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m WP:CHECKWIKI error fix for #03. Missing Reflist. Do general fixes if a problem exists. - using AWB (11377) |
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== Tensor Operators Expansion ==
Consider a quantum mechanical system with Hilbert space spanned by <math> |j,m_{j} \rangle </math>, where <math> j </math> is the total angular momentum and <math> m_{j} </math> is its projection on the quantization axis. Then any quantum operators can be represented using the basis set <math> \lbrace |j,m_{j} \rangle \rbrace </math> as a matrix with dimension <math> (2j+1) </math>. Therefore, one can define <math> (2j+1)^{2} </math> 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
:<math>
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The example tells us, for a <math> J </math>-multiplet problem, one will need all rank <math> 0 \sim 2J </math> tensor operators to form a complete super basis. Therefore, for a <math> J=1 </math> system, its density matrix must have quadrupole components. This is the reason why a <math> J > 1/2 </math> problem will automatically introduce high-rank multipoles to the system <ref name="multipolar exchange">S.-T. Pi, R. Nanguneri, and S. Savrasov, Phys. Rev. Lett 112, 077203 (2014); S.-T. Pi, R. Nanguneri, and S. Savrasov, Phys. Rev. B 90, 045148 (2014)</ref></ref>.
[[File:Tensor operator.png|thumb|frame|right|matrix elements and the real part of corresponding harmonic functions of cubic operator basis in J=1 case.<ref name="multipolar exchange"/>]]
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Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.
Using the addition rules of tensor operators, a product of a rank n tensor and a rank m tensor can generate a tensor with rank n+m ~ |n-m|. Therefore, a high rank tensor can be expressed as the product of low rank tensors. This convention is useful to interpret the high rank multipolar exchange terms as a "multi-exchange" process of dipoles (or pseudospins). For example, for the spherical harmonic tensor operators of <math> J=1 </math> case, we have
:<math> Y_{2}^{-2}=2Y_{1}^{-1}Y_{1}^{-1} </math>
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