Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors

Abstract

Four types of atomic-scale multipoles, electric, magnetic, magnetic toroidal, and electric toroidal multipoles, give a complete set to describe arbitrary degrees of freedom for coupled charge, spin, and orbital of electrons. We here present a systematic classification of these multipole degrees of freedom towards the application in condensed matter physics. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structures and linear responses, such as the magneto-electric effect, magneto-current (magneto-gyrotropic) effect, spin conductivity, Piezo-electric effect, and so on. Moreover, we exhibit a complete table to represent the active multipoles under 32 crystallographic point groups. Our comprehensive and systematic analyses will give a foundation to identify enigmatic electronic order parameters and a guide to evaluate peculiar cross-correlated phenomena in condensed matter physics from microscopic point of view.

arXiv comment37 pages, 4 figures, 32 tables

References in corpus (78)

• Anomalous Hall effect
• Quantized Electric Multipole Insulators
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Nematic order in iron superconductors - who is in the driver's seat?
• Hidden Order in the Cuprates
• Large anomalous Nernst effect at room temperature in a chiral antiferromagnet
• Spin-lattice coupling in frustrated antiferromagnets
• Topological Mott Insulators
• Spin Anisotropy and Quantum Hall Effect in the Kagome Lattice - Chiral Spin State based on a Ferromagnet-
• Theory of Electron Nematic Order in LaOFeAs
• Evidence for Magnetic Weyl Fermions in a Correlated Metal
• d-wave superconductivity and Pomeranchuk instability in the two-dimensional Hubbard model
• Large magneto-optical Kerr effect and imaging of magnetic octupole domains in an antiferromagnetic metal
• Iron-Based Superconductors: current status of materials and pairing mechanism
• Ferro-Orbital Order and Strong Magnetic Anisotropy in the Parent Compounds of Iron-Pnictide Superconductors
• Coupling the valley degree of freedom to antiferromagnetic order
• Cluster multipole theory for anomalous Hall effect in antiferromagnets
• Spin-orbital frustrations and anomalous metallic state in iron-pnictide superconductors
• Effects of nematic fluctuations on the elastic properties of iron arsenide superconductors
• Nematic order in square lattice frustrated ferromagnets
• Non-Trivial Non-Radiating Excitation as a Mechanism of Resonant Transparency in Toroidal Metamaterials
• Orbital-Ordering Induces Structural Phase Transition and the Resistivity Anomaly in Iron Pnictides
• Density Wave States of Non-Zero Angular Momentum
• Kondo Effects and Multipolar Order in the cubic PrTr2Al20 (Tr=Ti, V)
• Self-Consistent Vertex Correction Analysis for Iron-Based Superconductors: Mechanism of Coulomb-Interaction-Driven Orbital Fluctuations
• Heavy Fermion Superconductivity in the Quadrupole Ordered State of PrV2Al20
• Structural quantum criticality and superconductivity in iron-based superconductor Ba(Fe1-xCox)2As2
• Possible Quasi-One-Dimensional Fermi Surface in La_{2-x}Sr_{x}CuO_4
• Parity-breaking phases of spin-orbit-coupled metals with gyrotropic, ferroelectric and multipolar orders
• Superconductivity in the Ferroquadrupolar State in the Quadrupolar Kondo Lattice PrTi${}_2$Al${}_{20}$
• Toroidal order in metals without local inversion symmetry
• Theory of Unconventional Spin Density Wave: A Possible Mechanism of the Micromagnetism in U-based Heavy Fermion Compounds
• Orbital Edelstein effect as a condensed-matter analog of solenoid
• Magneto-Electric Effect in Three-dimensional Coupled Zigzag Chains
