Symmetry Classes Research Articles

The Morphology of the Regular & Semi-Regular Polyhedra and Tessellations According to the Separation of Facial Polytopes

In previous work, inspired by Critchlow, and by Grünbaum & Shephard, I proposed an integral 2.5D cubic schema of the regular and semi-regular polyhedra and polygonal tessellations of the plane for each class of symmetry, which could be differentiated into an upper and lower layer of 4 polytopes each, and characterized by corresponding pairs, so that upper polytope always corresponds to lower. I explored the motif of paired two-step sequences of first alternating facial separation and morphological transformation, and second facial morphological transformation and separation, which in the 2D consideration of the 2.5D schema are disposed about the vertical axis, as characterized by the correspondence between the  of the lower and upper squares (diamonds or rhombi). Following intensive research, I here focus on a deeper pattern of morphological transformation of the primary prototypes that is characterized by the separation of one gendered set of the negative (−ve), neutral (ntrl), or positive (+ve) facial polytopes along the Y, Z, & X axes of the cubic schema. As one set of faces separates, the other two sets morph/ project if polar/neutral, through nullregular or quasi-regulardouble facial levels (0α|β2) of the rhombic schema or its reflection. Each facial set separates just once: d=01. The cubic schema reveals significant three-fold symmetry by gender. The separation of faces provides an adequate schema for the morphology of the three classes of the regular and semi-regular polyhedra of {2,3,3}, {2,3,4}, and {2,3,5} symmetry, and two classes of polygonal tessellations (tilings) of {2,3,6} and {2,4,4} symmetry.

Universal topological marker

The topological order of Dirac models in any dimension and symmetry class can be mapped to lattice sites by a universal topological marker. From the topological operator that extracts this marker, one can further define a nonlocal marker that detects topological phase transitions. Several prototype models such as the Su-Schrieffer-Heeger model, Majorana chain, Chern insulator, Bernevig-Hughes-Zhang model, chiral and helical $p$-wave superconductors, three-dimensional time-reversal symmetric topological insulators, among many others, are employed to demonstrate the ubiquity of these markers.

On exotic linear materials: 2D elasticity and beyond

The symmetry classes of a linear constitutive law define the different types of anisotropy that can be modelled by it. However, the spaces of linear materials are very rich and a whole range of intermediate possibilities exists beyond symmetry classes. Geometric methods developed to characterise spaces of linear materials in a very fine way allow these intermediate possibilities to be detected. Materials with non-standard anisotropic properties associated with these intermediate possibilities are called exotic materials. In this paper, we provide a mathematical and mechanical definition of what an exotic material is. Using these definitions, we conclude that, for 2D linear elasticity, there is only one possible exotic material that meets our criteria. An example of a unit cell producing such exotic material is determined. Finally, the enumeration result obtained for elasticity is generalised to other bi-dimensional linear constitutive laws.

On the symmetries in the dynamics of wide two-layer neural networks

<abstract><p>We consider the idealized setting of gradient flow on the population risk for infinitely wide two-layer ReLU neural networks (without bias), and study the effect of symmetries on the learned parameters and predictors. We first describe a general class of symmetries which, when satisfied by the target function $ f^* $ and the input distribution, are preserved by the dynamics. We then study more specific cases. When $ f^* $ is odd, we show that the dynamics of the predictor reduces to that of a (non-linearly parameterized) linear predictor, and its exponential convergence can be guaranteed. When $ f^* $ has a low-dimensional structure, we prove that the gradient flow PDE reduces to a lower-dimensional PDE. Furthermore, we present informal and numerical arguments that suggest that the input neurons align with the lower-dimensional structure of the problem.</p></abstract>

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

An Urban Odyssey: City Beautiful to City Instagrammable

AbstractProfessor at the Polytechnic University of Catalonia Jordi Oliveras describes the centuries‐long history of ‘beautifying the city’. He discusses how utopian ideas of urban form and placement or architectural design can quickly become dystopian by blindly following rules of classical symmetry or ideas of modernist fashion.

GROUP ANALYSIS FOR KLEIN-GORDON EQUATION VIA THEIR SYMMETRIES

In the present paper we have obtained the one parameter groups and symmetry transformations associated to the classical symmetries of the Klein-Gordon (KG) equation, we have also constructed an optimal system of two dimensional sub-algebras of the KG equation which provides the preliminary classification of group invariant solutions and yield the most general group invariant solution.

