Symmetry Aspects Research Articles

Superconducting order parameter of Sr2RuO4 : A microscopic perspective

The character of the superconducting phase of Sr$_2$RuO$_4$, is topic of a longstanding discussion. The classification of the symmetry allowed order parameters has relied on the tetragonal symmetry of the lattice and on cylindrical Fermi surfaces, usually taken to be featureless, not including the non-trivial symmetry aspects related to their orbital content. Here we show how the careful account of the orbital degree of freedom in Sr$_2$RuO$_4$, leads to a much richer classification of order parameters. We analyse the stability and degeneracy of these new order parameters from the perspective of the concept of superconducting fitness and propose a new best order parameter candidate.

Potential-Energy Surfaces for Ring-Puckering Motions of Flexible Cyclic Molecules through Cremer-Pople Coordinates: Computation, Analysis, and Fitting.

Ring-puckering motion in 12 flexible cyclic molecules is investigated by calculation and analysis of two-dimensional potential-energy surfaces (PESs) using the so-called ring-puckering coordinates proposed by Cremer and Pople. The PESs are calculated by means of density-functional theory using a B2PLYP-D3BJ exchange-correlation functional with a maug-cc-pVTZ basis set, and results are compared to the available experimental and theoretical data. Special care is devoted to the aspect of symmetry in such two-dimensional PESs, which are here reported for the first time also for molecules whose planar form has symmetry lower than D5 h or C2 v. The issue of PES fitting and that of solving the nuclear dynamics using ring-puckering coordinates are also addressed. Analytical formulations of the computed PESs using suitable functional forms with a limited set of parameters are provided.

Development of the Local (Site) Symmetry Method in the Supercell Model for a Crystal with an Impurity

Symmetry aspects of the periodic model of a crystal with a point defect (a supercell model or a extended unit cell (EUC) model) are considered; splitting of the Wyckoff positions in a primitive crystal cell with the introduction of supercell and the change of the one-electron states classification over k-vector (Brillouin zone folding) are discussed. When considering a point defect in a crystal in the supercell model, it is necessary to take into account the symmetry of the one-electron states of the original crystal at the top of the valence band and at the bottom of the conduction band. The selected supercell should reproduce these states. Each specific selection of a supercell corresponds to a specific splitting of the Wyckoff positions of the original crystal and, as a result, the possibility to place a defect in positions with different point symmetry and to perform a calculation without taking into account the point symmetry of the crystal with a defect at all (site symmetry method). By the results of the calculation of a crystal with a defect in the supercell model, taking into account site symmetry, it is possible to determine the real symmetry of the crystal with a defect, which is essential for the interpretation of experimental data. A copper impurity in a lithium node in a LiCl crystal retains the cubic symmetry of the atom being replaced; the impurity of an iron atom in a titanium site with cubic local symmetry in a SrTiO3 crystal lowers the symmetry to tetragonal; and the H site in the CsPbI3 crystal is characterized by the complete removal of point symmetry with the formation of an $${\text{I}}_{2}^{ - }$$ ion.

Petroglyphs as ‘contraptions’ – animacy and vitalist technologies in a Bronze Age archipelago

ABSTRACTThe text discusses the possibilities of Bronze Age petroglyphs working as vitalist contraptions produced to harvest or affect certain animacies in coastal milieus. As primarily magical devices, a wider range of properties of the petroglyphs becomes important besides their figurative and representational facets. For example, the mediality and physicality of the motifs, including aspects of size, embellishment, and symmetry, as well as relations between the motifs, the rock and the water. A case study based on the dense concentration of rock art in the Boglösa area of central eastern Sweden suggests, considering these aspects in full, that the petroglyphs are most likely to have worked as magic contraptions. Cut into the bedrock at the water’s edge, they utilize the resilience of rock for permanence while either being consecrated by water or employed as contraptions to affect a particular body of water.