• The anapole moment and nucleon weak interactions
• Description of Multipole in f-Electron Systems
• Simultaneous Superconducting and Antiferroquadrupolar Transitions in PrRh${}_2$Zn${}_{20}$
• Microscopic theory of the nematic phase in Sr${}_3$Ru${}_2$O${}_7$
• Microscopic Description of Electric and Magnetic Toroidal Multipoles in Hybrid Orbitals
• First-principles theory of magnetic multipoles in condensed matter systems
• Nonreciprocal magnons and symmetry-breaking in the noncentrosymmetric antiferromagnet
• Magnetic hexadecapole order and magnetopiezoelectric metal in Ba${}_{1-x}$K${}_x$Mn${}_2$As${}_2$
• Bond order instabilities in a correlated two-dimensional metal
• Spin nematic state as a candidate of the hidden order phase of URu${}_2$Si${}_2$
• Evidence for a New Magnetoelectric Effect of Current-Induced Magnetization in a Toroidal Magnetic Ordered State of UNi${}_4$B
• Octupole Ordering Model for the Phase IV of CexLa1-xB6
• Detection of Neutron Scattering from Phase IV of Ce0.7La0.3B6: A Confirmation of the Octupole Order
• Emergent spin-valley-orbital physics by spontaneous parity breaking
• Theory of Orbital Magnetic Quadrupole Moment and Nonlinear Anomalous Thermoelectric Transport
• Quadrupole-Driven Non-Fermi Liquid and Magnetic-Field Induced Heavy Fermion States in a Non-Kramers Doublet System
• Asymmetric Magnon Excitation by Spontaneous Toroidal Ordering
• Unconventional metamagnetic electron states in orbital band systems
• Spontaneous Parity Breaking in Spin-Orbital Coupled Systems
• Spontaneous multipole ordering by local parity mixing
• Excitonic condensation of strongly correlated electrons: the case of Pr${}_{0.5}$Ca${}_{0.5}$CoO${}_3$
• Manipulating magnetoelectric effect -- Essence learned from Co${}_4$Nb${}_2$O${}_9$
• Magnetic structure and Dzyaloshinskii-Moriya interaction in the $S=1/2$ helical-honeycomb antiferromagnet $\alpha$-Cu${}_2$V${}_2$O${}_7$
• Superconductivity in magnetic multipole states
• Heat transport as torsional responses and Keldysh formalism in a curved spacetime
• Competition of Fermi surface symmetry breaking and superconductivity
• Classification of multipole superconductivity in multi-orbital systems and its implications
• Emergent odd-parity multipoles and magnetoelectric effects on a diamond structure: implication to 5$d$ transition metal oxides $A$OsO${}_4$ ($A=$ K, Rb, and Cs)
• Multipole superconductivity in nonsymmorphic Sr${}_2$IrO${}_4$
• Magnetic-Field-Induced 4f-Octupole in CeB6 Probed by Resonant X-ray Diffraction
• Orbital Nematic Instability in Two-Orbital Hubbard Model: Renormalization-Group + Constrained RPA Analysis
• Theory of orbital magnetic quadrupole moment and magnetoelectric susceptibility
• Van Hove singularity and spontaneous Fermi surface symmetry breaking in Sr3Ru2O7
• Nonreciprocal spin Seebeck effect in antiferromagnets
• Electric Octupole Order in Bilayer Ruthenate Sr${}_3$Ru${}_2$O${}_7$
• High-field magnetization and magnetic phase diagram of $\alpha$-Cu${}_2$V${}_2$O${}_7$
• Charge instabilities at the metamagnetic transition
• Symmetry Analysis of Spin-Dependent Electric Dipole and Its Application to Magnetoelectric Effects
• Electric octupole order in bilayer Rashba system
• Electric and magnetic multipoles and bond orders in excitonic insulators
• Optical Selection Rules in Spin-Orbit Coupled Systems on Honeycomb Lattice
• Toroidal order in a partially disordered state on a layered triangular lattice: implication to UNi4B
• Quantum spin Hall effect in a two-orbital model on a honeycomb lattice
• Local nodal Cooper pairs in multiorbital systems

Cited by in corpus (165)

• Altermagnetism: spin-momentum locked phase protected by non-relativistic symmetries
