page_title: Multipolar exchange interaction
section_title: Multipolar Exchange Interactions
hierarchical_section_title: Multipolar exchange interaction / Multipolar Exchange Interactions
caption_attribution_description: English: Examples of dipole-dipole and quadrupole-quadrupole exchange interactions in J=1 case. Ref: Phys. Rev. B 90, 045148 (2014)
context_page_description: 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
context_section_description: 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  are the site indexes and  is the coupling constant that couples two multipole moments  and . One can immediately find if  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  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  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.
caption: Examples of dipole-dipole and quadrupole-quadrupole exchange interactions in J=1 case. Blue arrow means the transition comes with a 
    
      
        
          π
        
      
      {\displaystyle \pi }
    
  phase shift.[21]
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