Role Of Symmetry Research Articles

Symmetry or Asymmetry? Complex Problems and Solutions by Computational Intelligence and Soft Computing

What is the role of symmetry in the seemingly far away topics of solving complex applied problems by approaches offered by Soft Computing and Computational Intelligence [...]

Strong spin filtering by silicon nanoparticles with adsorbed bismuth atom clusters: Role of symmetry and electrostatic control of the direction of spin polarization

We present a theoretical study, based on density functional theory and tight-binding modeling, of the electronic structure and spin transport properties of silicon nanoparticles with adsorbed bismuth atoms. We find the bismuth atoms to form clusters separated by quantum tunnel barriers. We predict strong spin filtering by these nanostructures in the high conductance regime when the source and drain leads are connected to the same bismuth cluster and in the low conductance regime when the source and drain leads are connected to different bismuth clusters. We relate the spin filtering to a symmetry obeyed by the spin transmission probability matrix of the system. We also predict that for such systems the direction of the spin polarization in the drain lead can be tuned through large angles and even reversed electrostatically simply by varying the voltage applied to a gate. Realization of these silicon-bismuth nanostructures in the laboratory is feasible. We expect the predicted spin filtering to be experimentally accessible and potentially relevant for device applications.

SPONTANEOUS SYMMETRY BREAKING ON THE EXAMPLE OF THE KLEIN —GORDON REAL FIELD

This scientific work examiner symmetry in nature and physics, its mathematical description. The spontaneous breaking of the Klein—Gordon real field is considered. The role of symmetry and spontaneous symmetry breaking in the development of physics is shown.

Long-Time Bit Storage and Retrieval without Cold Atom Technology

We report computer studies showing how the duration of memory for storage and retrieval of a classical bit can be increased to 100 times the decay time of an isolated atom, with no use of high-tech cold-atom preparations recently developed in the light-matter field. We suggest that our low-tech procedure can greatly enlarge the number of experimenters able to enter this field. The role of symmetry in this procedure arises in a careful interplay of incoherent and coherent excitations of a large collection of “two-level” atoms, the level separation being matched by the dominant frequency of the electromagnetic fields (short pulses and continuing field) applied to the system.

Instability of universal terms in the entanglement entropy

The role of symmetries in what concerns entanglement entropy has been extensively explored in the last years and revealed a profound connection with the quantum field theory's algebraic structure. Recently, it was found that some universal contributions to the entanglement entropy and mutual information may be non uniquely defined in theories with generalized symmetries. Here, we study this issue in detail in the particular case of the entanglement entropy of the Maxwell theory in $(2+1)$ dimensions for rotationally symmetric regions. In this setup, the problem can be dimensionally reduced to a half-line. We find that the only difference between the reduced problem for the Maxwell field and the reduced scalar free field stems from the Fourier angular $n=0$ mode. This simplification allows us to check explicitly the many issues that characterize models with broken global symmetries. Namely, we manifestly show that the additive algebras break Haag duality, and single out the non-local operators which are responsible for the failure of this property. More interestingly, we present concrete lattice realizations that confirm that the logarithmic "universal" term of the Maxwell entanglement entropy for disks depends on the details of the algebra assignation. This ambiguity hinders the identification of possible topological contributions characteristic of models with generalized symmetries and tarnishes its universal character. We further calculate the Maxwell mutual information for two nearly complementary concentric disks. We obtain the expected universal contribution with a log-log dependence and check that, unlike entropy, this is stable. Accordingly, this supports mutual information as the appropriate probe to sense additivity-duality breaking and the consequent universal topological contributions.

Revealing the Symmetry of Materials through Neutron Diffraction

Magnetic materials are used in many devices in everyday life. To control their properties, we must first understand how they are ordered. This can be accomplished through neutron diffraction measurements. However, in many cases, there are too many parameters to determine the structure uniquely. Fortunately, symmetry can greatly constrain the number of parameters. Symmetry can also allow us to determine which physical properties are possible. In this review, I discuss the role of symmetry in magnetic structure determination using neutron diffraction. In this review, I will discuss both representational analysis as well as the magnetic superspace formalism. I will also discuss where the magnetic structure has been critical to understanding the fundamental science of the problem.

Optical second harmonic generation: role of symmetry and local resonances (review)

We report some results of studying optical second-harmonic generation (SHG) on surfaces and at interfaces of centrosymmetric media, in resonant nano- and microstructures, and in ferroelectric materials. The research was carried out at the Department of Quantum Electronics of the Lomonosov Moscow State University under the supervision of Professor O.A. Aktsipetrov, and subsequently developed by his disciples. As examples that clearly demonstrate the possibilities of the SHG method for examining nonstandard objects of nonlinear optics, we discuss the behaviour of the nonlinear optical response of single-crystal silicon and germanium surfaces and their interfaces with oxides, as well as nonlinear electroreflection. Optical interferometry and its possibilities are briefly described using the example of these systems. Unique sensitivity of SHG to the symmetry and resonance properties of nanostructures, including magnetic ones, is shown, which determines the efficiency of this method for investigating such systems. Finally, we demonstrate that the SHG effect is a unique remote and sensitive method for studying ferroelectric structures.

