Multipolar exchange interaction: Difference between revisions

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Magnetic materials with strong [[spin-orbit interaction]], such as: LaFeAsO,<ref name="LaFeAsO">F. Cricchio, O. Granas, and L. Nordstrom, Phys. Rev. B. 81, 140403 (2010); R. S. Gonnelli, D. Daghero, M. Tortello, G. A. Ummarino, V. A. Stepanov, J. S. Kim, and R. K. Kremer, Phys. Rev. B 79, 184526 (2009)</ref> PrFe4P12<ref name="PrFe4P12">A. Kiss and Y. Kuramoto, J. Phys. Soc. Jpn. 74, 2530 (2005); H. Sato, T. Sakakibara, T. Tayama, T. Onimaru, H. Sugawara, and H. Sato, J. Phys. Soc. Jpn. 76, 064701 (2007)</ref>

, YbRu2Ge2,<ref name="YbRu2Ge2">T. Takimoto and P. Thalmeier, Phys. Rev. B 77, 045105 (2008)</ref> UO2,<ref name="UO2">S.-T. Pi, R. Nanguneri, and S. Savrasov, Phys. Rev. Lett. 112, 077203 (2014); P. Giannozzi and P. Erdos, J. Mag. Mag Mater. 67, 75 (1987). V. S. Mironov, L. F. Chibotaru, and A. Ceulemans, Adv. Quan. Chem. 44, 599 (2003); S. Carretta, P. Santini, R. Caciuffo, and G. Amoretti, Phys. Rev. Lett. 105, 167201 (2010); R. Caciuffo, P. Santini, S. Carretta, G. Amoretti, A. Hiess, N. Magnani, L. P. Regnault, and G. H. Lander, Phys. Rev. B 84, 104409 (2011)</ref> NpO2,<ref name="NpO2">P. Santini and G. Amoretti, Phys. Rev. Lett 85, 2188 (2000); P. Santini, S. Carretta, N. Magnani, G. Amoretti, and R. Caciuffo, Phys. Rev. Lett. 97, 207203 (2006); K. Kubo and T. Hotta, Phys. Rev. B 71, 140404 (2005)</ref> Ce1−xLaxB6,<ref name="Ce1−xLaxB6">D. Mannix, Y. Tanaka, D. Carbone, N. Bernhoeft, and S. Kunii, Phys. Rev. Lett. 95, 117206 (2005)</ref> URu2Si2<ref name="URu2Si2">P. Chandra, P. Coleman, J. A. Mydosh, and V. Tripathi, Nature (London) 417, 831 (2002); Francesco Cricchio, Fredrik Bultmark, Oscar Granas, and Lars Nordstrom, Phys. Rev. Lett. 103, 107202 (2009); Hiroaki Ikeda, Michi-To Suzuki, Ryotaro Arita, Tetsuya Takimoto, Takasada Shibauchi, and Yuji Matsuda, Nat. Phys. 8, 528 (2012); A. Kiss and P. Fazekas, Phys. Rev. B 71, 054415 (2005); J. G. Rau and H.-Y. Kee, Phys. Rev. B 85, 245112 (2012)</ref> and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple, octople, etc.<ref name="Review">R. Caciuffo et al., Rev. Mod. Phys. 81, 807 (2009)</ref> 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 <ref name="NpO2"/><ref name="Ce1−xLaxB6"/><ref name="URu2Si2"/>

== Tensor Operators Expansion ==

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If we extend the problem to <math> J=1 </math>, we will need 9 matrices to form a super basis. For transition super basis, we have <math> \lbrace L_{ij};i,j=1\sim 3 \rbrace </math>. For cubic super basis, we have <math>\lbrace T_{s}, T_{x}, T_{y}, T_{z}, T_{xy}, T_{yz}, T_{zx}, T_{x^{2}-y^{2}}, T_{3z^{2}-r^{2}} \rbrace</math>. For spherical super basis, we have <math>\lbrace Y^{0}_{0}, Y^{1}_{-1}, Y^{1}_{0}, Y^{1}_{-1}, Y^{2}_{-2}, Y^{2}_{-1}, Y^{2}_{0}, Y^{2}_{1}, Y^{2}_{2} \rbrace</math>. In group theory, <math> T_{s}/Y_{0}^{0} </math> are called scalar or rank 0 tensor, <math> T_{x,yz,}/Y^{1}_{-1,0,+1} </math> are called dipole or rank 1 tensors, <math> T_{xy,yz,zx,x^2-y^2,3z^2-r^2}/Y^{2}_{-2,-1,0,+1,+2} </math> are called quadrupole or rank 2 tensors.<ref name="Review"/>

The example tells us, for a <math> J </math>-multiplet problem, one will need all rank <math> 0 \sim 2J </math> tensor operators to form a complete super basis. Therefore, for a <math> J=1 </math> system, its density matrix must have quadrupole components. This is the reason why a <math> J > 1/2 </math> problem will automatically introduce high-rank multipoles to the system <ref name="multipolar exchange">S.-T. Pi, R. Nanguneri, and S. Savrasov, Phys. Rev. Lett 112, 077203 (2014); S.-T. Pi, R. Nanguneri, and S. Savrasov, Phys. Rev. B 90, 045148 (2014)</ref></ref>.

=== Formal Definitions ===

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Then, any quantum operator <math> A </math> defined in the <math> J </math>-multiplet Hilbert space can be expanded as

:<math> A=\sum_{K,Q}\alpha_{K}^{Q} Y_{K}^{Q}=\sum_{K,Q}\beta_{K}^{Q} T_{K}^{Q}=\sum_{i,j}\gamma_{i,j} L_{i,j} </math>

where the expansion coefficients can be obtained by taking the trace inner product, e.g. <math> \alpha_{K}^{Q}=Tr[AY_{K}^{Q\dagger}] </math>.

Apparently, one can make linear combination of these operators to form a new super basis that have different symmetries.

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

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

If so, a quadrupole-quadrupole interaction (see next section) can be considered as a two step dipole-dipole interaction, e.g. <math> Y_{2_{i}}^{+2_{i}}Y_{2_{j}}^{-2_{j}}=4Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}}Y_{1_{j}}^{-1_{j}}Y_{1_{j}}^{-1_{j}} </math>. However, it is not a perturbation expansion but just a mathematical technique. The high rank multi-exchange terms are not necessarily smaller than low rank terms. In many cases, high rank terms are more important than low rank terms.<ref name="Review"/>

== Multipolar Exchange Interactions ==

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[[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 aan 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 dipole moments is equivalent to flipping its magnetization axis. As for <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.

== Compute 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, proposals 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.<ref>R. Siemann and B. R. Cooper, Phys. Rev. Lett. 44, 1015 (1980)</ref> 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.<ref>J. M. Wills and B. R. Cooper, Phys. Rev. B 42, 4682 (1990)</ref> Another first-principle approach was also proposed recently.<ref name="multipolar exchange"/> It maps all the coupling constants induced by all static mechanisms to a series of LDA+U total energy calculations and got agreement with uranium dioxide.

== References ==

{{Reflist}}