Symmetry Classes Research Articles

Non-Hermitian Spatial Symmetries and Their Stabilized Normal and Exceptional Topological Semimetals.

We study non-Hermitian spatial symmetries-a class of symmetries that have no counterparts in Hermitian systems-and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. Inderiving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that a certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.

Forced convection flow over a moving porous and non-uniform cylinder of variable thickness

Forced convection flow of viscous fluid over a moving, porous, and heated cylinder of variable thickness is studied in this paper. The non-uniform cylinder is placed vertically in a quiescent fluid. All the field quantities (the normal and axial velocities and temperature), defined at the wall or surface of cylinder, are variable and may have non-linear form. An appropriate set of transformations is developed on the analogies of systematic classical symmetries for a system of partial differential equations (PDE’s), whereas, they are equipped with the representative of each velocity component, temperature function, and similarity variable. However, the PDE’s along with the boundary conditions (BC’s) are reduced into a system of boundary value problem (BVP) of ordinary differential equations (ODE’s) by using these new and unseen similarity variables. The system of non-linear coupled ODE’s, combined with the BC’s is solved, whereas, the numerical results are obtained for different values of the existing governing parameters. All the field quantities, skin friction, rate of heat transfer at the surface of the cylinder are evaluated with the help of [Formula: see text] package of MATLAB and the results are shown in different graphs. The present simulation generalizes all types of forced convection flow problems, over a porous and heated cylinder of variable (uniform) radius, when it is stretched (shrunk) with linear and non-linear (uniform) velocity. However, the simulated problem gives rise to a new set of problems and that they are valid for non-linear (linear and uniform) injection (suction) velocity.

Leibniz-type rule of variable-order fractional derivative and application to build Lie symmetry framework

In this paper, we first study some properties of the variable-order fractional derivative defined by the Caputo fractional derivative and particularly present a Leibniz-type rule, which makes the variable-order fractional derivative to be expressed as an infinite sum of integer-order derivatives. Then we use such properties to build a Lie symmetry framework for a class of scalar variable-order fractional partial differential equations and show that such type of equations has an elegant symmetry structure, which facilitates us to explore the symmetry properties. By means of the Lie symmetry framework and the symmetry structure, we perform a Lie symmetry classification of the variable-order fractional diffusion equations and construct the corresponding symmetry reductions and certain exact solutions.

Powder X‐Ray Diffraction Pattern Is All You Need for Machine‐Learning‐Based Symmetry Identification and Property Prediction

Herein, data‐driven symmetry identification, property prediction, and low‐dimensional embedding from powder X‐Ray diffraction (XRD) patterns of inorganic crystal structure database (ICSD) and materials project (MP) entries are reported. For this purpose, a fully convolutional neural network (FCN), transformer encoder (T‐encoder), and variational autoencoder (VAE) are used. The results are compared to those obtained from a well‐established crystal graph convolutional neural network (CGCNN). A task‐specified small dataset that focuses on a narrow material system, knowledge (rule)‐based descriptor extraction, and significant data dimension reduction are not the main focus of this study. Conventional powder XRD patterns, which are most widely used in materials research, can be used as a significantly informative material descriptor for deep learning. Both the FCN and T‐encoder outperform the CGCNN for symmetry classification. For property prediction, the performance of the FCN concatenated with multilayer perceptron reaches the performance level of CGCNN. Machine‐learning‐driven material property prediction from the powder XRD pattern deserves appreciation because no such attempts have been made despite common XRD‐driven symmetry (and lattice size) prediction and phase identification. The ICSD and MP data are embedded in the 2D (or 3D) latent space through the VAE, and well‐separated clustering according to the symmetry and property is observed.

Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables

This work extends the analysis of the generalized multifractality of critical eigenstates at the spin quantum Hall transition in two-dimensional disordered superconductors [J. F. Karcher et al, Annals of Physics, 435, 168584 (2021)]. A mapping to classical percolation is developed for a certain set of generalized-multifractality observables. In this way, exact analytical results for the corresponding exponents are obtained. Furthermore, a general construction of positive pure-scaling eigenfunction observables is presented, which permits a very efficient numerical determination of scaling exponents. In particular, all exponents corresponding to polynomial pure-scaling observables up to the order $q=5$ are found numerically. For the observables for which the percolation mapping is derived, analytical and numerical results are in perfect agreement with each other. The analytical and numerical results unambiguously demonstrate that the generalized parabolicity (i.e., proportionality to eigenvalues of the quadratic Casimir operator) does not hold for the spectrum of generalized-multifractality exponents. This excludes Wess-Zumino-Novikov-Witten models, and, more generally, any theories with local conformal invariance, as candidates for the fixed-point theory of the spin quantum Hall transition. The observable construction developed in this work paves a way to investigation of generalized multifractality at Anderson-localization critical points of various symmetry classes.

Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model

Spectral correlations are a powerful tool to study the dynamics of quantum many-body systems. For Hermitian Hamiltonians, quantum chaotic motion is related to random matrix theory spectral correlations. Based on recent progress in the application of spectral analysis to non-Hermitian quantum systems, we show that local level statistics, which probe the dynamics around the Heisenberg time, of a non-Hermitian $q$-body Sachdev-Ye-Kitev (nHSYK) model with $N$ Majorana fermions, and its chiral and complex-fermion extensions, are also well described by random matrix theory for $q > 2$, while for $q = 2$, they are given by the equivalent of Poisson statistics. For that comparison, we combine exact diagonalization numerical techniques with analytical results obtained for some of the random matrix spectral observables. Moreover, depending on $q$ and $N$, we identify $19$ out of the $38$ non-Hermitian universality classes in the nHSYK model, including those corresponding to the tenfold way. In particular, we realize explicitly $14$ out of the $15$ universality classes corresponding to non-pseudo-Hermitian Hamiltonians that involve universal bulk correlations of classes ${\rm AI}^\dagger$ and ${\rm AII}^\dagger$, beyond the Ginibre ensembles. These results provide strong evidence of striking universal features in non-unitary many-body quantum chaos, which in all cases can be captured by nHSYK models with $q > 2$.

Symmetry Properties of Modified Black-Scholes Equation

This paper concerns the classical and conditional symmetries of the Black-Scholes equation. Modifications of the Black-Scholes equation have also been considered and their maximal algebras of invariance have been found. Examples of creation operators for the Black-Scholes eigenvalue problem have been provided.

Lie symmetries and exact solutions for a fourth‐order nonlinear diffusion equation

In this paper, we consider a fourth‐order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one‐dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one‐dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves.

New Reductions of the Unsteady Axisymmetric Boundary Layer Equation to ODEs and Simpler PDEs

Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, authors have proposed a method for constructing reductions of PDEs with two independent variables based on the idea of invariance. The proposed method in this work is a modification of the Clarkson–Kruskal direct method and expands the possibilities for its application. The main result of this article consists of a method for constructing reductions that generalizes the previously proposed approach to the case of three independent variables. The proposed method is used to construct reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. All reductions of this equation were obtained.

INFLUENCE OF INVERSE PIEZOELECTRIC AND PHOTOELASTIC EFFECTS ON THE INDEX SURFACE OF THE NORMAL COMPONENT INVERSE TENSOR DIELECTRIC PERMEABILITY OF THE Bi12TiO20 CRYSTAL

The index surface of the normal component of variation of the inverse tensor dielectric constant of the photorefractive Bi12TiO20 crystal of 23 symmetry class is constructed and analyzed. The extremal values of the index surface of the normal component for Bi12TiO20 crystal with formed holographic grating are determined. It is shown that the combined action of the photoelastic and inverse piezoelectric effects leads to a change in the maximum and minimum values of the normal component of variation of the inverse tensor dielectric constant of the crystal with the wave vector of the holographic grating oriented along the crystallographic direction.

