Symmetry Operations Research Articles

IrRep: Symmetry eigenvalues and irreducible representations of ab initio band structures

We present IrRep – a Python code that calculates the symmetry eigenvalues of electronic Bloch states in crystalline solids and the irreducible representations under which they transform. As input it receives bandstructures computed with state-of-the-art Density Functional Theory codes such as VASP, Quantum Espresso, or Abinit, as well as any other code that has an interface to Wannier90. Our code is applicable to materials in any of the 230 space groups and double groups preserving time-reversal symmetry with or without spin-orbit coupling included, for primitive or conventional unit cells. This makes IrRep a powerful tool to systematically analyze the connectivity and topological classification of bands, as well as to detect insulators with non-trivial topology, following the Topological Quantum Chemistry formalism: IrRep can generate the input files needed to calculate the (physical) elementary band representations and the symmetry-based indicators using the ▪ routine of the Bilbao Crystallographic Server. It is also particularly suitable for interfaces with other plane-waves based codes, due to its flexible structure. Program summaryProgram Title:IrRepCPC Library link to program files::https://doi.org/10.17632/yznd2ky9r6.1Developer's repository link:https://github.com/stepan-tsirkin/irrepLicensing provisions: GPLv3Programming language: PythonNature of problem: Symmetry properties of electronic band structures in solids are tightly related to their topological features. This relation is set mathematically by the formalisms of Topological Quantum Chemistry [1,2] and symmetry-based indicators of topology [3], but their application requires knowledge of the irreducible representations of the bands. Therefore, a code to calculate irreducible representations of ab initio bands is essential for a systematic theoretical search and classification of topological materials.Solution method:IrRep reads wave functions from files generated by VASP, Abinit, or Quantum Espresso (also files written as input for Wannier90), determines the space group and symmetry operations via the spglib library, evaluates the eigenvalues of the symmetry operations for the selected bands, and applies Group Theory to determine their irreducible representations.

Understanding and Controlling Mode Hybridization in Multicavity Optical Resonators Using Quantum Theory and the Surface Forces Apparatus.

Optical fields in metal-dielectric multilayers display typical features of quantum systems, such as energy level quantization and avoided crossing, underpinned by an isomorphism between the Helmholtz and Schrödinger wave equations. This article builds on the fundamental concepts and methods of quantum theory to facilitate the understanding and design of multicavity resonators. It also introduces the surface forces apparatus (SFA) as a powerful tool for rapid, continuous, and extensive characterization of mode dispersion and hybridization. Instead of fabricating many different resonators, two equal metal-dielectric-metal microcavities were created on glass lenses and displaced relative to each other in a transparent silicone oil using the SFA. The fluid thickness was controlled in real time with nanometer accuracy from more than 50 μm to less than 20 nm, reaching mechanical contact between the outer cavities in a few minutes. The fluid gap acted as a third microcavity providing optical coupling and producing a complex pattern of resonance splitting as a function of the variable thickness. An optical wave in this symmetric three-cavity resonator emulated a quantum particle with nonzero mass in a potential comprising three square wells. Interference between the wells produced a 3-fold splitting of degenerate energy levels due to hybridization. The experimental results could be explained using the standard methods and formalism of quantum mechanics, including symmetry operators and the variational method. Notably, the interaction between square wells produced bonding, antibonding, and nonbonding states that are analogous to hybridized molecular orbitals and are relevant to the design of “epsilon-near-zero” devices with vanishing dielectric permittivity.

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

Spinor vacuum and C, P, T inversions

We have developed the theory of Clifford reflections and extended spacetime inversions. This extended spacetime has two additional dimensions associated with the presence of internal degrees of freedom of spinors. Inversions C, P, and T contain not only reflections of the basis Clifford vectors and transformations of basis spinors, but also transformations of the components of vector and spinor quantities. The research is carried out on the basis of algebraic quantum field theory using the superalgebraic representation of spinors as well as the 8-component matrix representation of spinors. We have proved that due to the presence of internal degrees of freedom of spinors, there are two vacua, the vacuum of our Universe and an alternative vacuum. The inversion operators C and T transform the vacuum into an alternative one, and therefore cannot be operators of the exact symmetry of our Universe.

