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.
Highlights
It was well-known that there are four types of fundamental multipoles according to their spatial inversion and timereversal properties [4,19,20,21]: electric (E: polar/true tensor with time-reversal even), magnetic (M: axial/pseudotensor with time-reversal odd), magnetic toroidal (MT: polar/true tensor with time-reversal odd), and electric toroidal (ET: axial/pseudotensor with time-reversal even) multipoles
We have presented a definition of four multipoles in both real and momentum spaces, and how to apply to 32 point groups in seven crystal systems
We have demonstrated which multipole degrees of freedom become active in the tetragonal D2d group as an example
Summary
The multipole moments characterize electric charge and current distributions, whose concept has been widely developed in various fields of physics at different length scales, such as classical electromagnetism [1,2,3,4], nuclear physics [5,6,7,8], solid-state physics [9,10,11,12,13], and metamaterials [14,15,16,17,18]. A recent study has shown that the four fundamental multipoles in atomic-scale constitute a complete set to span the Hilbert space under the space-time inversion group They can be applied to a classical and a quantum-mechanical picture [21]. The concept of multipoles is extended to multiple hybrid orbitals [21], and to the momentum space relevant with topologically nontrivial excitations through the Berry curvature [49,50,51,52,53] In this way, studies of the multipoles are useful to cover various unconventional order parameters in a systematic manner and understand/expect physical phenomena from the symmetry viewpoint. We use here some of our results previously reported in Ref. [21]
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