Symmetry Group Research Articles

Characteristics of the distribution of minerals among the space groups

Abstract The RRUFF online database (a project by Robert Downs, University of Arizona) catalogs the occurrence of point and space group symmetries of over 4800 of the nearly 6000 currently accepted mineral species. Of these, over 90% are monomorphic, while the remaining species have been recognized to be polymorphic. To avoid issues of reliability and inconsistency of characterization, the 4499 monomorphic minerals are the subject of this analysis. It has long been recognized that minerals are non-uniformly distributed among the space groups as well as their corresponding point groups and crystal systems. However, this non-uniformity is not non-systematic. A random distribution of minerals among the space groups would result in an average of about 20 species per group. In fact, the observed distribution of minerals is highly skewed. Analysis of the frequency distribution of minerals among the space groups reveals three distinct populations: those without currently known monomorphic species (52), those sparsely populated by minerals that exhibit an exponential distribution of species abundances (170), and those most populous space groups (8) that are differentiated from the others by an inflection in the slope of the species frequency distribution. Additionally, a series of relationships have been identified, and they collectively characterize crystallographic controls on and resultant distributions of mineral occurrence. First, there is a strong preference across all crystal systems for minerals to exhibit structures associated with the holohedral space groups relative to those pedial groups of the lowest symmetry. Second, mineral species exhibit a strong preference for the Laue class space groups over their paired Sohncke groups. Furthermore, it is recognized that the space groups that make up the 11 enantiomorphic pairs are either sparsely populated or devoid of known mineral species. Third, the centering types of the Bravais lattices and structures defined by the Wyckoff multiplicities of the various space groups are recognized to strongly influence the distribution of mineral species such that more structurally and symmetrically complex space groups (those with non-primitive lattice structures) tend to be represented by more minerals. Fourth, when considering the 73 arithmetic space group classes, in most cases non-symmorphic space groups are more richly populated than their linked symmorphic space groups, and in those cases where both hemisymmorphic and asymmorphic space groups are present in the same arithmetic class, the asymmorphic space groups typically contain more species. Together, these observations and interpretations augment and advance previously achieved understandings of mineral frequency distributions among the point groups and imply four future lines of inquiry: extending consideration to polymorphic species as well as relating distributions among the space groups to terrestrial mineral species abundance, mineral chemistry, as well as the evolution of mineral chemistry and paragenesis.

Unveiling Molecular Symmetry through Twisted X-Ray Diffraction.

The centrosymmetric constraint of Friedel's law in standard x-ray diffraction limits its ability to reveal complex molecular symmetry. In contrast, twisted x-ray diffraction with vortex beams, which carry orbital angular momentum, breaks Friedel's law yielding diffraction patterns that reflect the intrinsic symmetry of molecules. Through analytical derivations and numerical simulations, we demonstrate the enhanced sensitivity of twisted x-ray diffraction to the symmetry of M-fold symmetric molecules. Our results show that, while traditional standard x-ray diffraction struggles to distinguish between structurally similar molecules with different symmetries, twisted x-ray diffraction patterns can clearly differentiate these molecules. Additionally, we show that increasing the orbital angular momentum enhances the diffraction resolution and reveals finer symmetry-specific features of the molecules. This positions twisted x-ray diffraction as a promising tool for molecular imaging, capable of revealing intricate structures with complex symmetry.

Non-holomorphic modular S4 lepton flavour models

In the formalism of the non-supersymmetric modular invariance approach to the flavour problem the elements of the Yukawa coupling and fermion mass matrices are expressed in terms of polyharmonic Maaß modular forms of level N in addition to the standard modula forms of the same level and a small number of constant parameters. Non-trivial polyharmonic Maaß forms exist for zero, negative and positive integer modular weights. Employing the finite modula group S4 as a flavour symmetry group and assuming that the three left-handed lepton doublets furnish a triplet irreducible representation of S4, we construct all possible 7- and 8-parameter lepton flavour models in which the neutrino masses are generated either by the Weinberg effective operator or by the type-I seesaw mechanism. We identify the phenomenologically viable models and obtain predictions for each of these models for the neutrino mass ordering, the absolute neutrino mass scale, the Dirac and Majorana CP-violation phases and, correspondingly, for the sum of neutrino masses and the neutrinoless double beta decay effective Majorana mass. We comment on how these models can be tested and conclude that they are all falsifiable. Detailed analyses are presented in the case of three representative benchmark lepton flavour scenarios.

