Symmetry Group Research Articles

Quantum inspired kernel matrices: Exploring symmetry in machine learning

Quantum machine learning, an emerging field at the intersection of quantum computing and classical machine learning, has shown great promise in enhancing computational capabilities beyond classical bounds. A key element in this area of research is the utilization of Quantum Kernel Estimators, traditionally grounded in the symmetries of SU(2) groups associated with qubits. This study extends the conceptual framework of Quantum Kernel Estimators to incorporate a broader spectrum of symmetry groups. By harnessing the various structures of Lie groups, we develop novel quantum-inspired feature maps that offer more flexible and potentially powerful ways to encode and compress classical data into quantum states. We present a comprehensive theoretical introduction for this approach, followed by a methodology that integrates the developed feature maps into quantum-inspired kernel classifiers. Our results, derived from a series of computational experiments across various datasets, demonstrate the efficacy of this approach in comparison to traditional quantum and classical machine learning models. The findings not only underline the versatility of Lie-group theory in potentially enhancing quantum machine learning algorithms but also open new avenues for exploring complex symmetries in quantum information processing. This research bridges a gap between the study of symmetries and machine learning, paving the way for more sophisticated quantum algorithms capable of tackling complex, high-dimensional data in ways previously unattainable.

Synthesis and crystal structure of acentric anhydrous beryllium carbonate Be(CO3).

The anhydrous alkaline earth metal carbonate Be(CO3) was synthesized in a laser-heated diamond anvil cell at moderate pressures and temperatures (20(2) GPa and 1500(200) K) by a reaction of BeO with CO2. It crystallizes in the acentric, trigonal space group P3121 with Z = 3. The crystal structure was obtained from synchrotron single crystal X-ray diffraction data and confirmed by density functional theory-based calculations in combination with Raman spectroscopy. Second harmonic generation measurements were employed to verify the acentric space group symmetry. The crystal structure of Be(CO3) is characterized by the presence of isolated [CO3]2--groups and BeO4-tetrahedra. This is a new structure type and such a topology has not been observed in carbonates before. Be(CO3) can be recovered to ambient conditions and is not undergoing a phase transition during decompression.

Superconductivity in Monolayer Carbon Allotropes with High Thermal Stability.

Intrinsic superconductivity is rarely discovered in sp2-hybridized monolayer carbon allotropes. Here we design a carbon monolayer configured of pentagon, heptagon, and hexagon rings with p2 plane group symmetry. Full-sp2 hybridization is proposed to favor thermal metastability on a low Gibbs free energy. The extremely small thermal expansion coefficient is predicted to the turn negative value to positive with elevating temperature. Carbon polygon structures remain intact at a high thermal temperature of 3,000 K. The high specific surface area is found to approach 2,700 m2/g, with O2-adsorption being advantageous over pristine graphene. We reveal electronic Fermi surfaces mediated by phonon modes of carbon out-of-plane vibrations. By calculating the Eliashberg equation, we evaluate intrinsic superconductivity with a large electron-phonon coupling coefficient. The superconducting transition temperature is estimated to reach 20 K under a high logarithmic average frequency. These first-principles calculations shall stimulate experimentalists' interest in exploring low-dimensional carbon superconductors with gas sensitivity.

Cluster realisations of ıquantum$\imath {\rm quantum}$ groups of type AI

AbstractThe group of type is a coideal subalgebra of the quantum group , associated with the symmetric pair . In this paper, we give a cluster realisation of the algebra . Under such a realisation, we give cluster interpretations of some fundamental constructions of , including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of , which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form.

Emergent (2+1)D topological orders from iterative (1+1)D gauging

Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group symmetries and arranging the gauge fields in a 2D lattice, the local symmetries become the stabilizer of the XZZX-code for any Abelian group. By twisting the gauging map, we obtain codes that explicitly confine anyons, whose local creating operators violate an odd number of plaquettes. Their fusion results in either mobile dipole excitations twisting only half of the plaquette terms, or complete immobile Sierpiński-like excitations if we twist all the terms. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. In addition, our method also establishes a promising route to obtain higherdimensional topological codes from lower ones and to identify their gapped boundaries and their tensor network representations.