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Nilpotent (anti-)BRST and (anti-)co-BRST symmetries in 2D non-Abelian gauge theory: Some novel observations

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-) co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two [Formula: see text]-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

WannSymm: A symmetry analysis code for Wannier orbitals

We derived explicit expressions of symmetry operators on Wannier basis, and implemented these operators in WannSymm software. Based on this implementation, WannSymm can i) symmetrize the real-space Hamiltonian output from Wannier90 code, ii) generate symmetry operators of the little group at a specific k-point, and iii) perform symmetry analysis for Wannier band structure. In general, symmetrized Hamiltonians yield improved results compared with the original ones when they are employed for nodal structure searching, surface Green's function calculations, and other model calculations. Program summaryProgram Title: WannSymmCPC Library link to program files:https://doi.org/10.17632/zgf4p85pwx.1Licensing provisions: BSD 3-clauseProgramming language: CExternal libraries: spglib; BLAS and LAPACK; MPI libraries (optional)Nature of problem: Generate the symmetry operators in Wannier basis; calculate the symmetry eigenvalues and characters; symmetrize the real-space Hamiltonian according to the crystal structure. It deals with nonmagnetic or magnetic systems with or without spin-orbit coupling. For magnetic systems, it can deal with ferromagnetic, commensurate antiferromagnetic long range ordered collinear or non-collinear systems with spin-orbit coupling.Solution method: The code requires the real-space Hamiltonian from Wannier projection, the definition of Wannier basis and the crystal structure associated with the defined basis.

On doubly symmetric Dyck words

In this paper we consider doubly symmetric Dyck words, i.e. Dyck words which are fixed by two symmetry operations α and β introduced in [1]. We study combinatorial properties of doubly symmetric Dyck words, leading to the definition of two recursive algorithms to build these words. As a consequence we have a representation of doubly symmetric Dyck words as vectors of integers, called track vectors. Finally, we show some bijections between a subfamily of doubly symmetric Dyck words and a subfamily of integer partitions. The computation of the sequence fn of doubly symmetric Dyck words of semi-length n shows surprising properties giving rise to some conjectures.

Symmetry operators for the conformal wave equation in rotating black hole spacetimes

We present covariant symmetry operators for the conformal wave equation in the (off-shell) Kerr-NUT-AdS spacetimes. These operators, that are constructed from the principal Killing-Yano tensor, its `symmetry descendants', and the curvature tensor, guarantee separability of the conformal wave equation in these spacetimes. We next discuss how these operators give rise to a full set of conformally invariant mutually commuting operators for the conformally rescaled spacetimes and underlie the $R$-separability of the conformal wave equation therein. Finally, by employing the WKB approximation we derive the associated Hamilton-Jacobi equation with a scalar curvature potential term and show its separability in the Kerr-NUT-AdS spacetimes.

The Role of Symmetry in Non-Hermitian Scattering1

We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

Multiple Dirac nodal lines in an in-plane anisotropic semimetal TaNiTe5

Nodal-line semimetals (NLSMs) contains Dirac/Weyl type band-crossing nodes extending into shapes of line, loop and chain in the reciprocal space, leading to novel band topology and transport responses. Robust NLSMs against spin-orbit coupling typically occur in three-dimensional materials with more symmetry operations to protect the line nodes of band crossing, while the possibilities in lower-dimensional materials are rarely discussed. Here we demonstrate robust NLSM phase in a quasi-one-dimensional nonmagnetic semimetal TaNiTe5. Combining angle-resolved photoemission spectroscopy measurements and first-principles calculations, we reveal how reduced dimension can interact with nonsymmorphic symmetry and result into multiple Dirac-type nodal lines with four-fold degeneracy. Our findings suggest rich physics and application in (quasi-)one-dimensional topological materials and call for further investigation on the interplay between the quantum confinement and nontrivial band topology.