Crystallography of quasiperiodic moiré patterns in homophase twisted bilayers.

This paper discusses the geometric properties and symmetries of general moiré patterns generated by homophase bilayers twisted by rotation 2δ. These patterns are generically quasiperiodic of rank 4 and result from the interferences between two basic periodicities incommensurate to each other, defined by the sites in the layers that are kept invariant through the symmetry operations of the structure. These invariant sites are distributed on the nodes of a set of lattices called Φ-lattices - where Φ runs on the rotation operations of the symmetry group of the monolayers - which are the centers of rotation 2δ + Φ transforming a lattice node of the first layer into a node of the second. It is demonstrated that when a coincidence lattice exists, it is the intersection of all the Φ-lattices of the structure.

The Space of Continuous Linear Functionals on ℓ1 Approximated by Weakly Symmetric Continuous Linear Functionals

We study the space of continuous linear functionals approximated by weakly symmetric continuous linear functionals on the complex Banach space ℓ1 of all absolutely summing complex sequences. We construct the sequence of groups of symmetries on ℓ1, obtain the structure of corresponding weakly symmetric continuous linear functionals, find the completion of the space, and construct for it a Schauder basis.

Conditional symmetries and conditional constants of motion for dynamical systems

Conditional symmetries were introduced by Levi and Winternitz in their 1989 seminal paper to deal with nonlinear PDEs. Here we discuss their application in the framework of ODEs, and more specifically Dynamical Systems; it turns out they are closely related to two established -- albeit maybe less widely known -- concepts, i.e. orbital symmetries and configurational invariants. The paper is devoted to studying the interplay of these notions, and their application in the study of Dynamical Systems, with special attention to invariant manifolds of these.

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations ρ¯ of the Lie group Diffc(M) of compactly supported diffeomorphisms of a smooth manifold M that satisfy a so-called generalized positive energy condition. In particular, this captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by ρ¯. We show that if M is connected and dim(M)>1 1$$\\end{document}]]>, then any such representation is necessarily trivial on the identity component Diffc(M)0. As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology Hct2(Xc(M),R) of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand–Fuks cohomology in view of the compact support condition.

Critical spin chains and loop models with $PSU(n)$ symmetry

Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the O(n)O(n) model (symmetry group O(n)O(n)) and the Potts model (symmetry group S_QSQ). Both models make sense for n,Q∈ \mathbb{C}n,Q∈ℂ and not just n,Q∈ \mathbb{N}n,Q∈ℕ, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the group PSU(n)PSU(n). We focus on the two-dimensional case, where the models can be described either as gases of non-intersecting orientable loops, or as alternating spin chains. This allows us to determine their spectra either by computing a twisted torus partition function, or by studying representations of the walled Brauer algebra. In the critical limit, our models give rise to a CFT that exists for any n∈\mathbb{C}n∈ℂ and has a global PSU(n)PSU(n) symmetry. Its spectrum is similar to those of the O(n)O(n) and Potts CFTs, but a bit simpler. We conjecture that the O(n)O(n) CFT is a \mathbb{Z}_2ℤ2 orbifold of the PSU(n)PSU(n) CFT, where \mathbb{Z}_2ℤ2 acts as complex conjugation.

Crystalline-symmetry-protected Majorana modes in coupled quantum dots

We propose a minimalist architecture for achieving various crystalline-symmetry-protected Majorana modes in an array of coupled quantum dots. Our framework is motivated by the recent experimental demonstrations of two-site and three-site artificial Kitaev chains in a similar setup. We find that introducing a π-phase domain wall in the Kitaev chain leads to a pair of mirror-protected Majorana zero modes located at or near the junction. Joining two π junctions into a closed loop, we can simulate two distinct classes of two-dimensional higher-order topological superconducting phases, both carrying symmetry-protected Majorana modes around the sample corners. As an extension of the π junction, we further consider a general vertex structure where n Kitaev chains meet, i.e., a Kitaev n vertex. We prove that such an n vertex, if respecting a dihedral symmetry group Dn, necessarily carries n vertex-bound Majorana modes protected by the Dn symmetry. Resilience of the junction and vertex Majorana bound states against disorder and correlation effects is also discussed. Our architecture paves the way for designing, constructing, and exploring a wide variety of artificial topological crystalline phases in quantum-dot experiments. Published by the American Physical Society 2025