Two-point resistances in Archimedean resistor networks

The rise of novel materials such as graphene compounds or carbon nanotubes recently revived the interest on the role of symmetries upon electrical properties. Among possible solids, symmetrical polytopes, widely realized in natural and artificial matter, deserve particular attention. When equal resistors are attached between neighbouring nodes of Perfect (Platonic) solids (PS) along the edges, the two-point resistance between any pair of nodes was established in the 1990s by van Steenwijk using an elegant and efficient method. Using van Steenwijk’s method, Moddy and Aravind subsequently derived the resistance values for two Archimedean and two Catalan solids. Here, with the same method, we derive the exact expression for the two-point resistance between any two nodes for resistor networks made of equal resistors placed along the edges of the thirteen Archimedean solids (AS). While the calculation remains elementary for the simplest AS, a dedicated method using computer assistance was developed for the Snub Cube, the Snub Dodecahedron and the three AS with three different rotation symmetries (the Great Rhombicuboctahedron, the Small Rhombicosidodecahedron and the Great Rhombicosidodecahedron). The largest resistance value (191/90) is obtained between the two opposite nodes of the Truncated Dodecahedron, and the second largest (42815/21114) is obtained between the two opposite nodes of the Great Rhombicosidodecahedron. The smallest resistance value (14137/38016) is obtained between some of the neighbouring nodes of the Snub Cube, whereas the resistance between neighbouring nodes of the Icosahedron (11/30) is slightly smaller. Some general symmetry relations between two-point resistances in AS networks are also derived, as well as some relations with two-point resistances in PS networks. The complete set of exact two-point resistance values for PS and AS networks can be used to check numerical codes or to evaluate the capacity of regular solids to provide appropriate models for given experimental situations.

Interaction of nanoparticles with lipid films: the role of symmetry and shape anisotropy.

The bioactivity, biological fate and cytotoxicity of nanomaterials when they come into contact with living organisms are determined by their interaction with biomacromolecules and biological barriers. In this context, the role of symmetry/shape anisotropy of both the nanomaterials and biological interfaces in their mutual interaction, is a relatively unaddressed issue. Here, we study the interaction of gold nanoparticles (NPs) of different shapes (nanospheres and nanorods) with biomimetic membranes of different morphology, i.e. flat membranes (2D symmetry, representative of the most common plasma membrane geometry), and cubic membranes (3D symmetry, representative of non-lamellar membranes, found in Nature under certain biological conditions). For this purpose we used an ensemble of complementary structural techniques, including Neutron Reflectometry, Grazing Incidence Small-Angle Neutron Scattering, on a nanometer lengthscale and Confocal Laser Scanning Microscopy on a micrometer length scale. We found that the structural stability of the membrane towards NPs is dependent on the topological characteristic of the lipid assembly and of the NPs, where a higher symmetry gave higher stability. In addition, Confocal Laser Scanning Microscopy analyses highlighted that NPs interact with cubic and lamellar phases according to two distinct mechanisms, related to the different structures of the lipid assemblies. This study for the first time systematically addresses the role of NPs shape in the interaction with lipid assemblies with different symmetry. The results will contribute to improve the fundamental knowledge on lipid interfaces and will provide new insights on the biological function of phase transitions as a response strategy to the exposure of NPs.

The Development and Role of Symmetry in Ancient Scripts

This paper investigates various scripts regarding the occurrences of signs that have mir- ror symmetry along a central vertical line. The scripts belong to one of the following three script families: the Phoenician script family, the Minoan-Mycenaean script family, and the Indus Valley script family. It is shown that many scripts in these script families contain a high percentage of signs that have mirror symmetry. Moreover, the scripts within the Phoenician and the Minoan- Mycenaean script families show a tendency of increased percentage of mirror-symmetric signs over time. For example, while the Phoenician Alphabet contains 40.9 percent mirror-symmetric signs, one of its descendants, the Euclidean Greek Alphabet, contains 59.3 percent mirror-symmetric signs. The paper also identifies the boustrophedonic way of writing and religious writings with a deliberate mirroring as an afterlife symbolism as possible causes of the increased use of mirror- symmetric signs.

Providing theoretical insight into the role of symmetry in the photoisomerization mechanism of a non-symmetric dithienylethene photoswitch.

Dithienylethene (DTE) molecular photoswitches have shown to be excellent candidates in the design of efficient optoelectronic devices, due to their high photoisomerization quantum yield (QY), for which symmetry is suggested to play a crucial role. Here, we present a theoretical study on the photochemistry of a non-symmetric dithienylethene photoswitch, with a special emphasis on the effect of asymmetric substitution on the photocyclization and photoreversion mechanisms. We used the Spin-Flip Time Dependent Density Functional Theory (SF-TDDFT) method to locate and characterize the main structures (conical intersections and minima) of the ground state and the first two excited states, S1 and S2, along the ring-opening/closure reaction coordinate of the photocyclization and photoreversion processes, and to identify the important coordinates governing the radiationless decay pathways. Our results suggest that while the main features that characterize the photoisomerization of symmetric DTEs are also present for the photoisomerization of the non-symmetric DTE, the lower energy barrier on S1 along the cycloreversion reaction speaks in favor of a more efficient and therefore a higher cycloreversion QY for the non-symmetric DTEs, making them a better candidate for molecular optoelectronic devices than their symmetric counterparts.

