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{{Use American English|date=January 2019}}{{Short description|Higher-order interactions of magnetic moments of chemicals
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Magnetic materials with strong [[spin-orbit interaction]], such as: LaFeAsO,<ref name="LaFeAsO">{{cite journal | last1=Cricchio | first1=Francesco | last2=Grånäs | first2=Oscar | last3=Nordström | first3=Lars | title=Low spin moment due to hidden multipole order from spin-orbital ordering in LaFeAsO | journal=Physical Review B | publisher=American Physical Society (APS) | volume=81 | issue=14 | date=13 April 2010 | issn=1098-0121 | doi=10.1103/physrevb.81.140403 | page=140403(R)| bibcode=2010PhRvB..81n0403C }}</ref><ref>{{cite journal | last1=Gonnelli | first1=R. S. | last2=Daghero | first2=D. | last3=Tortello | first3=M. | last4=Ummarino | first4=G. A. | last5=Stepanov | first5=V. A. | last6=Kim | first6=J. S. | last7=Kremer | first7=R. K. | title=Coexistence of two order parameters and a pseudogaplike feature in the iron-based superconductor LaFeAsO<sub>1−x</sub>F<sub>x</sub> | journal=Physical Review B | volume=79 | issue=18 | date=29 May 2009 | issn=1098-0121 | doi=10.1103/physrevb.79.184526 | page=184526| arxiv=0807.3149 | s2cid=118546381 }}</ref> PrFe<sub>4</sub>P<sub>12</sub>,<ref name="PrFe4P12">{{cite journal | last1=Kiss | first1=Annamária | last2=Kuramoto | first2=Yoshio | title=On the Origin of Multiple Ordered Phases in PrFe<sub>4</sub>P<sub>12</sub> | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=74 | issue=9 | date=15 September 2005 | issn=0031-9015 | doi=10.1143/jpsj.74.2530 | pages=2530–2537| arxiv=cond-mat/0504014 | bibcode=2005JPSJ...74.2530K | s2cid=119350615 }}</ref><ref>{{cite journal | last1=Sato | first1=Hidekazu | last2=Sakakibara | first2=Toshiro | last3=Tayama | first3=Takashi | last4=Onimaru | first4=Takahiro | last5=Sugawara | first5=Hitoshi | last6=Sato | first6=Hideyuki | title=Angle-Resolved Magnetization Study of the Multipole Ordering in PrFe<sub>4</sub>P<sub>12</sub> | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=76 | issue=6 | date=15 June 2007 | issn=0031-9015 | doi=10.1143/jpsj.76.064701 | page=064701| bibcode=2007JPSJ...76f4701S }}</ref> YbRu<sub>2</sub>Ge<sub>2</sub>,<ref name="YbRu2Ge2">{{cite journal | last1=Takimoto | first1=Tetsuya | last2=Thalmeier | first2=Peter | title=Theory of induced quadrupolar order in tetragonal YbRu<sub>2</sub>Ge<sub>2</sub>| journal=Physical Review B | volume=77 | issue=4 | date=8 January 2008 | issn=1098-0121 | doi=10.1103/physrevb.77.045105 | page=045105| arxiv=0708.2872 | bibcode=2008PhRvB..77d5105T | s2cid=119203279 }}</ref> UO<sub>2</sub>,<ref name="UO2">{{cite journal | last1=Pi | first1=Shu-Ting | last2=Nanguneri | first2=Ravindra | last3=Savrasov | first3=Sergey | title=Calculation of Multipolar Exchange Interactions in Spin-Orbital Coupled Systems | journal=Physical Review Letters | volume=112 | issue=7 | date=20 February 2014 | issn=0031-9007 | doi=10.1103/physrevlett.112.077203 | page=077203| pmid=24579631 | arxiv=1308.1488 | bibcode=2014PhRvL.112g7203P | s2cid=42262386 }}</ref><ref>{{cite journal | last1=Giannozzi | first1=Paolo | last2=Erdös | first2=Paul | title=Theoretical analysis of the 3-k magnetic structure and distortion of uranium dioxide | journal=Journal of Magnetism and Magnetic Materials | publisher=Elsevier BV | volume=67 | issue=1 | year=1987 | issn=0304-8853 | doi=10.1016/0304-8853(87)90722-0 | pages=75–87| bibcode=1987JMMM...67...75G }}</ref><ref>{{cite book | last1=Mironov | first1=V.S | last2=Chibotaru | first2=L.F | last3=Ceulemans | first3=A | title=Advances in Quantum Chemistry | chapter=First-order Phase Transition in UO<sub>2</sub>: The Interplay of the 5f<sup>2</sup>–5f<sup>2</sup> Superexchange Interaction and Jahn–Teller Effect | publisher=Elsevier | year=2003 | isbn=978-0-12-034844-2 | issn=0065-3276 | doi=10.1016/s0065-3276(03)44040-9 | pages=599–616|volume=44}}</ref><ref>{{cite journal | last1=Carretta | first1=S. | last2=Santini | first2=P. | last3=Caciuffo | first3=R. | last4=Amoretti | first4=G. | title=Quadrupolar Waves in Uranium Dioxide | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=105 | issue=16 | date=11 October 2010 | issn=0031-9007 | doi=10.1103/physrevlett.105.167201 | page=167201| pmid=21231002 | bibcode=2010PhRvL.105p7201C }}</ref><ref>{{cite journal | last1=Caciuffo | first1=R. | last2=Santini | first2=P. | last3=Carretta | first3=S. | last4=Amoretti | first4=G. | last5=Hiess | first5=A. | last6=Magnani | first6=N. | last7=Regnault | first7=L.