Unifying the Anderson transitions in Hermitian and non-Hermitian systems

Non-Hermiticity enriches the 10-fold Altland-Zirnbauer symmetry class into the 38-fold symmetry class, where critical behavior of the Anderson transitions (ATs) has been extensively studied recently. Here, we propose a correspondence of the universality classes of the ATs between Hermitian and non-Hermitian systems. We illustrate that the critical exponents of the length scale in non-Hermitian systems coincide with the critical exponents in the corresponding Hermitian systems with additional chiral symmetry. A remarkable consequence of the correspondence is superuniversality, i.e., the ATs in some different symmetry classes of non-Hermitian systems are characterized by the same critical exponent. In addition to the comparisons between the known critical exponents for non-Hermitian systems and their Hermitian counterparts, we obtain the critical exponents in symmetry classes AI, AII, AII$^{\dagger}$, CII$^{\dagger}$, and DIII in two and three dimensions. Estimated critical exponents are consistent with the proposed correspondence. According to the correspondence, some of the exponents also give useful information of the unknown critical exponents in Hermitian systems, paving a way to study the ATs of Hermitian systems by the corresponding non-Hermitian systems.

Travelling wave solutions for a Zakharov–Kuznetsov modified equal width equations

We consider a family of generalized Zakharov–Kuznetsov modified equal width equations in (2＋1)-dimensions which involves an arbitrary function and three parameters. By applying the Lie group theory, we perform a complete classification of the Lie point symmetries of these equations depending on the arbitrary elements involved. We carry out symmetry reductions and obtain exact solutions by using the Lie point symmetries admitted by this family. Moreover, we show that this family of equations admits local low-order multipliers. Finally, we derive all local low-order conservation laws of this family through the multiplier approach.

Dzyaloshinskii-Moriya induced spin-transfer torques in kagome antiferromagnets

In recent years antiferromagnets (AFMs) have become very promising for nanoscale spintronic applications due to their unique properties, such as THz dynamics and the absence of stray fields. Manipulating antiferromagnetic textures is currently, however, limited to very few exceptional material symmetry classes allowing for staggered torques on the magnetic sublattices. In this work, we predict for kagome AFMs with broken mirror symmetry a new coupling mechanism between antiferromagnetic domain walls (DWs) and spin currents, produced by the relativistic Dzyaloshinskii-Moriya interaction (DMI). We microscopically derive the DMI's free-energy contribution for the kagome AFMs. Unlike ferromagnets and collinear AFMs, the DMI does not lead to terms linear in the spatial derivatives, but instead renormalizes the spin-wave stiffness and anisotropy energies. Importantly, we show that the DMI induces a highly nontrivial, twisted DW profile that is controllable via two linearly independent components of the spin accumulation. This texture manipulation mechanism goes beyond the concept of staggered torques and implies a higher degree of tunability for the current-driven DW motion compared to conventional ferromagnets and collinear AFMs.

RUScal: Software for the analysis of resonant ultrasound spectroscopy measurements.

Resonant ultrasound spectroscopy is used to nondestructively measure the elastic resonances of small solids to elucidate the material's elastic properties or other qualities like size, shape, or composition. Here, we introduce the software RUScal for the purpose of determining elastic properties by analyzing the eigenfrequencies of solid specimens with common shapes, such as rectangular parallelepipeds, cylinders (solid and hollow tube), ellipsoids, and octahedrons, as well as irregularly shaped ellipsoids that can be described analytically. All symmetry classes are supported, from isotropic to triclinic, along with the option to add or remove up to three orthogonal mirror planes as well as the ability to reorient the crystal axes with respect the sample edges via Euler angles. Additional features include tools to help find initial sets of elastic constants, including grid exploration and Monte Carlo methods, a tool to analyze frequencies as a function of sample length or crystal orientation, an error analysis tool to assess fit quality, and formatting of the input and output files for batch fitting, e.g., as a function of temperature. This software was validated with published resonant ultrasound spectroscopy data for various materials, shapes, and symmetries with noted improvements in calculation time compared to finite element methods.

Algebraic solutions for pricing American put options under the constant elasticity of variance (CEV) model: Application of the Lie group approach

In this work, the algebraic properties of the American put option under the constant elasticity of variance (CEV) model of financial markets are studied with the implementation of the symmetry approach based on Lie’s group theory. The CEV model is described mathematically in terms of the parabolic time–space partial differential equation (PDE) along with the suitable choice of a terminal condition. The classical Lie symmetry analysis is carried out for the governing PDE. Based on the infinitesimal Lie symmetry generators, new classes of group invariant solutions are derived from the mathematical model and the general as well as the specific cases of the CEV model are thoroughly examined. Finally, the results are plotted against the various emergent parameters and their effect are analyzed and discussed.

