Multipolar exchange interaction: Difference between revisions

Revision as of 02:34, 22 January 2019

Revision as of 16:22, 19 February 2019

Line 101:

=== Formal Definitions ===

[[File:Tensor operator.png|thumb|frame|right|matrix elements and the real part of corresponding harmonic functions of cubic operator basis in J=1 case.<ref name="multipolar exchange"/>]]

A general definition of spherical harmonic super basis of a <math> J </math>-multiplet problem can be expressed as <ref name="Review"/>

Line 132:

== Multipolar Exchange Interactions ==

[[File:Multipolar exchange interactions.png|thumb|frame|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"/>]]

There are four major mechanisms to induce exchange interactions between two magnetic moments in a system:<ref name="Review"/> 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<ref name="multipolar exchange"/>

Line 141:

== Antiferromagnetism of Multipolar Moments==

[[File:Flipping the phases of multipoles.png|thumb|frame|right|Flipping the phases of multipoles <ref name="multipolar exchange"/>]]

[[File:AFM multipole chain.png|thumb|frame|right|AFM ordering chains of different multipoles.<ref name="multipolar exchange"/>]]

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.