Structural and Magnetic Properties of B-Site-Ordered RE2CuTiO6 (RE = Dy, Ho, and Er) Double-Perovskite Oxides with Significant Cryogenic Magnetocaloric Performance.

The magnetocaloric effect (MCE) is currently intensively investigated in various rare earths (RE)-containing magnetic solids, not only for developing appropriate magnetocaloric materials (MCMs) for cryogenic magnetic cooling but also for deepening our understanding into the inherent physical properties of these materials. Here, we provide a systematic experimental investigation into a series of new RE2CuTiO6 (RE = Dy, Ho, Er) double-perovskite (DP) oxides regarding the structural and magnetic properties, especially for their cryogenic MCE and magnetic-phase transition (MPT). All of these RE2CuTiO6 oxides crystallize in a B-site-ordered hexagonal DP-type structure with the symmetry of the crystallographic space group P6̅m2. These DP oxides exhibit magnetic ordering, with MPT temperatures of approximately 2.7, 2.2, and 2.7 K for Dy2CuTiO6, Ho2CuTiO6, and Er2CuTiO6, respectively. The magnetocaloric performances of the RE2CuTiO6 DP oxides were characterized by the peak values of magnetic entropy change, the temperature-averaged magnetic entropy change (5K-lift), and relative cooling powers. These magnetocaloric parameters were deduced to be 18.7/17.9 J/kgK and 298.2 J/kg for Dy2CuTiO6, 12.5/12.2 J/kgK and 273.9 J/kg for Ho2CuTiO6, and 13.8/12.9 J/kgK and 188.4 J/kg for Er2CuTiO6 under a magnetic field change of 0-5 T. These values are comparable to those of most reported RE-containing magnetocaloric materials, indicating that these RE2CuTiO6 DP oxides are promising candidates for cryogenic magnetic cooling applications.

Symmetry-controlled SrRuO3/SrTiO3/SrRuO3 magnetic tunnel junctions: spin polarization and its relevance to tunneling magnetoresistance

Magnetic tunnel junctions (MTJs), that consist of two ferromagnetic electrodes separated by an insulating barrier layer, have non-trivial fundamental properties associated with spin-dependent tunneling. Especially interesting are fully crystalline MTJs where spin-dependent tunneling is controlled by the symmetry group of wave vector. In this work, using first-principles quantum-transport calculations, we explore spin-dependent tunneling in fully crystalline SrRuO3/SrTiO3/SrRuO3(001) MTJs and predict tunneling magnetoresistance (TMR) of nearly 3000%. We demonstrate that this giant TMR effect is driven by symmetry matching (mismatching) of the incoming and outcoming Bloch states in the SrRuO3(001) electrodes and evanescent states in the SrTiO3(001) barrier. We argue that under the conditions of symmetry-controlled transport, spin polarization, whatever definition is used, is not a relevant measure of spin-dependent tunneling. In the presence of diffuse scattering, however, e.g. due to localized states in the band gap of the tunnel barrier, symmetry matching is no longer valid and TMR in SrRuO3/SrTiO3/SrRuO3(001) MTJs is strongly reduced. Under these conditions, the spin polarization of the interface transmission function becomes a valid measure of TMR. These results provide an important insight into understanding and optimizing TMR in all-oxide MTJs.

Magnetic space groups versus representation analysis in the investigation of magnetic structures: the happy end of a strained relationship.

In recent decades, sustained theoretical and software developments have clearly established that representation analysis and magnetic symmetry groups are complementary concepts that should be used together in the investigation and description of magnetic structures. Historically, they were considered alternative approaches, but currently, magnetic space groups and magnetic superspace groups can be routinely used together with representation analysis, aided by state-of-the-art software tools. After exploring the historical antagonism between these two approaches, we emphasize the significant advancements made in understanding and formally describing magnetic structures by embracing their combined use.

Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions

The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations andnon-chiraltopological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials.

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

An energy-modified quantum defect method for the analysis of Rydberg spectra: Application to 2-butyne.

The high resolution Rydberg absorption spectrum of 2-butyne C4H6 recorded previously at the SOLEIL synchrotron facility has been interpreted using multichannel quantum defect theory (MQDT). The calculations are based on the continuum scattering calculations of Xu et al., J. Chem. Phys. 136, 154303 (2012) and of Jacovella et al., J. Phys. Chem. A 119, 12339 (2015) pertaining to the dipole-allowed excited state symmetries in absorption from the ground state. In contrast to the traditional approach of calculating low-lying electronic states first and then attempting to extend the calculations to ever higher energy, here the analysis proceeds through the extension of these detailed calculations of the electronic continuum scattering down into the discrete region of the spectrum. The continuum reaction matrices and dipole transition moments are adapted to the discrete Rydberg region via the use of an energy-modified formulation of MQDT theory and associated energy dependences of the quantum defects. The analysis reproduces more than 40 Rydberg states from n ≈ 10 down to the 3d and 4s levels with an rms error of better than 20 cm-1. These belong to five Rydberg series with three different molecular symmetries. While the approach predicts many additional series, most of these are calculated and observed to carry only little oscillator strength. The analysis shows that the Rydberg spectrum is dominated by the excitation of an e″ symmetry electron of fδ and gπ type, in line with what previous studies of the above-threshold shape resonance of 2-butyne have shown. The present study is intended to serve as an example showing how first principles continuum calculations may be useful for the interpretation of highly bound discrete states in a range that poses problems for the standard abinitio techniques. The quantitative treatment of the dipole absorption cross sections is deferred to a future paper.

Quantum reference frames from top-down crossed products

All physical observations are made relative to a reference frame, which is a system in its own right. If the system of interest admits a group symmetry, the reference frame observing it must transform commensurately under the group to ensure the covariance of the combined system. We point out that the crossed product is a way to realize quantum reference frames from the bottom-up; adjoining a quantum reference frame and imposing constraints generates a crossed product algebra. We provide a top-down specification of crossed product algebras and show that one cannot obtain inequivalent quantum reference frames using this approach. As a remedy, we define an abstract algebra associated to the system and symmetry group built out of relational crossed product algebras associated with different choices of quantum reference frames. We term this object the G-framed algebra, and show how potentially inequivalent frames are realized within this object. We comment on this algebra’s analog of the classical Gribov problem in gauge theory, its importance in gravity where we show that it is relevant for semiclassical de Sitter and potentially beyond the semiclassical limit, and its utility for understanding the frame dependence of physical notions like observables, density states, and entropies. Published by the American Physical Society 2024

Modeling Dipolar Molecules with PCP-SAFT: A Vector Group-Contribution Method.

Predicting thermodynamic equilibrium properties is essential to develop chemical and energy conversion processes in the absence of experimental data. For the modeling of thermodynamic properties, statistical associating fluid theory (SAFT)-based equations of state, such as perturbed-chain polar (PCP)-SAFT, have been proven powerful and found broad application. The PCP-SAFT parameters can be predicted by group-contribution (GC) methods. However, their application to the dipole term is substantially limited: current GC methods neglect the dipole term or only allow for a single dipolar group per substance to avoid handling the molecular dipole moment's symmetry effects. Still, substances with multiple dipolar groups are highly relevant, and their description substantially improves by including the dipole term in SAFT models. To overcome these limitations, this work proposes a vector-addition-based (Vector-)GC method for the dipole term of PCP-SAFT that accounts for molecular symmetry. The Vector-GC employs information on the substance's molecular 3D structure to predict the molecular dipole moment through a vector addition of bond contributions. Combining the proposed sum rule for dipole moments with established sum rules for the remaining parameters yields a consistent GC method for PCP-SAFT for dipolar substances. The prediction capabilities of the Vector-GC method are analyzed against experimental data for two substance classes: nonassociating oxygenated and halogenated substances. We demonstrate that the Vector-GC method improves vapor pressure and liquid density predictions compared to neglecting the dipole term. Moreover, we show that the Vector-GC method enables differentiation between cis- and trans-isomers. The Vector-GC method, hence, substantially increases the predictive capabilities and applicability domain of GC methods. All parameters are provided as JSON and CSV files, and the Vector-GC method is available through an open-source python package. Additionally, the developed regression framework for GC methods for PCP-SAFT is openly available. The regression framework can be employed to regress the Vector-GC method to other substance classes and is easily adaptable to other sum rules for PCP-SAFT.

