One of the most intriguing aspects of condensed matter physics is its diversity. Numerous materials with various physical properties have been discovered. They are often classified according to symmetry breaking in the electron system. For example, ferroelectricity is related to the breaking of spatial-inversion symmetry and ferromagnetism is related to the breaking of time-reversal symmetry.

Utilizing the symmetry breakings, various cross correlations and transport phenomena have been discovered. In this regard, electronic multipoles have been introduced as the symmetry-adapted complete basis set to describe any internal electronic degrees of freedom in solids, such as charge, spin, and orbital degrees of freedom, in a unified manner. The complete basis set includes four types of multipole bases, including electric, magnetic, electric toroidal, and magnetic toroidal multipoles.

In a new study published in Journal of the Physical Society of Japan, physicists reviewed the recent developments in the research of multipole representations and their applications to different kinds of materials. According to their review, multipole representation offers several advantages, including systematic identification and classification of electronic order parameters, predictability of overlooked physical phenomena under various electronic orderings, and the exploration of cross correlations and transport properties.

These advantages provide a comprehensive understanding of various physical phenomena observed in solids beyond the symmetry argument and can lead to bottom-up engineering of desired functionalities based on microscopic electronic degrees of freedom.

Lastly, the study also presented methods to identify active multipoles and expected physical phenomena in real materials using various examples. Thus, it serves as the foundation for studying unknown electronic phases and their related physical phenomena, pushing condensed matter physics onto the next stage.