-P. | last8=Lander | first8=G. H. | title=Multipolar, magnetic, and vibrational lattice dynamics in the low-temperature phase of uranium dioxide | journal=Physical Review B | volume=84 | issue=10 | date=6 September 2011 | issn=1098-0121 | doi=10.1103/physrevb.84.104409 | page=104409| arxiv=1312.5113 | bibcode=2011PhRvB..84j4409C | s2cid=118624728 }}</ref> NpO<sub>2</sub>,<ref name="NpO2">{{cite journal | last1=Santini | first1=P. | last2=Amoretti | first2=G. | title=Magnetic-Octupole Order in Neptunium Dioxide? | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=85 | issue=10 | date=4 September 2000 | issn=0031-9007 | doi=10.1103/physrevlett.85.2188 | pages=2188–2191| pmid=10970494 | bibcode=2000PhRvL..85.2188S }}</ref><ref>{{cite journal | last1=Santini | first1=P. | last2=Carretta | first2=S. | last3=Magnani | first3=N. | last4=Amoretti | first4=G. | last5=Caciuffo | first5=R. | title=Hidden Order and Low-Energy Excitations in NpO<sub>2</sub> | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=97 | issue=20 | date=14 November 2006 | issn=0031-9007 | doi=10.1103/physrevlett.97.207203 | page=207203| pmid=17155710 | bibcode=2006PhRvL..97t7203S }}</ref><ref>{{cite journal | last1=Kubo | first1=Katsunori | last2=Hotta | first2=Takashi | title=Microscopic theory of multipole ordering in NpO<sub>2</sub> | journal=Physical Review B | publisher=American Physical Society (APS) | volume=71 | issue=14 | date=29 April 2005 | issn=1098-0121 | doi=10.1103/physrevb.71.140404 | page=140404(R)| arxiv=cond-mat/0409116 | bibcode=2005PhRvB..71n0404K | s2cid=119391692 }}</ref> Ce<sub>1−x</sub>La<sub>x</sub>B<sub>6</sub>,<ref name="Ce1−xLaxB6">{{cite journal | last1=Mannix | first1=D. | last2=Tanaka | first2=Y. | last3=Carbone | first3=D. | last4=Bernhoeft | first4=N. | last5=Kunii | first5=S. | title=Order Parameter Segregation in Ce<sub>0.7</sub>La<sub>0.3</sub>B<sub>6</sub>:4f Octopole and 5d Dipole Magnetic Order | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=95 | issue=11 | date=8 September 2005 | issn=0031-9007 | doi=10.1103/physrevlett.95.117206 | page=117206| pmid=16197044 | bibcode=2005PhRvL..95k7206M }}</ref> URu<sub>2</sub>Si<sub>2</sub><ref name="URu2Si2">{{cite journal | last1=Chandra | first1=P. | last2=Coleman | first2=P. | last3=Mydosh | first3=J. A. | last4=Tripathi | first4=V. | title=Hidden orbital order in the heavy fermion metal URu<sub>2</sub>Si<sub>2</sub> | journal=Nature | publisher=Springer Nature | volume=417 | issue=6891 | year=2002 | issn=0028-0836 | doi=10.1038/nature00795 | pages=831–834| pmid=12075346 | arxiv=cond-mat/0205003 | bibcode=2002Natur.417..831C | s2cid=11902278 }}</ref><ref>{{cite journal | last1=Cricchio | first1=Francesco | last2=Bultmark | first2=Fredrik | last3=Grånäs | first3=Oscar | last4=Nordström | first4=Lars | title=Itinerant Magnetic Multipole Moments of Rank Five as the Hidden Order in URu<sub>2</sub>Si<sub>2</sub> | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=103 | issue=10 | date=1 August 2009 | issn=0031-9007 | doi=10.1103/physrevlett.103.107202 | page=107202| pmid=19792338 | arxiv=0904.3883 | bibcode=2009PhRvL.103j7202C | s2cid=20622071 }}</ref><ref>{{cite journal | last1=Ikeda | first1=Hiroaki | last2=Suzuki | first2=Michi-To | last3=Arita | first3=Ryotaro | last4=Takimoto | first4=Tetsuya | last5=Shibauchi | first5=Takasada | last6=Matsuda | first6=Yuji | title=Emergent rank-5 nematic order in URu<sub>2</sub>Si<sub>2</sub> | journal=Nature Physics | volume=8 | issue=7 | date=3 June 2012 | issn=1745-2473 | doi=10.1038/nphys2330 | pages=528–533| arxiv=1204.4016 | bibcode=2012NatPh...8..528I | s2cid=119108102 }}</ref><ref>{{cite journal | last1=Kiss | first1=Annamária | last2=Fazekas | first2=Patrik | title=Group theory and octupolar order in URu<sub>2</sub>Si<sub>2</sub> | journal=Physical Review B | publisher=American Physical Society (APS) | volume=71 | issue=5 | date=23 February 2005 | issn=1098-0121 | doi=10.1103/physrevb.71.054415 | page=054415| arxiv=cond-mat/0411029 | bibcode=2005PhRvB..71e4415K | s2cid=118892596 }}</ref><ref>{{cite journal | last1=Rau | first1=Jeffrey G. | last2=Kee | first2=Hae-Young | title=Hidden and antiferromagnetic order as a rank-5 superspin in URu<sub>2</sub>Si<sub>2</sub> | journal=Physical Review B | volume=85 | issue=24 | date=13 June 2012 | issn=1098-0121 | doi=10.1103/physrevb.85.245112 | page=245112| arxiv=1203.1047 | bibcode=2012PhRvB..85x5112R | s2cid=118313829 }}</ref> and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple,