QSPR-based prediction model for the melting point of polycyclic aromatic hydrocarbons using MLR and ANN methods

The melting point is an important property that helps generate specific compounds with desired thermos-physical properties. Much work has been done applying quantitative structure-property relationships to improve the melting-point correlations, but they are unreliable. This gap might come from the melting point's sensitivity for small molecular variations and descriptors, which currently do not fully consider all factors determining melting behavior. In this work, we provide a QSPR model for predicting the melting point of a heterogeneous polycyclic aromatic hydrocarbons dataset. The model was generated using a robust hybrid linear approach (Genetic Algorithm-Multiple Linear Regression) and a nonlinear approach named Artificial Neural Network (ANN). Three descriptors were chosen to explain the influence of molecular weight and symmetry on melting point. The resulting QSPR model can model melting-point behavior with an RMSE of 34.88K, a coefficient correlation value of R²=0.887, and a prediction coefficient of Q²LOO= 0.863. This study reveals that the results produced by MLR were appropriate and served to predict melting points. However, compared to the results obtained by the ANN model, we conclude that the latter is more effective and better than the MLR model. Based on the results, our suggested model may be effective in predicting melting points, and the selected descriptors play essential roles in determining melting points.

Photoionization of aziridine: Nonadiabatic dynamics of the first six low-lying electronic states of the aziridine radical cation.

In this article, the theoretical photoionization spectroscopy of the aziridine (C2H5N) molecule is investigated. To start with, we have optimized the geometry of this molecule at the neutral electronic ground state at the density functional theory/augmented correlation-consistent polarized valence triple zeta level of theory using the G09 program. The electronic structure calculations were restricted to the first six low-lying electronic states in order to account for the experimental photoelectron spectrum of the C2H5N molecule. The first six low-lying electronic states (X̃2A', Ã2A', B̃2A″, C̃2A″, D̃2A', and Ẽ2A') of the potential energy surfaces (PESs) are calculated by both equation of motion-ionization potential-coupled cluster singles and doubles and multi-configuration quasi-degenerate perturbation theory abinitio quantum chemistry methods along the dimensionless normal displacement coordinates in which multiple conical intersections were established among the considered electronic states. A (6 × 6) model vibronic Hamiltonian is constructed on a diabatic electronic basis, using the symmetry selection rules and Taylor series expansion. The Cs symmetry point group of the aziridine molecule leads to electronic states symmetry of either A' or A″, and these states are close in energy, due to which the same symmetry electronic states avoid each other. To get a smooth diabatic PES, a fourfold diabatization scheme is used, which is implemented in the General Atomic and Molecular Electronic Structure Systems suite of programs. All the parameters used in the diabatic vibronic coupling model Hamiltonian are calculated in terms of the normal modes of vibrational coordinates. Finally, the vibronic model Hamiltonian constructed for the coupled six electronic states is used to solve both time-independent and time-dependent Schrödinger equations using the multi-configuration time-dependent Hartree program module to obtain the dynamical observables. The theoretical vibronic band structure is found to be in good accord with the available experimental results.