Geometric transformations and Talavera tiles: a culturally responsive approach to teacher professional development and mathematics teaching

Research in mathematics education show that students learn more mathematics when it is connected to what they hear, see, and do outside of school; students’ cultural knowledge and experiences can serve as familiar, relatable contexts for learning mathematics. When teachers adopt a culturally responsive stance to teaching mathematics, they integrate the knowledge and experiences of students as situated in their cultures, homes, and communities. The following paper presents Talavera tiles as one example of a culturally responsive context for teaching geometry. The authors first briefly present the experiences of seven elementary teachers who participated in a professional development workshop, Te’ALaMo (Teachers, Art from Latin@ cultures, and Mathematical modelling) where they used Talavera tiles to explore aspects of symmetry. In the second half of the paper, the authors extend and unpack the mathematics of Talavera tiles in the context of teaching geometry for children and adolescents.

Understanding grain boundaries – The role of crystallography, structural descriptors and machine learning

Grain boundaries (GBs) influence a wide array of physical properties in polycrystalline materials and play an important role in governing microstructural evolution under extreme environments. While the importance of interfaces is well documented, their properties are among the least understood of all the defect types present in engineering material systems. This is due to the vast configurational space of interfaces, resulting in a diverse range of structures and properties. The complexity associated with GB structures is related to the different crystallographic degrees of freedom – the misorientation, the boundary-plane orientation and the relative translations between the adjoining crystals. These unique challenges can be addressed by leveraging high-throughput simulations of GB properties and developing machine learning algorithms grounded in the concepts of bicrystallography of interfaces. To demystify the relationships between crystallography and properties, the symmetry aspects of GBs are reviewed with an emphasis on boundary-plane orientations and disconnection line defects. To quantify structure-property relationships, recent advances in describing GB structures using the polyhedral unit model and the gaussian-based approximation of local atomic environments are discussed. Finally, examples of predicting GB structure-property relationships using machine learning techniques are summarized. As part of a special issue, the goal of this review article is to motivate machine learning strategies that are informed by bicrystallography and novel structural descriptors for developing reliable crystallography-structure-property relationships for grain boundaries.

Symmetry aspects of spin filtering in molecular junctions: Hybridization and quantum interference effects

Control and manipulation of electric current and, especially, its degree of spin polarization (spin filtering) across single molecules are currently of great interest in the field of molecular spintronics. We explore one possible strategy based on the modification of nanojunction symmetry which can be realized, for example, by a mechanical strain. Such modification can activate new molecular orbitals which were inactive before due to their orbital mismatch with the electrode's conduction states. This can result in several important consequences such as (i) quantum interference effects appearing as Fano-like features in electron transmission and (ii) the change in molecular level hybridization with the electrode's states. We argue that the symmetry change can affect very differently two majority- and minority-spin conductances and thus alter significantly the resulting spin-filtering ratio as the junction symmetry is modified. We illustrate the idea for two basic molecular junctions: Ni/benzene/Ni (perpendicular vs tilted orientations) and Ni/Si chain/Ni (zigzag vs linear chains). In both cases, one highest occupied molecular orbital (HOMO) and one lowest unoccupied molecular orbital (LUMO) (out of HOMO and LUMO doublets) are important. In particular, their destructive interference with other orbitals leads to dramatic suppression of majority-spin conductance in low-symmetry configurations. For a minority-spin channel, on the contrary, the conductance is strongly enhanced when the symmetry is lowered due to an increase in hybridization strength. We believe that our results may offer a potential route for creating molecular devices with a large on-off ratio of spin polarization via quantum interference effects.

Postural Symmetry Evaluation Based on the Analysis of Temporary and Average CoP Displacements Registered During the Follow-Up Posturography

This paper presents a method enabling a postural symmetry assessment based on the evaluation of similarity of temporary and average CoP (Center of Pressure) displacements registered in response to the clockwise and counter-clockwise visual stimuli applied while performing the so-called follow-up posturography. This kind of visual feedback diagnostics is an intermediary between the static posturography and the dynamic posturography. One of its advantages is the ability to evaluate the dynamic performance of the human balance and posture control mechanisms using relatively inexpensive and popular static posturography platforms. The method presented in this paper was developed as a means for measuring the effectiveness of the rehabilitation program following total hip arthroplasty. The postural symmetry is, in this case, evaluated as the degree of mutual symmetry of the visually stimulated loading exerted on the left and right lower limbs. Usability of the method was verified in the group of 30 patients rehabilitated after total hip arthroplasty. The statistical analysis confirmed a significant growth of the values of the proposed symmetry measure over the period of the 21-day rehabilitation program (p<; 0.001 ). There were, however, no significant correlations between that measure and the symmetry measures applied in the case of static posturography. The obtained results support the statement that the herein presented diagnostic approach enables the quantification of some other aspects of postural symmetry, namely the dynamic ones. This also corroborates the diagnostic value of the method discussed in this paper.

