Magnetic Point Group Symmetry Research Articles

Activating magnetoelectric optical properties by twisting antiferromagnetic bilayers

Twisting in bilayers introduces structural chirality with two enantiomers, i.e., left- and right-hand bilayers, depending on the oriented twist angle. The interplay between this global chirality and additional degrees of freedom, such as magnetic ordering and the local octahedral chirality arising from the geometry of the bonds, can yield striking phenomena. In this work, we focus on collinear antiferromagnetic CrI$_3$ twisted homo-bilayers, which are characterized by a staggered octahedral chirality in each monolayer. Using symmetry analysis, density functional theory and tight-binding model calculations we show that layers twisting can lower the structural and magnetic point-group symmetries, thus activating pyroelectricity and the magneto-optical Kerr effect, which would otherwise be absent in untwisted antiferromagnetic homo-bilayers. Interestingly, both electric polarization and Kerr angle are controllable by the twist angle and their sign is reversed when switching from left- to right-twisted bilayers. We further unveil the occurrence of unconventional vortices with spin textures that alternate opposite chiralities in momentum space. These findings demonstrate that the interplay between twisting and octahedral chirality in magnetic bilayers and related van der Waals heterostructures represents an extraordinary resource for tailoring their physical properties for spintronic and optoelectronic applications.

The classification of surface states of topological insulators and superconductors with magnetic point group symmetry

Abstract We present the exhaustive classification of surface states of topological insulators and superconductors protected by crystallographic magnetic point group symmetry in three spatial dimensions. Recently, Cornfeld and Chapman [Phys. Rev. B 99, 075105 (2019)] pointed out that the topological classification of mass terms of the Dirac Hamiltonian with point group symmetry is recast as an extension problem of the Clifford algebra, and we use their results extensively. Comparing two types of Dirac Hamiltonians with and without the mass-hedgehog potential, we establish an irreducible character formula to read off which Hamiltonian in the whole K-group belongs to 4th-order topological phases in three spatial dimensions, which are equivalent to atomic insulators consisting of atoms localized at the point group center.

Topological Magnons with Nodal-Line and Triple-Point Degeneracies: Implications for Thermal Hall Effect in Pyrochlore Iridates.

We analyze the magnon excitations in pyrochlore iridates with all-in-all-out (AIAO) antiferromagnetic order, focusing on their topological features. We identify the magnetic point group symmetries that protect the nodal-line band crossings and triple-point degeneracies that dominate the Berry curvature. We find three distinct regimes of magnon band topology, as a function of the ratio of Dzyaloshinskii-Moriya interaction to the antiferromagnetic exchange. We show how the thermal Hall response provides a unique probe of the topological magnon band structure in AIAO systems.

Axion coupling in the hybrid Wannier representation

Many magnetic point-group symmetries induce a topological classification on crystalline insulators, dividing them into those that have a nonzero quantized Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those that do not ("axion-even" or "trivial"). For time-reversal or inversion symmetries, the resulting topological state is usually denoted as a "strong topological insulator" or an "axion insulator" respectively, but many other symmetries can also protect this "axion $Z_2$" index. Topological states are often insightfully characterized by choosing one crystallographic direction of interest, and inspecting the hybrid Wannier (or equivalently, the non-Abelian Wilson-loop) band structure, considered as a function of the two-dimensional Brillouin zone in the orthogonal directions. Here, we systematically classify the axion-quantizing symmetries, and explore the implications of such symmetries on the Wannier band structure. Conversely, we clarify the conditions under which the axion$Z_2$ index can be deduced from the Wannier band structure. In particular, we identify cases in which a counting of Dirac touchings between Wannier bands, or a calculation of the Chern number of certain Wannier bands, or both, allows for a direct determination of the axion $Z_2$ index. We also discuss when such symmetries impose a "flow" on the Wannier bands, such that they are always glued to higher and lower bands by degeneracies somewhere in the projected Brillouin zone, and the related question of when the corresponding surfaces can remain gapped, thus exhibiting a half-quantized surface anomalous Hall conductivity. Our formal arguments are confirmed and illustrated in the context of tight-binding models for several paradigmatic axion-odd symmetries including time reversal, inversion, simple mirror, and glide mirror symmetries.