• Momentum-Dependent Spin Splitting by Collinear Antiferromagnetic Ordering
• Bottom-up design of spin-split and reshaped electronic band structures in spin-orbit-coupling free antiferromagnets: Procedure on the basis of augmented multipoles
• Multipolar origin of bound states in the continuum
• Unconventional Superconductivity in UTe2
• Landau Theory of Altermagnetism
• Chiral photocurrent in parity-violating magnet and enhanced response in topological antiferromagnet
• Magnetic octupoles as the order parameter for unconventional antiferromagnetism
• Topological transition from nodal to nodeless Zeeman splitting in altermagnets
• Spontaneous Antisymmetric Spin Splitting in Noncollinear Antiferromagnets without Spin-Orbit Coupling
• Minimal Models for Altermagnetism
• Multipole classification in 122 magnetic point groups for unified understanding of multiferroic responses and transport phenomena
• Multipole expansion for magnetic structures: A generation scheme for symmetry-adapted orthonormal basis set in crystallographic point group
• Loop currents in $A$V${}_3$Sb${}_5$ kagome metals: multipolar and toroidal magnetic orders
• Nonlinear electric transport in odd-parity magnetic multipole systems: Application to Mn-based compounds
• Symmetry analysis and multipole classification of eigenmodes in electromagnetic resonators for engineering their optical properties
• Electric Toroidal Quadrupoles in Spin-Orbit Coupled Metal Cd${}_2$Re${}_2$O${}_7$
• Complete Multipole Basis Set for Single-Centered Electron Systems
• Second-harmonic generation in Mie-resonant dielectric nanoparticles made of noncentrosymmetric materials
• Unified description of electronic orderings and cross correlations by complete multipole representation
• Essential role of anisotropic magnetic dipole in anomalous Hall effect
• Origin of the Magnetic Spin Hall Effect: Spin Current Vorticity in the Fermi Sea
• Nonlinear spin Hall effect in $\mathcal{P}\mathcal{T}$-symmetric collinear magnets
• Systematic analysis method for nonlinear response tensors
• Nonreciprocal Magnon by Symmetric Anisotropic Exchange Interaction in Honeycomb Antiferromagnet
• Analysis of model parameter dependences on the second-order nonlinear conductivity in $\mathcal{P}\mathcal{T}$-symmetric collinear antiferromagnetic metals with magnetic toroidal moment on zigzag chains
• On the definition of chirality and enantioselective fields
• Electric Ferro-Axial Moment as Nanometric Rotator and Source of Longitudinal Spin Current
• Fulde-Ferrell-Larkin-Ovchinnikov state induced by antiferromagnetic order in $\kappa$-type organic conductors
• Nonlinear nonreciprocal transport in antiferromagnets free from spin-orbit coupling
• Helicity locking of square skyrmion crystal in a centrosymmetric lattice system without vertical mirror symmetry
• Rotation and electric-field responses and absolute enantioselection in chiral crystals
• Spin Seebeck and Spin Nernst Effects of Magnons in Noncollinear Antiferromagnetic Insulators
• Second harmonic generation in dielectric nanoparticles with different symmetries
• Symmetry-adapted modeling for molecules and crystals
• Orbital magnetoelectric effect in zigzag nanoribbons of p-band systems
• Even-odd parity transition in strongly correlated locally noncentrosymmetric superconductors : An application to CeRh${}_2$As${}_2$
• Emergent Magneto-multipoles and Nonlinear Responses of a Magnetic Hopfion
• Symmetry analysis with spin crystallographic groups: Disentangling effects free of spin-orbit coupling in emergent electromagnetism
• Magnetoelectric Effect in the Antiferromagnetic Ordered State of Ce${}_3$TiBi${}_5$ with Ce Zig-Zag Chains
