Multipolar exchange interaction

Magnetic materials with strong spin-orbit interaction, such as: LaFeAsO, PrFe4P12 , YbRu2Ge2, UO2, NpO2, Ce1−xLaxB6, URu2Si2 and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple, octople, etc. Due to the strong spin-orbit coupling, multipoles are automatically introduced to the systems when the total angular momentum quantum number J is larger than 1/2. If those multipoles are coupled by some exchange mechanisms, those multipoles could tend to have some ordering as conventional spin 1/2 Heisenberg problem. Except the multipolar ordering, many hidden order phenomena are believed closely related to the multipolar interactions

Consider a quantum mechanical system with Hilbert space spanned by ${\displaystyle |j,m_{j}\rangle }$, where ${\displaystyle j}$ is the total angular momentum and ${\displaystyle m_{j}}$ is its projection on the quantization axis. Then any quantum operators can be represented using the basis set ${\displaystyle \lbrace |j,m_{j}\rangle \rbrace }$ as a matrix with dimension ${\displaystyle (2j+1)}$. Therefore, one can define ${\displaystyle (2j+1)^{2}}$ matrices to completely expand any quantum operator in this Hilbert space. Taking J=1/2 as an example, a quantum operator A can be expanded as

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