On the constitutive relations of linear insulating media and related magnetoelectric effects

General symmetry considerations are used to establish the type of constitutive relations that characterize the electromagnetic response of a material with a given magnetic point group. For clarity, the constitutive relations are generalized to explicitly include two additional secondary vector fields related to the magneto- and the electro-toroidal moments. Local energy conservation, as expressed by the Poynting-Heaviside theorem, is invoked to constrain the structure of the possible material tensors. In this way, the different types of non-dissipative materials with local interactions can be classified according to the different electromagnetic effects that are allowed by their magnetic point group symmetry. The electromagnetic response of some selected materials, chosen for illustrative purposes, is briefly discussed.

Singular angular magnetoresistance in a magnetic nodal semimetal.

Transport coefficients of correlated electron systems are often useful for mapping hidden phases with distinct symmetries. Here we report a transport signature of spontaneous symmetry breaking in the magnetic Weyl semimetal cerium-aluminum-germanium (CeAlGe) system in the form of singular angular magnetoresistance (SAMR). This angular response exceeding 1000% per radian is confined along the high-symmetry axes with a full width at half maximum reaching less than 1° and is tunable via isoelectronic partial substitution of silicon for germanium. The SAMR phenomena is explained theoretically as a consequence of controllable high-resistance domain walls, arising from the breaking of magnetic point group symmetry strongly coupled to a nearly nodal electronic structure. This study indicates ingredients for engineering magnetic materials with high angular sensitivity by lattice and site symmetries.

Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology

We classify time-reversal breaking (class A) spinful topological crystalline insulators with crystallographic nonmagnetic (32 types) and magnetic (58 types) point groups. The classification includes all possible magnetic topological crystalline insulators protected by point group symmetry. Whereas the classification of topological insulators is known to be given by the $K$-theory in the momentum space, computation of the $K$-theory has been a difficult task in the presence of complicated crystallographic symmetry. Here we consider the $K$-homology in the real space for this problem, instead of the $K$-theory in the momentum space, both of which give the same topological classification. We apply the Atiyah-Hirzebruch spectral sequence (AHSS) for computation of the $K$-homology, which is a mathematical tool for generalized (co)homology. In the real-space picture, the AHSS naturally gives the classification of higher-order topological insulators at the same time. By solving the group extension problem in the AHSS on the basis of physical arguments, we completely determine possible topological phases including higher-order ones for each point group. Relationships among different higher-order topological phases are argued in terms of the AHSS in the $K$-homology. We find that in some nonmagnetic and magnetic point groups, a stack of two ${\mathbb{Z}}_{2}$ second-order topological insulators can be smoothly deformed into nontrivial fourth-order topological insulators, which implies nontrivial group extensions in the AHSS.

Electrically induced and detected N\xe9el vector reversal in a collinear antiferromagnet

Antiferromagnets are enriching spintronics research by many favorable properties that include insensitivity to magnetic fields, neuromorphic memory characteristics, and ultra-fast spin dynamics. Designing memory devices with electrical writing and reading is one of the central topics of antiferromagnetic spintronics. So far, such a combined functionality has been demonstrated via 90° reorientations of the Néel vector generated by the current-induced spin orbit torque and sensed by the linear-response anisotropic magnetoresistance. Here we show that in the same antiferromagnetic CuMnAs films as used in these earlier experiments we can also control 180° Néel vector reversals by switching the polarity of the writing current. Moreover, the two stable states with opposite Néel vector orientations in this collinear antiferromagnet can be electrically distinguished by measuring a second-order magnetoresistance effect. We discuss the general magnetic point group symmetries allowing for this electrical readout effect and its specific microscopic origin in CuMnAs.

First-principles Theory of Magnetic Multipoles in Condensed Matter Systems

The multipole concept, which characterizes the spacial distribution of scalar and vector objects by their angular dependence, has already become widely used in various areas of physics. In recent years it has become employed to systematically classify the anisotropic distribution of electrons and magnetization around atoms in solid state materials. This has been fuelled by the discovery of several physical phenomena that exhibit unusual higher rank multipole moments, beyond that of the conventional degrees of freedom as charge and magnetic dipole moment. Moreover, the higher rank electric/magnetic multipole moments have been suggested as promising order parameters in exotic hidden order phases. While the experimental investigations of such anomalous phases have provided encouraging observations of multipolar order, theoretical approaches have developed at a slower pace. In particular, a materials' specific theory has been missing. The multipole concept has furthermore been recognized as the key quantity which characterizes the resultant configuration of magnetic moments in a cluster of atomic moments. This cluster multipole moment has then been introduced as macroscopic order parameter for a noncollinear antiferromagnetic structure in crystals that can explain unusual physical phenomena whose appearance is determined by the magnetic point group symmetry. It is the purpose of this review to discuss the recent developments in the first-principles theory investigating multipolar degrees of freedom in condensed matter systems. These recent developments exemplify that ab initio electronic structure calculations can unveil detailed insight in the mechanism of physical phenomena caused by the unconventional, multipole degree of freedom.

