Multipolar exchange interaction: Difference between revisions

Revision as of 18:00, 24 August 2015

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==== Multi-exchange Description ====

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 cubicharmonic tensorstensor operators of <math> J=1 </math> case, we have

:<math> Y_{2}^{-2}=2Y_{1}^{-1}Y_{1}^{-1} </math>

:<math> Y_{2}^{-1}=\sqrt{2}(Y_{1}^{-1}Y_{1}^{0}+Y_{1}^{0}Y_{1}^{-1}) </math>

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where <math> i,j </math> are the site indexes and <math> C_{K_{i}K_{j}}^{Q{i}Q_{j}} </math> is the coupling constant that couples two multipole moments <math> T_{K_{i}}^{Q_{i}} </math> and <math> T_{K_{j}}^{Q_{j}} </math>. One can immediately find if <math> K </math> is restricted to 1 only, the Hamiltonian reduces to conventional Heisenberg model.

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 ofbetween 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 Multipolar Moments==