Развитие метода локальной симметрии в модели суперъячейки для кристалла с примесью

The symmetry aspects of the supercell (extended primitive unit cell) model of the crystal with point defect are considered: Wyckoff positions splitting and the change of the one-electron states classification over k-vector (Brillouin zone folding). The supercell choice has to be made in a such a way that the one-electron states at the valence band top and the conduction band bottom for the perfect crystal are reproduced. Wyckoff positions splitting in the supercell model allows one to consider the defective crystal using different site symmetry point groups and also without use of the point symmetry at all. The site symmetry method allows one to predict the real symmetry of the defective crystal. This is demonstrated in the calculations of Cu impurity in LiCl crystal (the cubic symmetry of the perfect crystal is maintained for defective crystal), Fe impurity in SrTiO3 crystal (the point defect lowers the symmetry from cubic to tetragonal) and interstitial iodine atom in CsPbI3 crystal (the experimentally found I_2^- dumbell structure is confirmed only in the calculations without point symmetry restriction).

Hadron-quark phase transition: the QCD phase diagram and stellar conversion

Different extensions of the Nambu-Jona-Lasinio model, known to satisfy expected QCD chiral symmetry aspects, are used to investigate a possible hadron-quark phase transition at zero temperature and to build the corresponding binodal sections. We have shown that the transition point is very sensitive to the model parameters and that both pressure and chemical potential increase drastically with the increase of the vector interaction strength in the quark sector. Within the same framework, the possibility of quark and hybrid star formation is analyzed. The same conclusions drawn before with respect to the coexistence pressure and chemical potentials are reinforced. We conclude that even if a transition from a metastable hadronic star to a quark star is thermodinamically possible, it is either energetically forbidden or gives rise to a blackhole. Nevertheless, conversions from metastable to hybrid stars are possible, but the mass difference between both compact objects is very small, never larger than 0.2 M⊙.

Testing topological protection of edge states in hexagonal quantum spin Hall candidate materials

We analyze the detailed structure of topological edge mode protection occuring in hexagonal quantum spin Hall (QSH) materials. We focus on bismuthene, antimonene, and arsenene on a SiC substrate, which, due to their large bulk gap, may offer new opportunities for room-temperature QSH applications. While time reversal symmetry is responsible for the principal symmetry protected character of QSH states, the hexagonal edge terminations yield further aspects of crystal symmetry which affect the topological protection. We show that armchair QSH edge states remain gapless under an in-plane magnetic field in the direction along the edge, a hallmark of their topological crystalline protection. In contrast, an out-of-plane magnetic field opens a gap of the order of a few meV within realistic ranges of parameters. We use these intriguing signatures of armchair QSH edge states to predict experimentally testable fingerprints of their additional topological crystalline character and their helicity emerging in tunneling spectroscopy and ballistic magnetotransport.

Crystal symmetry aspects of materials with magnetic spin reorientation.

The symmetry of materials which undergo a continuous spin reorientation has been analysed. It is shown that continuous spin reorientation is possible only in materials with triclinic or monoclinic crystal structure symmetry, i.e. other symmetries - orthorhombic, tetragonal, trigonal, hexagonal and cubic - are forbidden.

Emergent multiplicities of self- and other-construction in Korean workplace-based television dramas

The Korean language contains a complexly intricate system of self- and other- reference marking that serves to designate multiple aspects of socio-cultural symmetry and asymmetry involving speakers, addressees, hearers, overhearers, and referents. In this article, we use a database of TV workplace-based contexts and analyze the intricately creative ways in which interactants position themselves vis à vis their interlocutors, with fine-grained, incrementally gauged bits of gap-creating and gap-effacing discourse, through the use or non-use of honorific markings We demonstrate, through an appeal to Positioning Theory and Indexicality that these fine-grained linguistic indicators of emergent multiplicities of self-and other-construction serve as metaphorical scalar points on the sociometer of interpersonal interaction. We uncover discursive tension within these interactional contexts whereby informal interpersonal relationships leak into the more formal workplace background. Having used data based on a workplace environment, we find elements of language choice that index the intrinsic institutional hierarchy, in addition to discursive features that index other aspects of the relationships, such as prior acquaintance, as well as varying degrees of intimacy, expertise, sarcasm, coercion, and so forth.

