Multipolar exchange interaction: Difference between revisions

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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"> 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>

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_1−xF_x | 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> PrFe₄P₁₂,<ref name="PrFe4P12">{{cite journal | last1=Kiss | first1=Annamária | last2=Kuramoto | first2=Yoshio | title=On the Origin of Multiple Ordered Phases in PrFe₄P₁₂ | 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 PrFe₄P₁₂ | 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> YbRu₂Ge₂,<ref name="YbRu2Ge2">{{cite journal | last1=Takimoto | first1=Tetsuya | last2=Thalmeier | first2=Peter | title=Theory of induced quadrupolar order in tetragonal YbRu₂Ge₂| 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> UO₂,<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 UO₂: The Interplay of the 5f²–5f² 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> NpO₂,<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 NpO₂ | 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 NpO₂ | 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_1−xLa_xB₆,<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_0.7La_0.3B₆:4f 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> URu₂Si₂<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 URu₂Si₂ | 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 URu₂Si₂ | 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 URu₂Si₂ | 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 URu₂Si₂ | 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 URu₂Si₂ | 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, octuple, etc.<ref name="Review">{{cite journal | last1=Santini | first1=Paolo | last2=Carretta | first2=Stefano | last3=Amoretti | first3=Giuseppe | last4=Caciuffo | first4=Roberto | last5=Magnani | first5=Nicola | last6=Lander | first6=Gerard H. | title=Multipolar interactions inf-electron systems: The paradigm of actinide dioxides | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=81 | issue=2 | date=2 June 2009 | issn=0034-6861 | doi=10.1103/revmodphys.81.807 | pages=807–863| bibcode=2009RvMP...81..807S | hdl=11381/2293903 | hdl-access=free }}</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" />

, 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 operator expansion ==

=== TensorBasic Operators Expansionconcepts ===

==== Basic Concepts ====

Consider a quantum mechanical system with Hilbert space spanned by <math> |j,m_{j} \rangle </math>, where <math> j </math> is the total angular momentum and <math> m_{j} </math> is its projection on the quantization axis. Then any [[quantum operatorsoperator]]s can be represented using the basis set <math> \lbrace |j,m_{j} \rangle \rbrace </math> as a matrix with dimension <math> (2j+1) </math>. Therefore, one can define <math> (2j+1)^{2} </math> matrices to completely expand any quantum operator in this Hilbert space. Taking J=1/2 as an example, a quantum operator A can be expanded as

:<math>

A=\begin{bmatrix}

Line 35:

=1L_{1,1}+2L_{1,2}+3L_{2,1}+4L_{2,2}

</math>

Obviously, the matrices: <math> L_{ij}=|i\rangle \langle j | </math> formsform a basis set in the operator space. Any quantum operator defined in this Hilbert can be expended by <math> \lbrace L_{ij} \rbrace </math> operators. In the following, let's call these matrices as a super basis to distinguish the eigen basis of quantum states. More speificallyspecifically the above super basis <math> \lbrace L_{ij} \rbrace </math> can be called a transition super basis because it describes the transition between states <math> |i\rangle </math> and <math> |j\rangle </math>. In fact, this is not the only super basis that does the trick. We can also use Pauli matrices plusand the identity matrix to form a super basis

:<math>

A=\begin{bmatrix}

Line 91:

=\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 spacialspatial functions.

