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{{Use American English|date=January 2019}}{{Short description|Higher-order interactions of magnetic moments of chemicals
}}
Magnetic materials with strong [[spin-orbit interaction]], such as: LaFeAsO,<ref name="LaFeAsO">{{cite journal |
== Tensor
=== Basic
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
:<math>
A=\begin{bmatrix}
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=\frac{5}{2}I+2\sigma_{+1}+\frac{3}{2} \sigma_{0}-3\sigma_{-1}
</math>
Again, <math> \sigma_{-1},\sigma_{0},\sigma_{+1} </math> share the same rotational properties as rank 1 spherical [[harmonic tensors]] <math> Y^{1}_{-1}, Y^{1}_{0}, Y^{1}_{-1} </math>, so it is called spherical super basis.
Because atomic orbitals <math> s,p,d,f </math> are also described by spherical or cubic harmonic functions, one can imagine or visualize these operators using the wave functions of atomic orbitals although they are essentially matrices not spatial functions.
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If we extend the problem to <math> J=1 </math>, we will need 9 matrices to form a super basis. For transition super basis, we have <math> \lbrace L_{ij};i,j=1\sim 3 \rbrace </math>. For cubic super basis, we have <math>\lbrace T_{s}, T_{x}, T_{y}, T_{z}, T_{xy}, T_{yz}, T_{zx}, T_{x^{2}-y^{2}}, T_{3z^{2}-r^{2}} \rbrace</math>. For spherical super basis, we have <math>\lbrace Y^{0}_{0}, Y^{1}_{-1}, Y^{1}_{0}, Y^{1}_{-1}, Y^{2}_{-2}, Y^{2}_{-1}, Y^{2}_{0}, Y^{2}_{1}, Y^{2}_{2} \rbrace</math>. In group theory, <math> T_{s}/Y_{0}^{0} </math> are called scalar or rank 0 tensor, <math> T_{x,yz,}/Y^{1}_{-1,0,+1} </math> are called dipole or rank 1 tensors, <math> T_{xy,yz,zx,x^2-y^2,3z^2-r^2}/Y^{2}_{-2,-1,0,+1,+2} </math> are called quadrupole or rank 2 tensors.<ref name="Review"/>
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">{{cite journal |
=== Formal
[[File:Tensor operator.png|thumb|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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A general definition of spherical harmonic super basis of a <math> J </math>-multiplet problem can be expressed as <ref name="Review"/>
:<math>
Y_{K}^{Q}(J) =\sum_{MM^{\prime }}(-1)^{J-M}(2K+1)^{1/2} \times \left(
\begin{matrix}
J & J & K \\
M^{^{\prime }} \end{matrix}
\right) |JM\rangle \langle JM^{^{\prime }}|,
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projection index of rank K which ranges from −K to +K. A cubic harmonic super basis where all the tensor operators are hermitian can be defined as
:<math> T_{K}^{Q} =\frac{1}{\sqrt{2}}[(-1)^{Q}Y_{K}^{Q}(J)+Y_{K}^{-Q}(J)] </math>
:<math> T_{K}^{-Q} =\frac{i}{\sqrt{2}}[Y_{K}^{
Then, any quantum operator <math> A </math> defined in the <math> J </math>-multiplet Hilbert space can be expanded as
:<math> A=\sum_{K,Q}\alpha_{K}^{Q} Y_{K}^{Q}=\sum_{K,Q}\beta_{K}^{Q} T_{K}^{Q}=\sum_{i,j}\gamma_{i,j} L_{i,j} </math>
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Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.
=== Multi-exchange
Using the addition theorem of tensor operators, the product of a rank n tensor and a rank m tensor can generate a new 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
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If so, a quadrupole-quadrupole interaction (see next section) can be considered as a two steps dipole-dipole interaction. For example, <math> Y_{2_{i}}^{+2_{i}}Y_{2_{j}}^{-2_{j}}=4Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}}Y_{1_{j}}^{-1_{j}}Y_{1_{j}}^{-1_{j}} </math>, so the one step quadrupole transition <math> Y_{2_{i}}^{+2_{i}} </math> on site <math> i </math> now becomes a two steps of dipole transition <math> Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}} </math>. Hence not only inter-site-exchange but also intra-site-exchange terms appear (so called multi-exchange). If <math> J </math> is even larger, one can expect more complicated intra-site-exchange terms would appear. However, one has to note that it is not a perturbation expansion but just a mathematical technique. The high rank terms are not necessarily smaller than low rank terms. In many systems, high rank terms are more important than low rank terms.<ref name="Review"/>
== Multipolar
[[File:Multipolar exchange interactions.png|thumb|right|Examples of dipole-dipole and quadrupole-quadrupole exchange interactions in J=1 case. Blue arrow means the transition comes with a <math> \pi </math>phase shift.<ref name="multipolar exchange"/>]]
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An important feature of the multipolar exchange Hamiltonian is its anisotropy.<ref name="multipolar exchange"/> The value of coupling constant <math> C_{K_{i}K_{j}}^{Q{i}Q_{j}} </math> is usually very sensitive to the relative angle between two multipoles. Unlike conventional spin only exchange Hamiltonian where the coupling constants are isotropic in a homogeneous system, the highly anisotropic atomic orbitals (recall the shape of the <math> s,p,d,f </math> wave functions) coupling to the system's magnetic moments will inevitably introduce huge anisotropy even in a homogeneous system. This is one of the main reasons that most multipolar orderings tend to be non-colinear.
== Antiferromagnetism of
[[File:Flipping the phases of multipoles.png|thumb|right|Flipping the phases of multipoles <ref name="multipolar exchange"/>]]
[[File:AFM multipole chain.png|thumb|right|AFM ordering chains of different multipoles.<ref name="multipolar exchange"/>]]
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Unlike magnetic spin ordering where the [[antiferromagnetism]] can be defined by flipping the magnetization axis of two neighbor sites from a [[ferromagnetic]] configuration, flipping of the magnetization axis of a multipole is usually meaningless. Taking a <math> T_{yz} </math> moment as an example, if one flips the z-axis by making a <math> \pi </math> rotation toward the y-axis, it just changes nothing. Therefore, a suggested definition<ref name="multipolar exchange"/> of antiferromagnetic multipolar ordering is to flip their phases by <math> \pi </math>, i.e. <math> T_{yz} \rightarrow e^{i\pi}T_{yz}=-T_{yz} </math>. In this regard, the antiferromagnetic spin ordering is just a special case of this definition, i.e. flipping the phase of a dipole moment is equivalent to flipping its magnetization axis. As for high rank multipoles, e.g. <math> T_{yz} </math>, it actually becomes a <math> \pi/2 </math> rotation and for <math> T_{3z^2-r^2} </math> it is even not any kind of rotation.
==
Calculation of multipolar exchange interactions remains a challenging issue in many aspects. Although there were many works based on fitting the model Hamiltonians with experiments, predictions of the coupling constants based on first-principle schemes remain lacking. Currently there are two studies implemented first-principles approach to explore multipolar exchange interactions. An early study was developed in 80's. It is based on a mean field approach that can greatly reduce the complexity of coupling constants induced by RKKY mechanism, so the multipolar exchange Hamiltonian can be described by just a few unknown parameters and can be obtained by fitting with experiment data.<ref>{{cite journal |
== References ==
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