A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity

Efficient conditions guaranteeing the existence of a T-periodic solution to the second order differential equation $$\begin{aligned} u''=\frac{h(t)}{u^{\lambda }},\quad \lambda \in (0,1), \end{aligned}$$ are established. Here, $$h\in L(\mathbb {R}/T\mathbb {Z})$$ is a rather general sign-changing function with $$\overline{h}<0$$ . In contrast with the results in Godoy and Zamora (Proc R Soc Edinb Sect A Math) and Hakl and Zamora (J Differ Equ 263:451–469, 2017), the key ingredient to solve the aforementioned problem seems to be connected more with the oscillation and the symmetry aspects of the weight function h than with the multiplicity of its zeroes. Roughly speaking, the solvability for the above-mentioned problem can be guaranteed when $$H_+\approx H_-$$ and $$H_+$$ is large enough.

Entanglement production by the magnetic dipolar interaction dynamics

We consider two qubits prepared in a product state and evolved under the magnetic dipolar interaction (MDI). We describe the dependence of the entanglement generated by the MDI with time, with the interaction parameters, and with the system’s initial state, identifying the symmetry and coherence aspects of those initial configurations that yield the maximal entanglement. We also show how one can obtain maximum entanglement from the MDI applied to some families of partially entangled initial states.

Topological node lines in mechanical metacrystals

Topological acoustic and elastic waves have recently emerged as an exciting interdisciplinary field which is still mainly focused on low-dimensional structures and model systems. Here we demonstrate numerically an elastic-wave analogue of topological node-lines in three-dimensional mechanical metacrystals with ribbon- or drumhead-like surface states. These two-dimensional topological surface states offer unprecedented, robust subwavelength confinement of elastic waves. Design principles for topological mechanical metamaterials, from both material and symmetry aspects, are unveiled and connected to fundamental conservation laws and nonsymmorphic space group. Our study paves the way toward the synergy between three-dimensional mechanical metamaterials and topological wave dynamics.

Julia fractal based multi-scroll memristive chaotic system

A memristor can be used in chaotic system as a nonlinear term, and thus enhancing the complexity of the chaotic system. Fractal theory is a leading and important branch of nonlinear science, and has been widely studied in many fields in the past few decades. The fractal and chaos are bound tightly and their relevant researches are well-established, but few of them focus on the research of the possibility of combining the fractal and the chaotic system. In order to obtain a multi scroll chaotic attractor, the fractal process is novelty introduced into the memristive chaotic system. In this paper, at first, a new memristive chaotic system is proposed. Then, the dynamic characteristics of the system are discussed from the aspects of symmetry, dissipation, stabilization of equilibrium points, power spectrum, Lyapunov exponent and fractional dimension. A mapping relationship based on classical Julia fractal is established. Through this mapping relationship, a multi-scroll memristive chaotic system based on the Julia fractal is obtained. Moreover, several deformed Julia fractal processes are applied to the memristive chaotic system, and abundant chaotic attractors are obtained. For example, the square term of the Julia fractal expression is multiplied by a coefficient, and according to the difference in coefficient, the resulting chaotic attractors have the same shape but different sizes. The exponent of the square term in the Julia fractal is changed into a variable, and the chaotic attractor of different scroll numbers is obtained according to the difference in power exponent. In addition, a rich multi-scroll chaotic attractor is obtained by using the fractal expression in the form of weighted sum polynomial. Finally, the influence of a complex constant in the fractal process on the system is discussed. The simulation results show that the combination of fractal process and chaotic system can obtain rich chaotic attractors, such as multi-scroll chaotic attractors. In general, compared with the single-scroll chaotic attractor, the multi-scroll chaotic attractor has a higher complexity and more adjustability. In addition, compared with other multi-scroll chaotic system, the proposed multi-scroll chaotic system is easy to adjust the number of the scrolls. To summarize, this work not only provides a new method of generating multi-scroll chaotic attractors, but also makes up for the lack of smoothness of the chaotic system caused by using functional methods.