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 S.-T.journal | last1=Pi, R.| first1=Shu-Ting | last2=Nanguneri, and| S.first2=Ravindra | last3=Savrasov, Phys.| Rev.first3=Sergey Lett| title=Calculation of Multipolar Exchange Interactions in Spin-Orbital Coupled Systems | journal=Physical Review Letters | volume=112, 077203| issue=7 | date=20 February (2014); S.| issn=0031-T9007 | doi=10.1103/physrevlett.112.077203 Pi,| page=077203| pmid=24579631 | Rarxiv=1308.1488 | bibcode=2014PhRvL.112g7203P | s2cid=42262386 }}</ref><ref>{{cite journal | last1=Pi | first1=Shu-Ting | last2=Nanguneri, and| S.first2=Ravindra | last3=Savrasov, Phys.| Rev.first3=Sergey | title=Anisotropic multipolar exchange interactions in systems with strong spin-orbit coupling | journal=Physical Review B | volume=90, 045148| issue=4 | date=31 July (2014) <| issn=1098-0121 | doi=10.1103/ref>physrevb.90.045148 | page=045148| arxiv=1406.0221 | bibcode=2014PhRvB..90d5148P | s2cid=118960388 }}</ref>.

==== Formal Definitionsdefinitions ====

[[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"/> ]]

[[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"/>

:<math>

Y_{K}^{Q}(J) =\sum_{MM^{\prime }}(-1)^{J-M}(2K+1)^{1/2} \times \left(

\begin{matrix}

J & J & K \\
M^{^{\prime }} \\& -M & Q

M^{^{\prime }} & M & Q

\end{matrix}

\right) |JM\rangle \langle JM^{^{\prime }}|,

</math>

where the where the parentheses denote a [[3-j symbol]]; K is the rank which ranges <math>0 \sim 2J</math>; Q is the

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

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}^{-Q}(J)-(-1)^{Q}Y_{K}^{-Q}(J)] </math>

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.

==== Multi-exchange Descriptiondescription ====

Using the addition rulestheorem of tensor operators, athe product of a rank n tensor and a rank m tensor can generate a new 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>

Line 127 ⟶ 129:

:<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 stepsteps dipole-dipole interaction. (seeFor next section)example, 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>, so the one step quadrupole transition <math> Y_{2_{i}}^{+2_{i}} </math> on site <math> i </math> now becomes a two steps of dipole transition <math> Y_{1_{i}}^{+1_{i}}Y_{1_{i}}^{+1_{i}} </math>. Hence not only inter-site-exchange but also intra-site-exchange terms appear (so called multi-exchange). If <math> J </math> is even larger, one can expect more complicated intra-site-exchange terms would appear. However, one has to note that it is not a perturbation expansion but just a mathematical technique. The high rank multi-exchange terms are necessarynot necessarily smaller than low rank terms. In many casessystems, high rank terms are often more important than low rank terms.<ref name="Review"/>.

== Multipolar Exchangeexchange Interactionsinteractions ==

[[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"/>

:<math> H =\sum_{ij}\sum_{KQ}C_{K_{i}K_{j}}^{Q{i}Q_{j}}T_{K_{i}}^{Q_{i}}T_{K_{j}}^{Q_{j}} </math>

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 Multipolarmultipolar Momentsmoments==

[[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 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 a dipole momentsmoment 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.

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 of 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.

== Computing coupling constants ==

== Antiferromagnetism of Multipolar Moments==

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, proposalspredictions 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 R.| last1=Siemann and| B.first1=Robert R.| last2=Cooper, Phys.| Rev.first2=Bernard LettR. 44,| 1015title=Planar Coupling Mechanism Explaining Anomalous Magnetic Structures in Cerium and Actinide Intermetallics | journal=Physical Review Letters | publisher=American Physical Society (1980APS) | 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 J.| last1=Wills | first1=John M. Wills| andlast2=Cooper B.| first2=Bernard R. Cooper,| Phys.title=First-principles Rev.calculations for a model Hamiltonian treatment of hybridizing light actinide compounds | journal=Physical Review B 42,| 4682publisher=American Physical Society (1990APS) | 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 LDADFT+U total energy calculations and got agreement with uranium dioxide.

[[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"/> ]]

== References ==

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 a 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.

{{Reflist}}

[[Category:Magnetic ordering]]

== Compute Coupling Constants ==

[[Category:Magnetic exchange interactions]]

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.

== Reference ==