramount roles in dynamics of physical systems. All theories of quantum physics and microworld including the fundamental Standard Model are constructed on the basis of symmetry principles. In classical physics, the importance and weight of these principles are the same as in quantum physics: dynamics of complex nonlinear statistical systems is straightforwardly dictated by their symmetry or its breaking, as we demonstrate on the example of developed (magneto)hydrodynamic turbulence and the related theoretical models. To simplify the problem, unbounded models are commonly used. However, turbulence is a mesoscopic phenomenon and the size of the system must be taken into account. It turns out that influence of outer length of turbulence is significant and can lead to intermittency. More precisely, we analyze the connection of phenomena such as behavior of statistical correlations of observable quantities, anomalous scaling, and generation of magnetic field by hydrodynamic fluctuations with symmetries such as Galilean symmetry, isotropy, spatial parity and their violation and finite size of the system. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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13 pages, 1356 KiB

Open AccessArticle

Structural Properties of Vicsek-like Deterministic Multifractals

by Eugen Mircea Anitas, Giorgia Marcelli, Zsolt Szakacs, Radu Todoran and Daniela Todoran

Symmetry 2019, 11(6), 806; https://doi.org/10.3390/sym11060806 - 18 Jun 2019

Cited by 6 | Viewed by 2613

Abstract

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle [...] Read more.

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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23 pages, 319 KiB

Open AccessArticle

Particle Fluctuations in Mesoscopic Bose Systems

by Vyacheslav I. Yukalov

Symmetry 2019, 11(5), 603; https://doi.org/10.3390/sym11050603 - 01 May 2019

Cited by 6 | Viewed by 1802

Abstract

Particle fluctuations in mesoscopic Bose systems of arbitrary spatial dimensionality are considered. Both ideal Bose gases and interacting Bose systems are studied in the regions above the Bose–Einstein condensation temperature

T_{c}

, as well as below this temperature. The strength of particle [...] Read more.

Particle fluctuations in mesoscopic Bose systems of arbitrary spatial dimensionality are considered. Both ideal Bose gases and interacting Bose systems are studied in the regions above the Bose–Einstein condensation temperature

T_{c}

, as well as below this temperature. The strength of particle fluctuations defines whether the system is stable or not. Stability conditions depend on the spatial dimensionality d and on the confining dimension D of the system. The consideration shows that mesoscopic systems, experiencing Bose–Einstein condensation, are stable when: (i) ideal Bose gas is confined in a rectangular box of spatial dimension

d > 2

above

T_{c}

and in a box of

d > 4

below

T_{c}

; (ii) ideal Bose gas is confined in a power-law trap of a confining dimension

D > 2

above

T_{c}

and of a confining dimension

D > 4

below

T_{c}

; (iii) the interacting Bose system is confined in a rectangular box of dimension

d > 2

above

T_{c}

, while below

T_{c}

, particle interactions stabilize the Bose-condensed system, making it stable for

d = 3

; (iv) nonlocal interactions diminish the condensation temperature, as compared with the fluctuations in a system with contact interactions. Full article

(This article belongs to the Special Issue Symmetry and Mesoscopic Physics)

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