== Tensor operator expansion ==
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=\frac{5}{2}I+2\sigma_{+1}+\frac{3}{2} \sigma_{0}-3\sigma_{-1}
</math>
Again, <math> \sigma_{-1},\sigma_{0},\sigma_{+1} </math> share the same rotational properties as rank 1 spherical [[harmonic tensors]] <math> Y^{1}_{-1}, Y^{1}_{0}, Y^{1}_{-1} </math>, so it is called spherical super basis.
Because atomic orbitals <math> s,p,d,f </math> are also described by spherical or cubic harmonic functions, one can imagine or visualize these operators using the wave functions of atomic orbitals although they are essentially matrices not spatial functions.
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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">{{cite journal | last1=Pi | first1=Shu-Ting | last2=Nanguneri | first2=Ravindra | last3=Savrasov | first3=Sergey | title=Calculation of Multipolar Exchange Interactions in Spin-Orbital Coupled Systems | journal=Physical Review Letters | volume=112 | issue=7 | date=20 February 2014 | issn=0031-9007 | doi=10.1103/physrevlett.112.077203 | page=077203| pmid=24579631 | arxiv=1308.1488 | bibcode=2014PhRvL.112g7203P | s2cid=42262386 }}</ref><ref>{{cite journal | last1=Pi | first1=Shu-Ting | last2=Nanguneri | first2=Ravindra | last3=Savrasov | first3=Sergey | title=Anisotropic multipolar exchange interactions in systems with strong spin-orbit coupling | journal=Physical Review B | volume=90 | issue=4 | date=31 July 2014 | issn=1098-0121 | doi=10.1103/physrevb.90.045148 | page=045148| arxiv=1406.0221 | bibcode=2014PhRvB..90d5148P | s2cid=118960388 }}</ref>
=== Formal definitions ===
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A general definition of spherical harmonic super basis of a <math> J </math>-multiplet problem can be expressed as <ref name="Review"/>
:<math>
Y_{K}^{Q}(J) =\sum_{MM^{\prime }}(-1)^{J-M}(2K+1)^{1/2} \times \left(
\begin{matrix}
J & J & K \\
M^{^{\prime }} \end{matrix}
\right) |JM\rangle \langle JM^{^{\prime }}|,
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projection index of rank K which ranges from −K to +K. A cubic harmonic super basis where all the tensor operators are hermitian can be defined as
:<math> T_{K}^{Q} =\frac{1}{\sqrt{2}}[(-1)^{Q}Y_{K}^{Q}(J)+Y_{K}^{-Q}(J)] </math>
:<math> T_{K}^{-Q} =\frac{i}{\sqrt{2}}[Y_{K}^{
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>
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== Computing 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, 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. 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>{{cite journal | last1=Siemann | first1=Robert | last2=Cooper | first2=Bernard R. | title=Planar Coupling Mechanism Explaining Anomalous Magnetic Structures in Cerium and Actinide Intermetallics | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=44 | issue=15 | date=14 April 1980 | issn=0031-9007 | doi=10.1103/physrevlett.44.1015 | pages=1015–1019| bibcode=1980PhRvL..44.1015S }}</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>{{cite journal | last1=Wills | first1=John M. | last2=Cooper | first2=Bernard R. | title=First-principles calculations for a model Hamiltonian treatment of hybridizing light actinide compounds | journal=Physical Review B | publisher=American Physical Society (APS) | volume=42 | issue=7 | date=1 August 1990 | issn=0163-1829 | doi=10.1103/physrevb.42.4682 | pages=4682–4693| pmid=9996001 | bibcode=1990PhRvB..42.4682W }}</ref> Another first-principle approach was also proposed recently.<ref name="multipolar exchange"/> 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.
== References ==
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