Transformation operators based grey wolf optimizer for travelling salesman problem

In the field of swarm intelligence, the Grey Wolf Optimizer (GWO) is a popular algorithm based on leadership hierarchy. Primarily, GWO was proposed to solve continuous optimization problem. However, in recent years, GWO has been extensively explored to deal with a wide variety of real world problem regardless the nature of problem. GWO has received a lot of attention from researchers because of its advantages over other swarm intelligence approaches and its simplicity. The classical GWO is redesigned in this paper by incorporating the swap, shift and symmetry transformation operators to solve permutation-coded travelling salesman problem (TSP), and it is named as transformation operator based grey wolf optimizer (TO-GWO). In TO-GWO, each wolf represents a possible solution of TSP and using swap, shift and symmetry operators wolves interact with leader wolves in order to obtain optimal solution for TSP. In order to improve the proposed algorithm’s local search capability when solving discrete problems, 2-opt algorithm have also been adapted. The TO-GWO is implemented in MATLAB environment. In this study, the TO-GWO is tested over 50 TSP instances. Also, the results of proposed algorithm are compared with 12 state-of-the-art algorithms for TSP instances with various numbers of cities in order to evaluate its performance. For the majority of the TSP instances used in the experiment, the TO-GWO significantly outperforms other algorithms in terms of quality of solutions and efficiency.

Different shapes of spin textures as a journey through the Brillouin zone

Crystallographic space group symmetry (CPGS) such as polar and nonpolar crystal classes have long been known to classify compounds that have spin-orbit-induced spin splitting. While taking a journey through the Brillouin Zone (BZ) from one k-point to another for a fixed CPGS, it is expected that the wavevector point group symmetry (WPGS) can change, and consequently a qualitative change in the texture of the spin polarization (the expectation value of spin operator $\vec{S}^{nk_{0}}$ in Bloch state $u(n,k)$ and the wavevector $k_0$). However, the nature of the spin texture (ST) change is generally unsuspected. In this work, we determine a full classification of the linear-in-$k$ spin texture patterns based on the polarity and chirality reflected in the WPGS at $k_0$. The spin-polarization vector $\vec{S}^{nk_{0}}$ controlling the ST is bound to be parallel to the rotation axis and perpendicular to the mirror planes and hence, symmetry operation types in WPGSs impose symmetry restriction to the ST. For instance, the ST is always parallel to the wavevector $k$ in non-polar chiral WPGSs since they contain only rotational symmetries. Some consequences of the ST classification based on the symmetry operations in the WPGS include the observation of ST patterns that are unexpected according to the symmetry of the crystal. For example, it is usually established that spin-momentum locking effect requires the crystal inversion symmetry breaking by an asymmetric electric potential. However, we find that polar WPGS can have this effect even in compounds without electric dipoles or external electric fields. We use the determined relation between WPGS and ST as a design principle to select compounds with multiple ST near band edges at different $k$-valleys. Based on high-throughput calculations for 1481 compounds, we find 37 previously fabricated materials with different ST near band edges.

Hidden symmetry operators for asymmetric generalized quantum Rabi models

The hidden symmetry of the asymmetric quantum Rabi model (AQRM) has recently been revealed via a systematic construction of the underlying symmetry operator. Based on the AQRM result, we propose an ansatz for the general form of the symmetry operators for AQRM-related models. Applying this ansatz we obtain the symmetry operator for three models: the anisotropic AQRM, the asymmetric Rabi–Stark model (ARSM), and the anisotropic ARSM.

MagneticTB: A package for tight-binding model of magnetic and non-magnetic materials

We present a Mathematica program package MagneticTB, which can generate the tight-binding model for arbitrary magnetic space group. The only input parameters in MagneticTB are the (magnetic) space group number and the orbital information in each Wyckoff position. Some useful functions including getting the matrix expression for symmetry operators, manipulating the energy band structure by parameters, and interfacing with other software are also developed. MagneticTB can help to investigate the physical properties in both magnetic and non-magnetic system, especially for topological properties. Program summaryProgram Title: MagneticTBCPC Library link to program files:https://doi.org/10.17632/kws9xbvz3y.1Developer's repository link:https://github.com/zhangzeyingvv/MagneticTBLicensing provisions: GNU General Public Licence 3.0Programming language: MathematicaExternal routines/libraries: ISOTROPY (iso.byu.edu)Nature of problem: Construct the symmetry adapted tight-binding model for the system with arbitrary magnetic space group.Solution method: The symmetry-adapted tight-binding model in this code is based on the Bloch theorem and group representation theory.Additional comments including restrictions and unusual features: This code supports the spinless system and spinful system with collinear or non-collinear magnetization configuration.