Topology of crystalline insulators and superconductors

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by non-interacting Bloch and Bogoliubov de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of ten classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes $Z_2$ global symmetry, mirror reflection, two-fold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a novel periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott-periodicity in the space dimensions. Various symmetry protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry protected Fermi points. The bulk classification and the surface Fermi point classification provide a novel realization of the bulk-boundary correspondence in terms of the K-theory.

Spiral-Spin-Driven Ferroelectricity in a Multiferroic DelafossiteAgFeO2

We have performed dielectric measurements and neutron diffraction experiments on the delafossite AgFeO2. A ferroelectric polarization P is approximately equal to 300 μC/m2 was observed in a powder sample, below 9 K. The neutron diffraction experiment demonstrated successive magnetostructural phase transitions at T(N1)=15 K and T(N2)=9 K. The magnetic structure for 9 K≤T≤15 K is a spin-density wave with a temperature dependent incommensurate modulation k=(-1, q, 1/2), q is approximately equal to 0.384. Below 9 K, the magnetic structure turns into elliptical cycloid with the incommensurate propagation vector k=(-1/2,q,1/2), q is approximately equal to 0.2026 Based on the deduced magnetic point-group symmetry m1' of the low-temperature polar phase, we conclude that the ferroelectric polarization in AgFeO2 is perpendicular to the monoclinic b axis and is driven by the inverse Dzyaloshinskii-Moriya effect with two orthogonal components p1 is proportional to r(ij)×(S(i)×S(j)) and p2 is proportional to S(i)×S(j).

Magneto-structural properties and magnetic anisotropy of small transition-metal clusters: afirst-principles study

Ab initio density-functional calculations including spin–orbit coupling (SOC) havebeen performed for Ni and Pd clusters with three to six atoms and for 13-atomclusters of Ni, Pd, and Pt, extending earlier calculations for Pt clusters with upto six atoms (2011 J. Chem. Phys. 134 034107). The geometric and magneticstructures have been optimized for different orientations of the magnetization withrespect to the crystallographic axes of the cluster. The magnetic anisotropy energies(MAE) and the anisotropies of spin and orbital moments have been determined.Particular attention has been paid to the correlation between the geometric andmagnetic structures. The magnetic point group symmetry of the clusters varies withthe direction of the magnetization. Even for a 3d metal such as Ni, the changein the magnetic symmetry leads to small geometric distortions of the clusterstructure, which are even more pronounced for the 4d metal Pd. For a 5d metal theSOC is strong enough to change the energetic ordering of the structural isomers.SOC leads to a mixing of the spin states corresponding to the low-energy spinisomers identified in the scalar-relativistic calculations. Spin moments are isotropiconly for Ni clusters, but anisotropic for Pd and Pt clusters, orbital momentsare anisotropic for the clusters of all three elements. The magnetic anisotropyenergies have been calculated. The comparison between MAE and orbital anisotropyinvalidates a perturbation analysis of magnetic anisotropy for these small clusters.

Visualization of Time-Reversal Symmetry in Magnetic Point Groups

The direct visualization of crystallographic and magnetic point group symmetry by means of computer graphics simplifies the teaching of point group symmetry at the undergraduate and graduate levels. Still images and animated movies for all 32 crystallographic point groups, and for the 122 magnetic point groups, are presented. For each point group, the action of the symmetry operators on scalar and pseudo-scalar objects, as well as on polar and axial vectors, is represented as a three-dimensional rendered image. All images and movies are made available as supplementary educational material via a dedicated web site.

Magnetic Point Group Symmetries of Spontaneously Polarized and/or Magnetized Domain Walls

We present a list of the 125 magnetic point groups associated with possible layer groups of domain walls. For each group we give the direction of the spontaneous polarization and/or magnetization allowed by the magnetic point group symmetry.

Magnetic Point Group Symmetries of Spontaneously Polarized and/or Magnetized Domain Walls

We present a list of the 125 magnetic point groups associated with possible layer groups of domain walls. For each group we give the direction of the spontaneous polarization and/or magnetization allowed by the magnetic point group symmetry.