• Mechanism of antisymmetric spin polarization in centrosymmetric multiple-$Q$ magnets based on bilinear and biquadratic spin cross products
• Dynamic paramagnon-polarons in altermagnets
• Theory of electric, magnetic, and toroidal polarizations in crystalline solids with applications to hexagonal lonsdaleite and cubic diamond
• Odd-Parity Multipoles by Staggered Magnetic Dipole and Electric Quadrupole Orderings in CeCoSi
• First-principles investigation of magnetic and transport properties in hole-doped shandite compounds Co${}_3$In${}_x$Sn${}_{2-x}$S${}_2$
• Quantification of chirality based on electric toroidal monopole
• Microscopic mechanism for intrinsic nonlinear anomalous Hall conductivity in noncollinear antiferromagnetic metals
• Magnetoelectric Effects and Charge-Imbalanced Solenoids: Antiferro Quadrupole Orders in a Diamond Structure
• Antisymmetric Thermopolarization by Electric Toroidicity
• Theory of Energy Dispersion of Chiral Phonon
• Orbital magnetoelectric effect in nanoribbons of transition metal dichalcogenides
• Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca${}_5$Ir${}_3$O${}_{12}$
• Superconductivity induced by fluctuations of momentum-based multipoles
• Ultrafast control of material optical properties via the infrared resonant Raman effect
• Benchmark for Ab Initio Prediction of Magnetic Structures based on Cluster-Multipole Theory
• Skyrmion crystals in centrosymmetric triangular magnets under hexagonal and trigonal single-ion anisotropy
• Spin Conductivity Based on Magnetic Toroidal Quadrupole Hidden in Antiferromagnets
• Vector, Bidirector and Bloch Skyrmion Phases Induced by Structural Crystallographic Symmetry Breaking
• Thermodynamic approach to electric quadrupole moments
• NQR and NMR spectra in odd-parity multipole material CeCoSi
• Essential model parameters for nonreciprocal magnons in multisublattice systems
• Electronic multipoles and multiplet pairs induced from Pomeranchuk and Cooper instabilities of Bogoliubov Fermi surfaces
• Electric Quadrupolar Contributions in the Magnetic Phases of UNi${}_4$B
• Nonlinear Transverse Magnetic Susceptibility under Electric Toroidal Dipole Ordering
• Photocurrent response in parity-time symmetric current-ordered states
• Orbitronics in Two-dimensional Materials
• Nonreciprocal Transport in Noncoplanar Magnetic Systems without Spin-Orbit Coupling, Net Scalar Chirality, or Magnetization
• Spin-orbital-momentum locking under odd-parity magnetic quadrupole ordering
• Structural Transition in the Hidden Ordered Phase of CeCoSi
• Generation of modulated magnetic structure based on cluster multipole: Application to $\alpha$-Mn and Co$M_3$S${}_6$
• Multiferroic collinear antiferromagnet with hidden altermagnetic split
• Chiral charge as hidden order parameter in URu${}_2$Si${}_2$
• Ferroaxial moment induced by vortex spin texture
• Nonlinear Magnetoelectric Effect under Magnetic Octupole Order: Its Application to a $d$-Wave Altermagnet and a Pyrochlore Lattice with All-In/All-Out Magnetic Order
• Time-reversal switching responses in antiferromagnets
• Triple-$\mathcal{Q}$ partial magnetic orders induced by quadrupolar interactions: Triforce order scenario for UNi${}_4$B
• Triple-$q$ quadrupole-octupole order scenario for PrV${}_2$Al${}_{20}$
• Quasi-symmetry Constrained Spin Ferromagnetism in Altermagnets
• Hidden orders in spin-orbit entangled correlated insulators
• Spin current and chirality degrees of freedom inherent in localized electron orbitals
• Direct Observation of Antiferromagnetic Parity Violation in the Electronic Structure of Mn${}_2$Au
• Magnetic Toroidal Moment under Partial Magnetic Order in Hexagonal Zigzag-Chain Compound Ce${}_3$TiBi${}_5$