Painlevé properties, gauge invariance and solitons of some classes of reaction-diffusion equations

The role of symmetries and conservation laws are well known mechanisms for the reduction of systems of differential equations and, used in conjunction, lead to double reductions of the underlying models. We show here that variational and gauge symmetries have additional applications in the integrability of differential equations. In particular, we present a broad class of diffusion type equations, viz., the Fisher–Kolmorov and Fitzhugh–Nagumo equations, that satisfy Pinlevé properties, of their respective travelling wave forms and solitons, under the existence of gauge symmetries following a variational principle.

Symmetry-protected solitons and bulk-boundary correspondence in generalized Jackiw\u2013Rebbi models

We investigate the roles of symmetry and bulk-boundary correspondence in characterizing topological edge states in generalized Jackiw–Rebbi (JR) models. We show that time-reversal (T), charge-conjugation (C), parity (P), and discrete internal field rotation (Z_n) symmetries protect and characterize the various types of edge states such as chiral and nonchiral solitons via bulk-boundary correspondence in the presence of the multiple vacua. As two representative models, we consider the JR model composed of a single fermion field having a complex mass and the generalized JR model with two massless but interacting fermion fields. The JR model shows nonchiral solitons with the Z_2 rotation symmetry, whereas it shows chiral solitons with the broken Z_2 rotation symmetry. In the generalized JR model, only nonchiral solitons can emerge with only Z_2 rotation symmetry, whereas both chiral and nonchiral solitons can exist with enhanced Z_4 rotation symmetry. Moreover, we find that the nonchiral solitons have C, P symmetries while the chiral solitons do not, which can be explained by the symmetry-invariant lines connecting degenerate vacua. Finally, we find the symmetry correspondence between multiply-degenerate global vacua and solitons such that T, C, P symmetries of a soliton inherit from global minima that are connected by the soliton, which provides a novel tool for the characterization of topological solitons.

The Role of Symmetry in Non-Hermitian Scattering1

We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

A somewhat random walk through nuclear and particle physics

These notes are an outgrowth of an advanced undergraduate course taught at the University of Maryland, College Park. They are intended as an introduction to various aspects of particle and nuclear physics with an emphasis on the role of symmetry. The basic philosophy is to introduce many of the fundamental ideas in nuclear and particle physics using relatively sophisticated mathematical tools -- but to do so in as a simplified a context to explain the underlying ideas. Thus, for example, the Higgs mechanism is discussed in terms of an Abelian Higgs model. The emphasis is largely, but not entirely theoretical in orientation. The goal is for readers to develop an understanding of many of the underlying issues in a relatively sophisticated way.

Role of Symmetry in Quantum Search via Continuous-Time Quantum Walk

For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.

A Certain Structure of Bipolar Fuzzy Subrings

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

Symmetries in geometric control theory using Maple

We focus on the role of symmetries in geometric control theory from the Hamiltonian viewpoint. We demonstrate the power of Ian Anderson’s package DifferentialGeometry in CAS software Maple for dealing with control problems on Lie groups. We apply the tools to the problem of vertical rolling disc, however, anyone can modify our approach and tools to other control problems.

A Role of Symmetries in Evaluation of Fundamental Bounds

A problem of the erroneous duality gap caused by the presence of symmetries is solved in this paper utilizing point group theory. The optimization problems are first divided into two classes based on their predisposition to suffer from this deficiency. Then, the classical problem of Q-factor minimization is shown in an example where the erroneous duality gap is eliminated by combining solutions from orthogonal sub-spaces. Validity of this treatment is demonstrated in a series of subsequent examples of increasing complexity spanning the wide variety of optimization problems, namely minimum Q-factor, maximum antenna gain, minimum total active reflection coefficient, or maximum radiation efficiency with self-resonant constraint. They involve problems with algebraic and geometric multiplicities of the eigenmodes, and are completed by an example introducing the selective modification of modal currents falling into one of the symmetry-conformal sub-spaces. The entire treatment is accompanied with a discussion of finite numerical precision, and mesh grid imperfections and their influence on the results. Finally, the robust and unified algorithm is proposed and discussed, including advanced topics such as the uniqueness of the optimal solutions, dependence on the number of constraints, or an interpretation of the qualitative difference between the two classes of the optimization problems.

Singular loci of reflection arrangements and the containment problem

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals. Our work ties together and generalizes results in Bauer et al. (Int Math Res Not IMRN 24:7459–7514, 2019), Dumnicki et al. (J Algebra 393:24–29, 2013), Harbourne and Seceleanu (J Pure Appl Algebra 219(4):1062–1072, 2015) and Malara and Szpond (J Pure Appl Algebra 222(8):2323–2329, 2018) under a unified approach.