Parity-time symmetry meets photonics: A new twist in non-Hermitian optics

In the past decade, the concept of parity-time symmetry, originally introduced in non-Hermitian extensions of quantum mechanical theories, has come into thinking of photonics, providing a fertile ground for studying, observing, and utilizing some of the peculiar aspects of symmetry in optics. Together with related concepts of non-Hermitian physics of open quantum systems, such as non-Hermitian degeneracies (exceptional points) and spectral singularities, symmetry represents one among the most fruitful ideas introduced in optics in the past few years. Judicious tailoring of optical gain and loss in integrated photonic structures has emerged as a new paradigm in shaping the flow of light in unprecedented ways, with major applications encompassing laser science and technology, optical sensing, and optical material engineering. In this perspective, I review some of the main achievements and emerging areas of -symmetric and non-Hermtian photonics, and provide an outline of challenges and directions for future research in one of the fastest growing research area of photonics.

Backscattering in a helical liquid induced by Rashba spin-orbit coupling and electron interactions: Locality, symmetry, and cutoff aspects

The combination of the time-reversal-symmetric single-particle backscattering field (commonly known as Rashba spin-orbit coupling) and non-backscattering electron interactions is generally expected to produce inelastic backscattering in 1D helical electron liquids at the edge of 2D topological insulators, as theoretically predicted in a number of works. An opposite conclusion of absent backscattering was reached in a recent work [H.-Y. Xie et al., Phys. Rev. Lett. 116, 086603 (2016)] for the "local" model of the backscattering field and interactions. Motivated to resolve this potential controversy, in the present work, we study backscattering effects employing fermionic perturbation theory and considering quite general forms of the backscattering field and electron interactions. We discover that backscattering effects are crucially sensitive to the locality properties of the backscattering field and electron interactions, to the symmetry of the latter, as well as to the presence or absence of the cutoff of the electron spectrum. We find that backscattering is indeed absent under the following assumptions: (i) local backscattering field; (ii.a) local or (ii.b) SU(2)-symmetric interactions; (iii) absent cutoff of the edge-state spectrum. However, violation of any of these conditions leads to backscattering. This also reconciles with the results based on the bosonization technique. We calculate the associated backscattering current, establish its low-bias scaling behavior, and predict a crossover between two different scaling regimes. The main implication of our findings is that backscattering of some magnitude is inevitable in a real system, although could be quite suppressed for nearly local backscattering field and interactions.

Temperature dependence of local structural changes around transition metal centers Cr3+ and Mn2+ in RAl3(BO3)4 crystals studied by EMR

Theoretical investigations are carried out to determine the temperature dependence of the local structural parameters of Cr3+ and Mn2+ ions doped into RAl3(BO3)4 (RAB, R = Y, Eu, Tm) crystals. The zero-field splitting (ZFS) parameters (ZFSPs) obtained from the spin Hamiltonian (SH) analysis of EMR (EPR) spectra serve for fine-tuning the theoretically predicted ZFSPs obtained using the semi-empirical superposition model (SPM). The SPM analysis enables to determine the local structure changes around Cr3+ and Mn2+ centers in RAB crystals and explain the observed temperature dependence of the ZFSPs. The local monoclinic C2 site symmetry of all Al sites in YAB necessitates consideration of one non-zero monoclinic ZFSP (in the Stevens notation, b21) for Cr3+ ions. However, the experimental second-rank ZFSPs (D=b20, E=1/3b22) were expressed in a nominal principal axis system. To provide additional insight into low symmetry aspects, the distortions (ligand's distances ΔRi and angular distortions Δθi) have been varied while preserving monoclinic site symmetry, in such way as to obtain the calculated values (D, E) close to the experimental ones, while keeping b21 close to zero. This procedure yields good matching of the calculated ZFSPs and the experimental ones, and enables determination of the corresponding local distortions. The present results may be useful in future studies aimed at technological applications of the Huntite-type borates with the formula RM3(BO3)4. The model parameters determined here may be utilized for ZFSP calculations for Cr3+ and Mn2+ ions at octahedral sites in single-molecule magnets and single-chain magnets.