• High-Field Ultrasonic Study of Quadrupole Ordering and Crystal Symmetry Breaking in CeRhIn5
• Acoustic resonators: symmetry classification and multipolar content of the eigenmodes
• Revealing hidden magneto-electric multipoles using Compton scattering
• Magnetic field-Temperature Phase Diagram of CeCoSi Constructed by Specific Heat, Magnetoresistivity, and Magnetization Measurements for a Single Crystal
• Uncovering the Hidden Ferroaxial Density Wave as the Origin of the Axial Higgs Mode in RTe${}_3$
• Atomic-Scale Magnetic Toroidal Dipole under Odd-Parity Hybridization
• Phase-modulated superconductivity via altermagnetism
• Kramers' degenerate magnetism and superconductivity
• Quadrupole partial orders and triple-$q$ states on the face-centered cubic lattice
• Ultrathin films of black phosphorus as suitable platforms for unambiguous observation of the orbital Hall effect
• Rotational Response Induced by Electric Toroidal Dipole
• Controlling the orbital Hall effect in gapped bilayer graphene in the terahertz regime
• Charge Transfer Effect under Odd-Parity Crystalline Electric Field: Divergence of Magnetic Toroidal Fluctuation in $\beta$-YbAlB${}_4$
• Classification of electronic nematicity in three-dimensional crystals and quasicrystals
• Analysis of photo-induced chirality and magnetic toroidal moment based on Floquet formalism
• Cycloidal Magnetic Ordering in Noncentrosymmetric EuIrGe${}_3$
• Spontaneous spin selectivity in chiral molecules at the interface
• Magnetic parity violation and parity-time-reversal-symmetric magnets
• Toroidal optical transitions in hydrogen-like atoms
• Nematic Bogoliubov Fermi surfaces from magnetic toroidal order in FeSe${}_{1-x}$S${}_x$
• Unconventional Hall effect and magnetoresistance induced by metallic ferroaxial ordering
• Mean-field Study of Antiferromagnetic and Antiferroquadrupolar Orderings in Tetragonal CeCoSi
• Multipolar ordering in the three-orbital Hubbard model
• Purely Electronic Chirality without Structural Chirality
• Magnetic structures and electronic properties of cubic-pyrochlore ruthenates from first principles
• Multipolar nematic state of nonmagnetic FeSe based on the DFT+$U$
• Multiple-$q$ dipole-quadrupole instability in spin-1 triangular-lattice systems
• Unconventional orbital ordering and emergent dimensional reduction in fulleride superconductors
• Symmetry-adapted closest Wannier modeling based on complete multipole basis set
• Piezoresistivity as a Fingerprint of Ferroaxial Transitions
• Bulk photovoltaic effect in antiferromagnet: Role of collective spin dynamics
• Observation of spin splitting in the surface electronic structure of antiferromagnet NdBi
• Quantification of electronic asymmetry: chirality and axiality in solids
• Magnetoelectric Response in Electric Octupole State: Possible Hidden Order in Cuprate Superconductors
• Quadrupolar Susceptibility and Magnetic Phase Diagram of PrNi${}_2$Cd${}_{20}$ with Non-Kramers Doublet Ground State
• Current-induced quasiparticle magnetic multipole moments
• Noncollinear Magnetic Multipoles in Collinear Altermagnets
• Field-induced superconductivity mediated by odd-parity multipole fluctuation
• Anisotropic Piezomagnetism in Noncollinear Antiferromagnets
• Designing Non-Relativistic Spin Splitting in Oxide Perovskites
• Multipole representation for anisotropic Coulomb interactions
• Magnetic Vortex induced by Nonmagnetic Impurity in Ferromagnets: Magnetic Multipole and Toroidal around the Vacancy
• Predominant Electronic Order Parameter for Structural Chirality -- Role of Spinless Electronic Toroidal Multipoles
• Exploration of Altermagnetism in $\mathrm{Ru}{\mathrm{O}}_2$