Multi-toric geometries with larger compact symmetry

ABSTRACT We study complete, simply connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For $ \mathrm{Spin}(7) $-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $ T^{3} \times \mathrm{SU}(2) $. We then specialize the analysis to holonomy G2, to Calabi-Yau geometries in real dimension six and to hyperKähler four-manifolds. Finally, we consider weakly coherent triples on $ \mathbb{R} \times \mathrm{SU}(2) $, and their extensions over singular orbits, to give local examples in the $ \mathrm{Spin}(7) $-case that have singular orbits where the stabilizer is of rank one.

A Density Functional Valence Bond Study on the Excited States

The accurate description of excited states is crucial for the development of electronic structure theory. In addition to determining excitation energies, strong state interactions arise when electronic states with the same symmetry are degenerate or nearly degenerate, often requiring a multi-state treatment. These strong correlation effects and state interactions can be effectively handled by the Hamiltonian matrix correction-based density functional valence bond (hc-DFVB) method, a multi-reference density functional theory capable of accurately describing electronic state interactions. In this paper, we explore the low-lying excited states of four isoelectronic systems (C2H, CN, CO⁺, BO) using valence bond methods, including the valence bond self-consistent field (VBSCF) and hc-DFVB methods. Our results show that the hc-DFVB method provides significantly better excitation energies compared to VBSCF. Furthermore, hc-DFVB can reliably predict the correct ordering of excited states, whereas VBSCF shows some ordering inconsistencies. By categorizing the VB structures into groups based on point group symmetry, we can extract the key structural contributions and bonding pictures of each state from the weight distribution of these groups. Additionally, we study the potential energy curves for lithium fluoride (LiF) and a mixed-valence spiro cation, demonstrating the superior performance of hc-DFVB when applied to the study of near-degenerate excited states in the avoided crossing region.

Physical origin and control of exciton spatial localization in high-κ MOene monolayers under external perturbations

Two-dimensional (2D) materials hold great promise for the next-generation optoelectronics applications, many of which, including solar cell, rely on the efficient dissociation of exciton into free charge carriers. However, photoexcitation in atomically thin 2D semiconductors typically produces exciton with a binding energy of ∼500 meV, an order of magnitude larger than thermal energy at room temperature. This inefficient exciton dissociation can limit the efficiency of photovoltaics. In this study, employing the first principles approach-DFT, GW + BSE, and analytical model, we demonstrate the role of asymmetric halogenation, dielectric environment, and magnetic field in 2D Ti2O MOene as an efficient strategy for regulating exciton binding energy (EBE) towards spontaneous exciton dissociation. Our study goes beyond the exciton ground state and quantifies the degree of spatial delocalization of exciton in excited states as well. We determine the quantitative impact of varying dielectric screening and magnetic field strength on EBE for different excited states (1 s, 2 s, 3 s, 4 s, and so on). Importantly, we reveal the significant role of orbital orientation (whether in-plane or out-of-plane) and symmetry (related to the angular momentum quantum number) in understanding the spatial localization of excitons and their binding energy. Additionally, a high dielectric constant in 2D MOene enables easier exciton dissociation, similar to that observed in 3D bulk semiconductors, while also harnessing the advantages of 2D materials. This makes it an effective material that combines the best of both 3D bulk and 2D structures. The study offers a promising strategy for designing next-generation optoelectronic devices.

Rashba effect originates from the reduction of point-group symmetries.

