Complete Multipole Basis Set for Single-Centered Electron Systems

Abstract

A whole series of expressions for four species of multipoles (electric, magnetic, magnetic toroidal, and electric toroidal) is provided as a complete basis set to describe arbitrary single-centered spinful electron systems. A compact formula to calculate matrix elements of these multipoles is also derived. A visualization method of an electronic state characterized in terms of multipoles is proposed. The complete basis set is useful to narrow down a candidate order parameter of electron systems in phase transition, to describe a property of cross-correlated phenomena, to analyze spectra of x-ray scattering in magnetically ordered state, and so on. We demonstrate a usage of the complete basis set by taking monopole and toroidal dipole orderings, and the mutual relationship among three distinct magnetic dipoles (orbital, spin angular momenta and anisotropic dipole) in a spin-orbit coupled system as prime examples.

10 pages, 4 figures, 3 tables, plus 27 pages supplemental materials (some corrections are made)

References in corpus (14)

• Cluster multipole theory for anomalous Hall effect in antiferromagnets
• Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors
• Towards a microscopic theory of toroidal moments in bulk periodic crystals
• Group-theoretical classification of multipole order: emergent responses and candidate materials
• Magneto-Electric Effect in Three-dimensional Coupled Zigzag Chains
• Description of Multipole in f-Electron Systems
• Multipole expansion for magnetic structures: A generation scheme for symmetry-adapted orthonormal basis set in crystallographic point group
• Microscopic Description of Electric and Magnetic Toroidal Multipoles in Hybrid Orbitals
• First-principles theory of magnetic multipoles in condensed matter systems
• Electric Toroidal Quadrupoles in Spin-Orbit Coupled Metal Cd${}_2$Re${}_2$O${}_7$
• Emergent spin-valley-orbital physics by spontaneous parity breaking
• Magnetoelectric multipoles in metals
• Symmetry Analysis of Spin-Dependent Electric Dipole and Its Application to Magnetoelectric Effects
• Augmented Magnetic Octupole in Kagome Antiferromagnets Detectable via X-ray Magnetic Circular Dichroism

Cited by in corpus (45)

• Bottom-up design of spin-split and reshaped electronic band structures in spin-orbit-coupling free antiferromagnets: Procedure on the basis of augmented multipoles
• Multipole classification in 122 magnetic point groups for unified understanding of multiferroic responses and transport phenomena
• Unified description of electronic orderings and cross correlations by complete multipole representation
• Essential role of anisotropic magnetic dipole in anomalous Hall effect
• Systematic analysis method for nonlinear response tensors
• On the definition of chirality and enantioselective fields
• Electric Ferro-Axial Moment as Nanometric Rotator and Source of Longitudinal Spin Current
• Rotation and electric-field responses and absolute enantioselection in chiral crystals
• Symmetry-adapted modeling for molecules and crystals
• Quantification of chirality based on electric toroidal monopole
• Emergence of chirality by multipole interconversion
• Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca${}_5$Ir${}_3$O${}_{12}$
• Spin Conductivity Based on Magnetic Toroidal Quadrupole Hidden in Antiferromagnets
• Essential model parameters for nonreciprocal magnons in multisublattice systems
• NQR and NMR spectra in odd-parity multipole material CeCoSi
• Nonlinear Transverse Magnetic Susceptibility under Electric Toroidal Dipole Ordering
• Spin-orbital-momentum locking under odd-parity magnetic quadrupole ordering
• Ferroaxial moment induced by vortex spin texture
• Chiral charge as hidden order parameter in URu${}_2$Si${}_2$
• Time-reversal switching responses in antiferromagnets
• Spin current and chirality degrees of freedom inherent in localized electron orbitals
• Magnetic Toroidal Moment under Partial Magnetic Order in Hexagonal Zigzag-Chain Compound Ce${}_3$TiBi${}_5$
• Rotational Response Induced by Electric Toroidal Dipole
• Analysis of photo-induced chirality and magnetic toroidal moment based on Floquet formalism
• Unconventional Hall effect and magnetoresistance induced by metallic ferroaxial ordering
• Multipolar ordering in the three-orbital Hubbard model
• Purely Electronic Chirality without Structural Chirality
• Symmetry-adapted closest Wannier modeling based on complete multipole basis set
• Sum rules for x-ray circular and linear dichroism based on complete magnetic multipole basis
• Quantification of electronic asymmetry: chirality and axiality in solids
• Predominant Electronic Order Parameter for Structural Chirality -- Role of Spinless Electronic Toroidal Multipoles
• Multipole representation for anisotropic Coulomb interactions
• Uniform and Staggered electric axial moment in zigzag chain
• Magnetic instability under ferroaxial moment
• Classification of multiorbital superconducting state based on augmented multipoles
• Chirality/Axiality-Induced Axiality/Chirality via Surface Polarization
• Electron chirality and hydrodynamic helicity: Analysis in the atomic limit
• Multipole analysis of spin currents in altermagnetic MnTe
• DiracBilinears.jl: A package for computing Dirac bilinears in solids
• X-ray magnetic circular dichroism originating from the $T_z$ term in collinear altermagnets under trigonal crystal field
• Odd-Parity Crystalline Electric Field and Multipoles in Icosahedral Quasicrystal and Approximant: Magnetic Toroidal and Electric Dipoles in Ce-Based Systems
• 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
• Symmetry Rules on Multipole Interactions under Crystallographic Point Groups and Application to Multiple-$Q$ Multipole States
• Detection of ferroic octupole ordering in $d$-wave altermagnetic rutile-type compounds