• Efficient Method for Prediction of Meta-stable/Ground Multipolar Ordered States and its Application in Monolayer $\alpha$-\ce{RuX3} (X=Cl,I)
• Relativistic Spin-momentum locking in altermagnets
• Optical Hall response in spin-orbit coupled metals: Comparative study of magnetic cluster monopole, quadrupole, and toroidal orders
• Finite-$q$ antiferrotoroidal and ferritoroidal order in a distorted kagome structure
• RKKY Interactions and Multipole Order in Ab initio Wannier Model of CeCoSi
• Chirality/Axiality-Induced Axiality/Chirality via Surface Polarization
• Unified Magnetoelectric Mechanism for Spin Splitting in Magnets
• Unique properties of the optical activity in noncentrosymmetric superconductors: sum rule, missing area, and relation with the superconducting Edelstein effect
• Domain Control by Adjusting Anisotropic Stress in Pyrochlore Oxide Cd2Re2O7
• Magnetic instability under ferroaxial moment
• Uniform and Staggered electric axial moment in zigzag chain
• Classification of multiorbital superconducting state based on augmented multipoles
• DiracBilinears.jl: A package for computing Dirac bilinears in solids
• Electron chirality and hydrodynamic helicity: Analysis in the atomic limit
• Field-induced electric polarization and elastic softening caused by parity-mixed $d$-$p$ hybridized states with electric multipoles in Ba${}_2$CuGe${}_2$O${}_7$
• Bulk inversion asymmetry induced magnetogyrotropic reflection from quantum wells
• Quantum spin models of commensurate $p$-wave magnets
• Multipole analysis of spin currents in altermagnetic MnTe
• X-ray magnetic circular dichroism originating from the $T_z$ term in collinear altermagnets under trigonal crystal field
• Nonlinear Hall effect driven by spin-charge-coupled motive force
• Nonlinear frequency-asymmetric optical response in chiral systems
• Standard model of electromagnetism and chirality in crystals
• Orbital altermagnetism on the kagome lattice and possible application to $A$V${}_3$Sb${}_5$
• Nonlinear orbital response across topological phase transition in centrosymmetric materials
• Theory of Many-Body Multipole Operators in Single-Centered Electron Systems: Two-Body Toroidal Monopoles in Spinless Orbitals
• Nonrelativistic Functional Properties in Collinear Antiferromagnets Based on Multipole Representation Theory
• Microscopic Determination of the c-axis-Oriented Antiferromagnetic Structure in LaMnSi by ${}^{55}$Mn and ${}^{139}$La NMR
• Elastic Softening in Quantum Critical Yb-Al-Au Approximant Crystal and Quasicrystal
• Nonreciprocal magnon excitations by the Dzyaloshinskii-Moriya interaction on the basis of bond magnetic toroidal multipoles
• Dirac bilinears in condensed matter physics: Relativistic correction for observables and conjugate electromagnetic fields
• Odd-Parity Crystalline Electric Field and Multipoles in Icosahedral Quasicrystal and Approximant: Magnetic Toroidal and Electric Dipoles in Ce-Based Systems
• Quantum geometric magnetic monopole and two-phase superconductivity in CeRh${}_2$As${}_2$
• Stacking-dependent nonreciprocal transport in magnetic skyrmions
• Detection of ferroic octupole ordering in $d$-wave altermagnetic rutile-type compounds
• Multipolar ferroelectricity in the Mott regime
• Symmetry Rules on Multipole Interactions under Crystallographic Point Groups and Application to Multiple-$Q$ Multipole States
• Fourth-order and six-order nonlinear spin current rectifier in three-dimensional $h$-wave and $j$-wave odd-parity magnet
• Charge Transfer Effect under Odd-Parity Crystalline Electric Field
• Orbital Hall effect from orbital magnetic moments of Bloch states: the role of a new correction term