The Rashba effect in a nonmagnetic condensed-matter system is described by the reduction of point-group symmetries. The inversion, two-fold rotation, and reflection symmetries transforming the wavevector k to -k are identified as the origin of a degenerate state according to the time-reversal symmetry. The lack of these symmetries in a bulk system or the breaking of these in a surface system is then identified as the origin of a nondegenerate state. The surface systems Au(111), Au(110), and W(110) are assessed. The bulk system BiTeI is demonstrated for the existence of a nondegenerate state on the basis of first-principles calculations. The related issues of the heterostructure GaAs/AlGaAs and the spin Hall effect are also presented.

Perturbation to μ–τ symmetry using type I and type II seesaw mechanisms under SU(2)L×Δ(27)×Z2 flavor symmetry

We construct the neutrino mass matrix by using the [Formula: see text] flavor symmetry group and apply the perturbation to [Formula: see text]–[Formula: see text] symmetric neutrino mass matrix generated from type I seesaw mechanism with the contribution from type II seesaw. The symmetric neutrino mass matrix is obtained by transforming the right-handed neutrinos [Formula: see text] as [Formula: see text], SM Higgs doublets H as [Formula: see text], scalar singlet [Formula: see text] as [Formula: see text] under [Formula: see text](27) symmetry whereas contribution from type II is obtained by transforming the scalar triplet [Formula: see text] as 3 under [Formula: see text](27) symmetry. We obtain non-diagonal charged-lepton mass matrix by transforming scalar doublets [Formula: see text] as [Formula: see text], scalar singlet [Formula: see text] as [Formula: see text] under [Formula: see text](27) symmetry. The detailed numerical analysis is carried out by scanning the type II seesaw strength and other input parameters. The neutrino oscillation parameters are found to be consistent with the 3[Formula: see text] bound of the latest experimental neutrino oscillation data. The analysis also focuses on the Planck cosmological bound on the sum of the three absolute neutrino masses, [Formula: see text] [Formula: see text]eV. We obtain baryon asymmetry via lepton asymmetry for both normal and inverted hierarchy in agreement with the Planck bound on the baryon-to-photon ratio.

Breakdown of the Kasha-Vavilov's rule in low-symmetry porphyrazines: Ultrafast intersystem crossing via high vibronic state.

Recently (Photochem Photobiol. 2023;100:1277-1289. doi:10.1111/php.13898), we described the anti-Kasha effect in tribenzo-6H-1,4-diazepinoporphyrazins with C2v symmetry, where the ultrafast spin changes successfully compete with the internal conversion. In this study, we show the presence of this effect in 2 (3),9 (10),16(17),23(24)-tetra-tert-butyl-29H,31H-phthalocyanine (1) and 1,4-di-[2-(2-methoxyethoxy)ethoxy]-29H,31H-phthalocyanine (2), which also possess reduced molecular symmetry and do not bear 6H-1,4-diazepine fragments. The anti-Kasha effect in 1 and 2 supplemented by Mg(II) tribenzo-6H-1,4-diazepinoporphyrazinates 3 and 4 exhibits a close-to-linear dependence on energy gap value between the zero vibrational levels of two lowest singlet excited states S1 0 and S2 0 (these states are degenerate in D4h symmetry) and enhances with increase. The theoretical kinetic model of excited state dynamics, which takes into account the observed effects and follows Fermi's golden rule, predicts the presence of an additional excited state with enhanced spin-orbit coupling compared to S1 0, S2 0 and the corresponding triplet states, which is not predicted by TDDFT calculations in the Born-Oppenheimer approximation. The combination of the above indicates that the key role in the observed anti-Kasha effect and the mechanism of dissipation of the excited state in porphyrazines and their analogs is played by vibronic excited states, which requires theoretical research methods beyond the Born-Oppenheimer approximation.

Structured Dynamics in the Algorithmic Agent.

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether's theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent's constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain.

On momentum map for the Maxwell–Lorentz equations with spinning particle

We develop the theory of the momentum map for the Maxwell–Lorentz equations for a spinning extended charged particle. The development relies on the Hamilton–Poisson structure of the system. This theory is indispensable for the study of long-time behavior and radiation of the solitons of this system, in particular for the proof of orbital stability of the corresponding solitons moving with constant speed and rotating with constant angular velocity. We apply the theory to the translation and rotation symmetry groups of the system. The general theory of the momentum map results in formal equations for the corresponding invariants. We solve these equations obtaining expressions for the momentum and the angular momentum. We check the conservation of these expressions and their coincidence with known classical invariants. For the first time, we specify a general class of finite energy initial states which provide the conservation of angular momentum.

