Spin-Derived Electric Polarization and Chirality Density Inherent in Localized Electron Orbitals

In solid state physics, any phase transition is commonly observed as a change in the microscopic distribution of charge, spin, or current. However, there is an exotic order parameter inherent in the localized electron orbitals that cannot be primarily captured by these three fundamental quantities. This order parameter is described as the electric toroidal multipoles connecting different total angular momenta under the spin-orbit coupling. The corresponding microscopic physical quantity is the spin current tensor on an atomic scale, which induces spin-derived electric polarization aligned circularly and the chirality density of the Dirac equation. Here, elucidating the nature of this exotic order parameter, we obtain the following general consequences that are not restricted to localized electron systems; chirality density is indispensable to unambiguously describe electronic states and it is a species of electric toroidal multipoles, just as the charge density is a species of electric multipoles. Furthermore, we derive the equation of continuity for chirality and discuss its relation to chiral anomaly and optical chirality. These findings link microscopic spin currents and chirality in the Dirac theory to the concept of multipoles and provide a new perspective for quantum states of matter.