Multipolar exchange interaction
Magnetic materials with strong spin-orbit interaction, such as: LaFeAsO,[1][2] PrFe4P12,[3][4] YbRu2Ge2,[5] UO2,[6][7][8][9][10] NpO2,[11][12][13] Ce1−xLaxB6,[14] URu2Si2[15][16][17][18][19] and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple, octople, etc.
[20] 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.
Except the multipolar ordering, many hidden order phenomena are believed closely related to the multipolar interactions [11][14][15] Consider a quantum mechanical system with Hilbert space spanned by
Then any quantum operators can be represented using the basis set
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:
form a basis set in the operator space.
We can also use Pauli matrices and the identity matrix to form a super basis Since the rotation properties of
follow the same rules as the rank 1 tensor of cubic harmonics
share the same rotational properties as rank 1 spherical harmonic tensors
tensor operators to form a complete super basis.
problem will automatically introduce high-rank multipoles to the system [21][22] A general definition of spherical harmonic super basis of a
-multiplet problem can be expressed as [20] where the parentheses denote a 3-j symbol; K is the rank which ranges
-multiplet Hilbert space can be expanded as where the expansion coefficients can be obtained by taking the trace inner product, e.g.
Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.
This convention is useful to interpret the high rank multipolar exchange terms as a "multi-exchange" process of dipoles (or pseudospins).
[20] There are four major mechanisms to induce exchange interactions between two magnetic moments in a system:[20] 1).
No matter which one is dominated, a general form of the exchange interaction can be written as[21] where
is restricted to 1 only, the Hamiltonian reduces to conventional Heisenberg model.
An important feature of the multipolar exchange Hamiltonian is its anisotropy.
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
This is one of the main reasons that most multipolar orderings tend to be non-colinear.
Therefore, a suggested definition[21] of antiferromagnetic multipolar ordering is to flip their phases by
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.
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.
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.
[23] 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.
[21] 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.