Extensions to the Maxwell boundary conditions: Reflection from uniaxial and cubic antiferromagnets

Standard Maxwell boundary conditions do not always apply at the surface of an anisotropic magnetic medium in a vacuum, even when the $\mathbf{D}$ and $\mathbf{H}$ fields that describe the properties of the medium are taken in covariant form. It is shown that discontinuities may arise in these fields due to bound surface current and electric-dipole moment, and the leading multipole forms of these discontinuities are identified. Reflection matrices for normal incidence are derived for all uniaxial and cubic antiferromagnetic crystals. These matrices satisfy reciprocity (time-reversal symmetry) only when the extended boundary conditions are used. Various magnetic point groups are identified that exhibit nonreciprocal effects in reflection, but not in transmission. It is also shown that when nonreciprocal birefringence exists in transmission, nonreciprocal effects are absent in reflection at normal incidence. The magnetic point-group symmetry assigned to ${\mathrm{Nd}}_{2}{\mathrm{CuO}}_{4}$ is questioned on the basis of the theory.

Effect of higher-order electric and magnetic multipolar transitions on Rayleigh light scattering in liquids

We calculate the effect of magnetic dipolar transitions, related with the polarizabilities B″, B′, B′ and B″ and electric-quadrupole transitions, related with the polarizabilities C′, C″, C″ and C′, on dynamical Rayleigh light scattering by isotrophic media. To this aim, we introduce molecular scattering factors of the fourth, fifth and sixth orders related with these polarizabilities. For molecules with 76 magnetic point group symmetries these transitions are found to affect the time-dependent intensities and depolarization ratios of scattered light. Among the 76 groups, the molecules belonging to 23 groups exhibit natural (transmission and Rayleigh) optical activity, whereas the point group 2mm shows Rayleigh activity only. The effects in question result moreover from the contribution of magnetic-dipole and electric-quadrupole transitions. For isotropic molecules, the above transitions lead to partial depolarization of the scattered light wave. Our use of i- and c-tensors as well as Hermitian and anti-Hermitian components permits the extension of our discussion to regions of absorption as well as cases of full symmetry of the molecules.

Electronic current density induced by nuclear magnetic dipoles

Magnetic point-group symmetries of high order in molecules can be described via the electronic current density vector field induced by a set of ideal magnetic dipoles properly placed in space. The expression for the diamagnetic Larmor-type current density induced in the electrons of a molecule by intrinsic nuclear magnetic dipole moments can be obtained via classical electrodynamics, but quantum mechanics is necessary to account for other terms: the Ramsey relations for paramagnetic spin-orbit, spinn-dipolar, Fermi contact, and cross Fermi contact/spin-dipolar contributions to (indirect) spin—spin nuclear magnetic coupling can be rewritten in terms of second-rank current density tensors. Maps of current densities induced by two nuclear dipoles could be used to visualize the electron coupled interaction between two nuclei and to rationalize the coupling mechanism.

Tables of spectroscopic coefficients for magnetic point group symmetry

For the first time symmetrized Clebsch-Gordan coefficients for corepresentations, the so called 2 D- and 3 D-symbols (analogous to the Wigner 2j- and 3j-symbols correspondingly) are calculated, and tabulated for all single- and double-valued corepresentations of all 90 antiunitary magnetic point groups.

Role of molecular symmetry and molecular interactions in the Faraday effect of fluids

General expressions are developed for assessing the effects of molecular symmetry and molecular interactions on the rotation of plane polarized light in the presence of a static magnetic field. This is accomplished by writing the Verdet constant V as a sum of four terms : V = VD O + VPO + VDINT + VPINT where the subscripts 0 and INT refer, respectively, to the absence and presence of molecular interactions in the presence of the external magnetic field and the superscripts D and P denote, respectively, diamagnetic and paramagnetic contributions. The results, which are given in terms of c- and i-tensors and radial and radial-angular correlation functions, are analyzed in terms of magnetic point group symmetries and applied to rarefied and dense systems. From the experimental Verdet constants at atmospheric pressure we calculate the mean electromagnetic hyperpolarizability tensor α* 123(ω) for the molecules He, H2, N2, Cl 2, CO2, HCl, CO, NH3 and CH4. We also calculate α*123(ω) for liquid CS2 and C 6H6 and the contribution due to nearest neighbour interaction to the Verdet constant It is found that VDINT/ VDO = 22.8 % for CS2 and 8.4 % for C 6H6. We also list, for all magnetic point groups, the tensors which describe the Faraday effect.