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 
Tensor operator expansion.
Basic concepts.
Consider a quantum mechanical system with Hilbert space spanned by formula_1, where formula_2 is the total angular momentum and formula_3 is its projection on the quantization axis. Then any quantum operators can be represented using the basis set formula_4 as a matrix with dimension formula_5. Therefore, one can define formula_6 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 
Obviously, the matrices: formula_8 form a basis set in the operator space. Any quantum operator defined in this Hilbert can be expended by formula_9 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 formula_9 can be called a transition super basis because it describes the transition between states formula_11 and formula_12. 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
Since the rotation properties of formula_14 follow the same rules as the rank 1 tensor of cubic harmonics formula_15 and the identity matrix formula_16 follows the same rules as the rank 0 tensor formula_17, the basis set formula_18 can be called cubic super basis. Another commonly used super basis is spherical harmonic super basis which is built by replacing the formula_19 to the raising and lowering operators formula_20 
Again, formula_22 share the same rotational properties as rank 1 spherical harmonic tensors formula_23, so it is called spherical super basis.
Because atomic orbitals formula_24 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.
If we extend the problem to formula_25, we will need 9 matrices to form a super basis. For transition super basis, we have formula_26. For cubic super basis, we have formula_27. For spherical super basis, we have formula_28. In group theory, formula_29 are called scalar or rank 0 tensor, formula_30 are called dipole or rank 1 tensors, formula_31 are called quadrupole or rank 2 tensors.
The example tells us, for a formula_32-multiplet problem, one will need all rank formula_33 tensor operators to form a complete super basis. Therefore, for a formula_25 system, its density matrix must have quadrupole components. This is the reason why a formula_35 problem will automatically introduce high-rank multipoles to the system 
Formal definitions.
A general definition of spherical harmonic super basis of a formula_32-multiplet problem can be expressed as 
where the parentheses denote a 3-j symbol; K is the rank which ranges formula_38; Q is the
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 
Then, any quantum operator formula_41 defined in the formula_32-multiplet Hilbert space can be expanded as
where the expansion coefficients can be obtained by taking the trace inner product, e.g. formula_44.
Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.
Multi-exchange description.
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 formula_25 case, we have
If so, a quadrupole-quadrupole interaction (see next section) can be considered as a two steps dipole-dipole interaction. For example, formula_51, so the one step quadrupole transition formula_52 on site formula_53 now becomes a two steps of dipole transition formula_54. Hence not only inter-site-exchange but also intra-site-exchange terms appear (so called multi-exchange). If formula_32 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.
Multipolar exchange interactions.
There are four major mechanisms to induce exchange interactions between two magnetic moments in a system: 1). Direct exchange 2). RKKY 3). Superexchange 4). Spin-Lattice. No matter which one is dominated, a general form of the exchange interaction can be written as
where formula_57 are the site indexes and formula_58 is the coupling constant that couples two multipole moments formula_59 and formula_60. One can immediately find if formula_61 is restricted to 1 only, the Hamiltonian reduces to conventional Heisenberg model.
An important feature of the multipolar exchange Hamiltonian is its anisotropy. The value of coupling constant formula_58 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 formula_24 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 multipolar moments.
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 formula_64 moment as an example, if one flips the z-axis by making a formula_65 rotation toward the y-axis, it just changes nothing. Therefore, a suggested definition of antiferromagnetic multipolar ordering is to flip their phases by formula_65, i.e. formula_67. 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. formula_64, it actually becomes a formula_69 rotation and for formula_70 it is even not any kind of rotation.
Computing coupling constants.
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. Later on, a first-principles approach to estimate the unknown parameters was further developed and got good agreements with a few selected compounds, e.g. cerium momnpnictides. Another first-principle approach was also proposed recently. It maps all the coupling constants induced by all static exchange mechanisms to a series of DFT+U total energy calculations and got agreement with uranium dioxide.