Outlier Rejection by Means of Median Matrices for Polarimetric SAR Covariance Symmetry Classification

Outlier Rejection by Means of Median Matrices for Polarimetric SAR Covariance Symmetry Classification

Anisotropic structure of two-dimensional linear Cosserat elasticity

In the present contribution the anisotropic structure of the two-dimensional linear Cosserat elasticity is investigated. The symmetry classes of this model are derived and detailed in a synthetic way. Particular attention is paid to specific features of Cosserat Elasticity which are the sensitivity to non-centrosymmetry as well as to chirality. These aspects are important for the application of this continuum theory to the mechanical modelling of lattices and metamaterials. In order to give a parameterisation to the Cosserat constitutive law, an explicit harmonic decomposition of its constitutive tensors is provided. Finally, using an algorithm introduced in a side paper, a minimal integrity basis, which is the minimal set of polynomial invariants generating the algebra of O(2)-invariant polynomials, is finally reported.

Quantum Wasserstein isometries on the qubit state space

We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. We derive a Wigner-type result for the cost operator involving all the Pauli matrices: in this case, the isometry group consists of unitary or anti-unitary conjugations. In the Bloch sphere model this means that the isometry group coincides with the classical symmetry group O(3). On the other hand, for the cost generated by the qubit ‘‘clock” and ‘‘shift” operators, we discovered non-surjective and non-injective isometries as well, beyond the regular ones. This phenomenon mirrors certain surprising properties of the quantum Wasserstein distance.

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries, symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics. In this paper, the finite point symmetry group of the combined KP3 and KP4 (CKP34) equation is found by means of a direct method. The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group. The point symmetries of the CKP34 equation constitute an infinite dimensional Kac-Moody–Virasoro algebra. The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.

All hourglass bosonic excitations in the 1651 magnetic space groups and 528 magnetic layer groups

The band connectivity as imposed by the compatibility relations between the irreducible representations of little groups can give rise to the exotic hourglasslike shape composed of four branches of bands and five band crossings (BCs). Such an hourglass band connectivity could enforce the emergence of nontrivial excitations like Weyl fermion, Dirac fermion, or even beyond them. On the other hand, the bosons, like phonons, magnons, and photons, were also shown to possess nontrivial topology and a comprehensive symmetry classification of the hourglass bosonic excitations would be of great significance to both materials design and device applications. Here we first list all concrete positions and representations of little groups in the Brillouin zone (BZ) related to the hourglass bosonic excitations in all 1651 magnetic space groups and 528 magnetic layer groups, applicable to three dimensional (3D) and two dimensional (2D) systems, respectively. 255 (42) MSGs (MLGs) are found to essentially host such hourglass BCs: Here ``essentially'' means that the bosonic hourglass BC exists definitely as long as the studied system is crystallized in the corresponding MSG/MLG. We also perform first-principles calculations on hundreds of 3D nonmagnetic materials, essentially hosting hourglass phonons and propose that the 2D material AlI can host hourglass phonons. We choose AuX (X = Br and I) as illustrative examples to demonstrate that two essential hourglass band structures can coexist in the phonon spectra for both materials while for AuBr, an accidental band crossing sticking two hourglasses is found, interestingly. Our results of symmetry conditions for hourglass bosonic excitations can provide a useful guide of designing artificial structures with hourglass bosonic excitations.

Fundamental constraints on the observability of non-Hermitian effects in passive systems

Utilizing scattering theory, we quantify the consequences of physical constraints that limit the visibility of non-Hermitian effects in passive devices. The constraints arise from the fundamental requirement that the system obeys causality, and can be captured concisely in terms of an internal time-delay operator, which furthermore provides a direct quantitative measure of the visibility of specific non-Hermitian phenomena in the density of states. We illustrate the implications by contrasting different symmetry classes and non-Hermitian effects, including exceptional points and the non-Hermitian skin effect, whose underlying extreme mode nonorthogonality turns out to be effectively disguised.

Whole-grain Petri Nets and Processes

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for P a Petri net, the Segal space of P -processes is shown to be the free coloured prop-in-groupoids on P . There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, every Petri net admits a universal unfolding, which in turn has associated an event structure and a Scott domain. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native. (Collective-token semantics emerges from rather drastic quotient constructions à la Best–Devillers, involving taking π 0 of the groupoids of states.)

Wronskian structures of planar symplectic ensembles

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel–Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane.