Directed Electrostatics Strategy Integrated as a Graph Neural Network Approach for Accelerated Cluster Structure Prediction.

We present a directed electrostatics strategy integrated as a graph neural network (DESIGNN) approach for predicting stable nanocluster structures on their potential energy surfaces (PESs). The DESIGNN approach is a graph neural network (GNN)-based model for building structures of large atomic clusters with specific sizes and point-group symmetry. This model assists in the structure building of atomic metal clusters by predicting molecular electrostatic potential (MESP) topography minima on their structural evolution paths. The DESIGNN approach is benchmarked on the prototype Mgn clusters with n < 150. The predicted MESP topography minima of Mgn clusters, n < 70, fairly agrees with the whole-molecule MESP topography calculations. Moreover, the ground-state structures of Mgn (n = 4-32) clusters generated through the DESIGNN approach corroborate well with the global minimum structures reported in the literature. Furthermore, this approach could generate novel symmetric isomers of medium to large Mgn clusters in the size regime, n < 150, by constraining the point-group symmetry of the parent clusters. The parent growth potential (GP) of a cluster gives a measure of its parent cluster to accommodate more atoms and characterize the structures on the DESIGNN-guided path. The GP of a cluster can also be interpreted as a measure of the cooperative interaction relative to its parent cluster. Along the highest GP paths, the DESIGNN approach is further employed to generate stable Mgn nanoclusters with n = 228, 236, 257, 260. Therefore, the DESIGNN approach holds great promise in accelerating the structure search and prediction of large metal clusters guided through MESP topography.

Inverse Design of 2D Altermagnetic Metal-Organic Framework Monolayers from Hückel Theory of Nonbonding Molecular Orbitals.

Altermagnets, characterized by spontaneous spin-splitting without net magnetization, are challenging to realize due to their unique spin group symmetries. Two-dimensional (2D) magnetic metal-organic frameworks (MOFs), with tunable topologies and spins, offer promising platforms for achieving altermagnetism. In this study, we propose a general strategy to create 2D altermagnetic monolayers by bridging Cr with organic ligands exhibiting nonbonding molecular orbitals (NBMOs) based on the Hückel molecular orbital theory and first-principles calculations. Three 2D MOFs, namely, Cr(diz)2, Cr(c-pyr)2, and Cr(f-pid)2 (diz = 1,3-diazete, c-pyr = pyrrolo[3,4-c]pyrrole, f-pid = pyrrolo[3,4-f]isoindole), are constructed using this strategy and exhibit the altermagntic ground state. These MOFs possess the spin point group 24̅1 m 22 and exhibit critical temperatures reaching up to 183 K. Analyses of orbital symmetry and energy levels rationalize the presence of altermagnetism. Our findings highlight the critical role of NBMOs in realizing 2D-MOF-based altermagnets with enhanced critical temperatures.

Crystal structure and Hirshfeld surface analysis of a new polymorph of chloridobis(1,10-phenanthroline-κ2 N,N′)copper(II) perchlorate

The title salt (systematic name: 2-methyl-4-oxo-3,4-dihydroquinazolin-1-ium chloride), [CuCl(C12H8N2)2](ClO4), is comprised of a mononuclear complex cation [Cu(phen)2Cl]+ (phen is 1,10-phenanthroline) and a perchlorate anion, ClO4 −, both with point group symmetry 2. The CuII atom has a slightly distorted trigonal–bipyramidal coordination environment, defined by a N4Cl coordination set with the Cl atom and two N atoms at the equatorial sites. In the crystal, each phen ring is parallel to neighboring phen rings. The resulting significant π–π stacking interactions lead to zigzag chains extending parallel to [001]. Hirshfeld surface analysis suggests that the most important contributions to the surface contacts are from H...H (32.1%), H...C/C...H (18.2%), H...O/O...H (14.6%), H...Cl/Cl...H (12.7%) and C...C (10.6%) interactions.