Advanced modeling of materials with PAOFLOW 2.0: New features and software design

Recent research in materials science opens exciting perspectives to design novel quantum materials and devices, but it calls for quantitative predictions of properties which are not accessible in standard first principles packages. PAOFLOW, is a software tool that constructs tight-binding Hamiltonians from self-consistent electronic wavefunctions by projecting onto a set of atomic orbitals. The electronic structure provides numerous materials properties that otherwise would have to be calculated via phenomenological models. In this paper, we describe recent re-design of the code as well as the new features and improvements in performance. In particular, we have implemented symmetry operations for unfolding equivalent k-points, which drastically reduces the runtime requirements of first principles calculations, and we have provided internal routines of projections onto atomic orbitals enabling generation of real space atomic orbitals. Moreover, we have included models for non-constant relaxation time in electronic transport calculations, doubling the real space dimensions of the Hamiltonian as well as the construction of Hamiltonians directly from analytical models. Importantly, PAOFLOW has been now converted into a Python package, and is streamlined for use directly within other Python codes. The new object oriented design treats PAOFLOW’s computational routines as class methods, providing an API for explicit control of each calculation.

States of self-stress in symmetric frameworks and applications

We use the symmetry-extended Maxwell rule established by Fowler and Guest to detect states of self-stress in symmetric planar frameworks. The dimension of the space of self-stresses that are detectable in this way may be expressed in terms of the number of joints and bars that are unshifted by various symmetry operations of the framework. Therefore, this method provides an efficient tool to construct symmetric frameworks with many ‘fully-symmetric’ states of self-stress, or with ‘anti-symmetric’ states of self-stress. Maximising the number of independent self-stresses of a planar framework, as well as understanding their symmetry properties, has important practical applications, for example in the design and construction of gridshells. We show the usefulness of our method by applying it to some practical examples.

Thermally Induced Reversible Metal-to-Metal Charge Transfer in Mixed-Valence {Fe III 4 Fe II 4 } Cubes

Thermally Induced Reversible Metal-to-Metal Charge Transfer in Mixed-Valence {Fe ^III ₄ Fe ^II ₄ } Cubes

Modeling and Structure Determination of Homo-Oligomeric Proteins: An Overview of Challenges and Current Approaches.

Protein homo-oligomerization is a very common phenomenon, and approximately half of proteins form homo-oligomeric assemblies composed of identical subunits. The vast majority of such assemblies possess internal symmetry which can be either exploited to help or poses challenges during structure determination. Moreover, aspects of symmetry are critical in the modeling of protein homo-oligomers either by docking or by homology-based approaches. Here, we first provide a brief overview of the nature of protein homo-oligomerization. Next, we describe how the symmetry of homo-oligomers is addressed by crystallographic and non-crystallographic symmetry operations, and how biologically relevant intermolecular interactions can be deciphered from the ordered array of molecules within protein crystals. Additionally, we describe the most important aspects of protein homo-oligomerization in structure determination by NMR. Finally, we give an overview of approaches aimed at modeling homo-oligomers using computational methods that specifically address their internal symmetry and allow the incorporation of other experimental data as spatial restraints to achieve higher model reliability.

Stable valley-layer coupling and design principle in 2D lattice

Stable valley-layer coupling, which can be against spin–orbit coupling (SOC), is of both fundamental and technological importance as it offers a design principle for 2D valleytronics; however, a reliable mechanism to achieve the goal is missing. In this Letter, a general rule to design such valley-layer coupling is mapped out from symmetry analysis. The degenerate valleys with valley-contrasted layer physics and protected valley-layer coupling can be present in bilayer lattice, when special symmetry operations between layers depending on the inversion center are satisfied. Such valley-layer coupling and its stability against SOC are further revealed in a real material of MnF4 based on first-principles. The distinctive properties, such as optical selection of valley and electric polarization of interlayer excitons, are observed in such a unique system. Our results not only provide a feasible principle to design materials with stable valley-layer coupling but also greatly enrich the physics and broaden the scientific impact of 2D valleytronics.