Two approximate symmetry frameworks for nonlinear partial differential equations with a small parameter: Comparisons, relations, approximate solutions

AbstractThe frameworks of Baikov–Gazizov–Ibragimov (BGI) and Fushchich–Shtelen (FS) approximate symmetries are used to study symmetry properties of partial differential equations with a small parameter. In general, it is shown that unlike the case of ordinary differential equations (ODEs), unstable BGI point symmetries of unperturbed partial differential equations (PDEs) do not necessarily yield local approximate symmetries for the perturbed model. While some relations between the BGI and FS approaches can be established, the two methods yield different approximate symmetry classifications. Detailed classifications are presented for two nonlinear PDE families. The second family includes a one-dimensional wave equation describing the wave motion in a hyperelastic material with a single family of fibers. For this model, approximate symmetries can be used to compute approximate closed-form solutions. Wave breaking times are found numerically and using the approximate solutions, which yield comparable results.

Bounded from below conditions on a class of symmetry constrained 3HDM

We study the bounded from below (BFB) conditions on a class of three Higgs doublet models (3HDM) constrained by the symmetry groups $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$, $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$, and ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$. These constraints must be implemented on both the neutral (BFB-n) and charged (BFB-c) directions. The exact necessary and sufficient BFB conditions are unknown in the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ case. We develop a general strategy using lower bounds to find sufficient conditions for BFB-n and BFB-c and apply it to these symmetries. In addition, we investigate the concern that the use of safe sufficient conditions can ignore valid points which would yield distinct physical consequences. This is done by performing a full phenomenological simulation of the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ models, where exact necessary and sufficient BFB conditions are possible. We look specifically at the points allowed by exact solutions but precluded by safe lower bounds. We found no evidence of remarkable new effects, partly reassuring the use of the lower bounds we propose here, for those potentials where no exact necessary and sufficient BFB conditions are known.

Enhanced topological superconductivity in an asymmetrical planar Josephson junction

As a platform for holding Majorana zero models (MZMs), the two-dimensional planar topological Josephson junction that can be used as carriers for topological quantum computing faces some challenges. One is a combination of mirror and time-reversal symmetries may make the system hold multiple pairs of MZMs. The other is that a soft gap dominated by a large momentum occurs in a clean system. To solve these problems, asymmetric junction can be introduced. Breaking this symmetry changes the symmetry class from class BDI to class D, and only a single pair of MZMs can be left at the boundary of the system. We numerically study four cases that create an asymmetric system and find out different superconducting pairing potential, different coupling coefficients between two-dimensional electron gases (2DEGs) and two superconducting bulks, different widths of two superconducting bulks make the gap of the system decrease at the optimal value, but make the gap at the minimum value increases. And the zigzag-shape quasi-one-dimensional junction eliminates the large momentum parallel to the junction and enhances the gap at the large momentum. However, the zigzag-shape junction cannot increase the gap at the region of multiple pairs of MZMs in a symmetric system. We show that by combining zigzag-shape junction with different coupling coefficients, the system can maintain a large gap (≈0.2Δ) in a wide region of the parameter space.

Fermiology of Topological Metals

The modern scope of fermiology encompasses not just the classical geometry of Fermi surfaces but also the geometry of quantum wave functions over the Fermi surface. This enlarged scope is motivated by the advent of topological metals—metals whose Fermi surfaces are characterized by a robustly nontrivial Berry phase. We review the extent to which topological metals can be diagnosed from magnetic-field-induced quantum oscillations of transport and thermodynamic quantities. A holistic analysis of the oscillatory wave form is proposed, in which different characteristics of the wave form (e.g., phase offset, high-harmonic amplitudes, temperature-dependent frequency) encode different aspects of a topologically nontrivial Fermi surface. Which characteristic to focus on depends on ( a) the orientation of the magnetic field relative to certain crystallographic axes, ( b) the symmetry class of the topological metal, and ( c) the separation of Fermi-surface pockets in quasimomentum [Formula: see text] space. Closely proximate pockets arise when (1) spin–split pockets are nearly overlapping due to a weak spin–orbit force or when (2) two pockets touch at an isolated [Formula: see text] point, which can be a topological band-touching point or a saddlepoint in the energy-momentum dispersion. The emergence of a pseudospin degree of freedom (in case 1) and the implications of magnetic breakdown (in case 2) are reviewed, with emphasis on new aspects originating from the (nonabelian) Berry connection of the Fermi surface. Future extensions of topofermiology are suggested in the directions of interaction-induced Fermi-liquid instabilities and two-dimensional electron liquids.

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman’s approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether’s theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.