Peierls Transition in Gross-Neveu Model from Bethe Ansatz

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg–de Vries (KdV) of the mean-field analysis. Published by the American Physical Society 2024

Non-relativistic torque and Edelstein effect in non-collinear magnets

The Edelstein effect is the origin of the spin-orbit torque: a current-induced torque that is used for the electrical control of ferromagnetic and antiferromagnetic materials. This effect originates from the relativistic spin-orbit coupling, which necessitates utilizing materials with heavy elements. Here, we show that in magnetic materials with non-collinear magnetic order, the Edelstein effect and, consequently, a current-induced torque can exist even in the absence of the spin-orbit coupling. Using group symmetry analysis, model calculations, and realistic simulations on selected compounds, we identify large classes of non-collinear magnet candidates and demonstrate that the current-driven torque is of similar magnitude as the celebrated spin-orbit torque in conventional transition metal structures. We also show that this torque can exist in an insulating material, which could allow for highly efficient electrical control of magnetic order.

Three-dimensional topological crystalline insulator without spin–orbit coupling in nonsymmorphic photonic metacrystal

Abstract By including certain point group symmetry in the classification of band topology, Fu proposed a class of three-dimensional topological crystalline insulators (TCIs) without spin–orbit coupling in 2011. In Fu’s model, surface states (if present) doubly degenerate at Γ ¯ and M ¯ when time-reversal and C 4 symmetries are preserved. The analogs of Fu’s model with surface states quadratically degenerate at M ¯ are widely studied, while surface states with quadratic degeneracy at Γ ¯ are rarely reported. In this study, we propose a three-dimensional TCI without spin–orbit coupling in a judiciously designed nonsymmorphic photonic metacrystal. The surface states of photonic TCIs exhibit quadratic band degeneracy in the (001) surface Brillouin zone (BZ) center ( Γ ¯ point). The gapless surface states and their quadratic dispersion are protected by C 4 and time-reversal symmetries, which correspond to the nontrivial band topology characterized by Z 2 topological invariant. Moreover, the surface states along lines from Γ ¯ to the (001) surface BZ boundary exhibit zigzag feature, which is interpreted from symmetry perspective by building composite operators constructed by the product of glide symmetries with time-reversal symmetry. The metacrystal array surrounded with air possesses high order hinge states with electric fields highly localized at the hinge that may apply to optical sensors. The gapless surface states and hinge states reside in a clean frequency bandgap. The topological surface states emerge at the boundary of the metacrystal and perfect electric conductor (PEC), which provide a pathway for topologically manipulating light propagation in photonic devices.

Symmetries in Riemann-Cartan Geometries

Riemann-Cartan geometries are geometries that admit non-zero curvature and torsion tensors. These geometries have been investigated as geometric frameworks for potential theories in physics including quantum gravity theories and have many important differences when compared to Riemannian geometries. One notable difference, is the number of symmetries for a Riemann-Cartan geometry is potentially smaller than the number of Killing vector fields for the metric. In this paper, we will review the investigation of symmetries in Riemann-Cartan geometries and the mathematical tools used to determine geometries that admit a given group of symmetries. As an illustration, we present new results by determining all static spherically symmetric and all stationary spherically symmetric Riemann-Cartan geometries. Furthermore, we have determined the subclasses of spherically symmetric Riemann-Cartan geometries that admit a seven-dimensional group of symmetries.

Non-magnetic layers with a single symmetry-protected Dirac cone: Which additional dispersions must appear?

In two-dimensional (2D), non-magnetic materials, a single Dirac cone at high-symmetry point (HSP) of the Brillouin zone (BZ), akin to the one in graphenes’ band structure, cannot appear as the only quasiparticle at the Fermi level. Here we found two layer groups with time-reversal symmetry, among all possible both without and with spin-orbit coupling, that host one Dirac cone at HSP and we show which additional dispersions appear: a pair of Dirac lines on opposite BZ edges and a pair of Dirac cones that can be moved but not removed by symmetry preserving perturbations, on the other two BZ edges. We illustrate our theory by a tight-binding band structure and discuss real 2D materials that belong to one of the two symmetry groups. Finally, we single out smaller or bigger discrepancies among the published papers on the same or related topic.