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

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 structure and linear responses, such as the magnetoelectric effect, magnetocurrent (magnetogyrotropic) effect, spin conductivity, piezoelectric 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 the microscopic point of view.

Figure 1

Four types of multipoles under the space-time inversion group. The E (M) multipole transforms the MT (ET) multipole by reversing the time-reversal property, while the E (M) multipole transforms ET (MT) multipole by reversing the spatial-inversion property. Schematic pictures of the wave functions for the E dipole, M dipole, MT dipole, and ET quadrupole are shown in each panel. The shape and color map of the pictures represent the electric charge density and the $z$ component of the orbital angular-momentum density, respectively.

Figure 2

Examples of the typical electronic band modulations in the $k_x-k_y$ plane at $k_z=0$ in the presence of the E, M, MT, and ET multipoles up to the rank $l=3$ in the single-band picture. The blue (red) boxes stand for the band modulations by the even-parity (odd-parity) multipoles. The green arrows represent the in-plane spin moments at $\mathit{k}$, while the red (blue) curves show the up (down)-spin moments along the out-of-plane direction. The dashed circles in each box represent the energy contours in the absence of the multipole, which corresponds to the energy contour for $Q_0^{\mathrm{ext}}$. The superscript “ext” of $X_{lm}^{\mathrm{ext}}$ is omitted for simplicity.

Figure 3

Schematic picture of a revised Heckmann diagram showing the relation among electric, magnetic, and mechanical properties in solids. $\mathit{E},\mathit{H},\mathit{J}$, and ${\zeta }_{kl}={\varepsilon }_{kl}+{\sum }_m{\epsilon }_{klm}{\omega }_m$ are the external electric field, magnetic field, electric current, and strain-rotation field, while $\mathit{P},\mathit{M}$, and ${\sigma }_{ij}$ represent the electric polarization, magnetization, and stress tensor, respectively. The relevant multipoles in the linear-response tensor for each cross-correlated coupling are also shown.

Figure 4

Odd-parity CEF dependencies of the atomic energy level for $p-d$ hybrid orbitals under the tetragonal $D_{2\mathrm{d}}$ group. The colors show the weights for $p$ and $d$ orbitals. The difference of the atomic levels for $p$ and $d$ orbitals is taken as (a) $\mathrm{\Delta }=-0.7$ and (b) $\mathrm{\Delta }=0.2$, and the CEF parameters are $c_{20}=1,c_{40}=0.5$, and $c_{44}=0.4$.