Objective crystallographic symmetry classifications of a noisy crystal pattern with strong Fedorov-type pseudosymmetries and its optimal image-quality enhancement.

Statistically sound crystallographic symmetry classifications are obtained with information-theory-based methods in the presence of approximately Gaussian distributed noise. A set of three synthetic patterns with strong Fedorov-type pseudosymmetries and varying amounts of noise serve as examples. Contrary to traditional crystallographic symmetry classifications with an image processing program such as CRISP, the classification process does not need to be supervised by a human being and is free of any subjectively set thresholds in the geometric model selection process. This enables crystallographic symmetry classification of digital images that are more or less periodic in two dimensions (2D), also known as crystal patterns, as recorded with sufficient structural resolution from a wide range of crystalline samples with different types of scanning probe and transmission electron microscopes. Correct symmetry classifications enable the optimal crystallographic processing of such images. That processing consists of the averaging over all asymmetric units in all unit cells in the selected image area and significantly enhances both the signal-to-noise ratio and the structural resolution of a microscopic study of a crystal. For sufficiently complex crystal patterns, the information-theoretic symmetry classification methods are more accurate than both visual classifications by human experts and the recommendations of one of the popular crystallographic image processing programs of electron crystallography.

Acoustic resonators: Symmetry classification and multipolar content of the eigenmodes

Acoustics recently became a versatile platform for discovering novel physical effects and concepts at a relatively simple technological level. On this way, single resonators and the structure of their resonant modes play a central role and define the properties of complex acoustic systems such as acoustic metamaterials, phononic crystals, and topological structures. In this paper, we present a powerful method allowing a qualitative analysis of eigenmodes of resonators in the linear monochromatic acoustic domain based on multipole classification of eigenmodes. Using the apparatus of group theory, we explain and predict the structure of the scattered field knowing only the symmetry group of the resonator by connecting the multipolar content of incident and scattered fields. Such an approach can be utilized for developing resonators with predesigned properties avoiding time-consuming simulations. We have performed full multipole symmetry classification for a number of resonators geometries, and tightened it with scattering spectra profiles.

Universal topological quench dynamics for [formula omitted] topological phases

The free-fermion topological phases with Z2 invariants cover a broad range of topological states, including the time-reversal invariant topological insulators, and are defined on the equilibrium ground states. Whether such equilibrium topological phases have universal correspondence to far-from-equilibrium quantum dynamics is a fundamental issue of both theoretical and experimental importance. Here we uncover the universal topological quench dynamics linking to these equilibrium topological phases of different dimensionality and symmetry classes in the tenfold way, with a general framework being established. We show a novel result that a generic d-dimensional topological phase represented by Dirac type Hamiltonian and with Z2 invariant defined on high symmetry momenta can be characterized by topology reduced to certain arbitrary discrete momenta of Brillouin zone called the highest-order band-inversion surfaces. Such dimension-reduced topology has unique correspondence to the topological pattern emerging in far-from-equilibrium quantum dynamics by quenching the system from trivial phase to the topological regime, rendering the dynamical hallmark of the equilibrium topological phase. This work completes the dynamical characterization for the full tenfold classes of topological phases, which can be partially extended to even broader topological phases protected by lattice symmetries and in non-Dirac type systems, and shall advance widely the research in theory and experiment.

Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution.

New γ-soft rotation in the interacting boson model with SU(3) higher-order interactions * *Supported by Science and Technology Research Planning Project of Education Department of Jilin Province (JJKH20210526KJ)

The interacting boson model with higher-order interactions offers a new route to enhance our understanding on γ-soft rotation. In this paper, -like and -like new γ-softness are observed, in which the corresponding energy levels in the ground and quasi-γ bands can be exactly degenerate and have a partial dynamical symmetry. The spherical-like γ-softness is not related to the classical dynamical symmetry. The transitional behaviors of values of the low-lying levels and quadrupole moment of the state are also discussed. Spherical-like γ-softness can be used to explain the low-lying spectra and values in 110Cd normal states.