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.

Second-harmonic generation in dielectric nanoparticles with different symmetries

“…Thus, it is necessary to consider the behavior of vector harmonics with respect to the intersection of the particle's and lattice's groups, and to determine whether their product contains an invariant representation. Symmetry classification and irreducible representations of vector spherical harmonics for finite groups are given, for example, in [29,44].…”

Second-harmonic generation in Mie-resonant dielectric nanoparticles made of noncentrosymmetric materials

“…While the selection rules in spherical BaTiO 3 nanoparticles can be determined just from the conservation of the angular momentum projection quantum number m, the table can be also useful for the nanoparticles of the pyramidal shape. Similar classifications for other symmetries can be found in [61] Appendix D: Anisotropy of the linear material parameters…”

Multipole Description of Emergent Spin–Orbit Interaction in Organic Antiferromagnet \(\kappa \)-(BEDT-TTF)2Cu[N(CN)2]Cl

“…xy ) for X = Q and T , respectively [31,32]. In particular, as we consider the s-orbital-like atomic wave function at each sublattice, the onsite degrees of freedom can be described by the distribution of the electric monopole Q 0 on a sublattice site, while the electric (magnetic) bond degrees of freedom can be described by the distribution of the electric monopole Q 0 (magnetic toroidal dipole T x , T y , T z ) on the bond center [33].…”

“…In the present study, we give a systematic investigation why the collinear antiferromagnet ordering in κ-Cl exhibits the momentum-dependent spin splitting in the band structure, based on a microscopic multipole description [20,[31][32][33]. By applying the multipole description to a sublattice cluster in the unit cell and a tight-binding model for κ-Cl, we show that the xy-type electric quadrupole degree of freedom becomes active under the collinear-type antiferromagnetic ordering in κ-Cl.…”