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, octuple, 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

Tensor operator expansion

Basic concepts Consider a quantum mechanical system with Hilbert space spanned by | j , m j ⟩ {\displaystyle |j,m_{j}\rangle } , where j {\displaystyle j} is the total angular momentum and m j {\displaystyle m_{j}} is its projection on the quantization axis. Then any quantum operators can be represented using the basis set { | j , m j ⟩ } {\displaystyle \lbrace |j,m_{j}\rangle \rbrace } as a matrix with dimension ( 2 j + 1 ) {\displaystyle (2j+1)} . Therefore, one can define ( 2 j + 1 ) 2 {\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

A = [ 1 2 3 4 ] = 1 [ 1 0 0 0 ] + 2 [ 0 1 0 0 ] + 3 [ 0 0 1 0 ] + 4 [ 0 0 0 1 ] = 1 L 1 , 1 + 2 L 1 , 2 + 3 L 2 , 1 + 4 L 2 , 2 {\displaystyle A={\begin{bmatrix}1&2\\3&4\end{bmatrix}}=1{\begin{bmatrix}1&0\\0&0\end{bmatrix}}+2{\begin{bmatrix}0&1\\0&0\end{bmatrix}}+3{\begin{bmatrix}0&0\\1&0\end{bmatrix}}+4{\begin{bmatrix}0&0\\0&1\end{bmatrix}}=1L_{1,1}+2L_{1,2}+3L_{2,1}+4L_{2,2}}

Obviously, the matrices: L i j = | i ⟩ ⟨ j | {\displaystyle L_{ij}=|i\rangle \langle j|} form a basis set in the operator space. Any quantum operator defined in this Hilbert can be expended by { L i j } {\displaystyle \lbrace L_{ij}\rbrace } operators. In the following, let's call these matrices as a super basis to distinguish the eigen basis of quantum states. More specifically the above super basis { L i j } {\displaystyle \lbrace L_{ij}\rbrace } can be called a transition super basis because it describes the transition between states | i ⟩ {\displaystyle |i\rangle } and | j ⟩ {\displaystyle |j\rangle } . In fact, this is not the only super basis that does the trick. We can also use Pauli matrices and the identity matrix to form a super basis

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