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">{{cite journal | lastlast1=Cricchio | firstfirst1=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)}}</ref><ref>{{cite journal | lastlast1=Gonnelli | firstfirst1=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 | publisher=American Physical Society (APS) | 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 | lastlast1=Kiss | firstfirst1=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 | s2cid=119350615 }}</ref><ref>{{cite journal | lastlast1=Sato | firstfirst1=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}}</ref> YbRu₂Ge₂,<ref name="YbRu2Ge2">{{cite journal | lastlast1=Takimoto | firstfirst1=Tetsuya | last2=Thalmeier | first2=Peter | title=Theory of induced quadrupolar order in tetragonal YbRu₂Ge₂| journal=Physical Review B | publisher=American Physical Society (APS) | volume=77 | issue=4 | date=8 January 2008 | issn=1098-0121 | doi=10.1103/physrevb.77.045105 | page=045105| arxiv=0708.2872 | s2cid=119203279 }}</ref> UO₂,<ref name="UO2">{{cite journal | lastlast1=Pi | firstfirst1=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 | publisher=American Physical Society (APS) | volume=112 | issue=7 | date=20 February 2014 | issn=0031-9007 | doi=10.1103/physrevlett.112.077203 | page=077203| arxiv=1308.1488 | s2cid=42262386 }}</ref><ref>{{cite journal | lastlast1=Giannozzi | firstfirst1=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}}</ref><ref>{{cite book | lastlast1=Mironov | firstfirst1=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 | lastlast1=Carretta | firstfirst1=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}}</ref><ref>{{cite journal | lastlast1=Caciuffo | firstfirst1=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 | publisher=American Physical Society (APS) | volume=84 | issue=10 | date=6 September 2011 | issn=1098-0121 | doi=10.1103/physrevb.84.104409 | page=104409| arxiv=1312.5113 | s2cid=118624728 }}</ref> NpO₂,<ref name="NpO2">{{cite journal | lastlast1=Santini | firstfirst1=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}}</ref><ref>{{cite journal | lastlast1=Santini | firstfirst1=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}}</ref><ref>{{cite journal | lastlast1=Kubo | firstfirst1=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 | s2cid=119391692 }}</ref> Ce_1−xLa_xB₆,<ref name="Ce1−xLaxB6">{{cite journal | lastlast1=Mannix | firstfirst1=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}}</ref> URu₂Si₂<ref name="URu2Si2">{{cite journal | lastlast1=Chandra | firstfirst1=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| arxiv=cond-mat/0205003 | s2cid=11902278 }}</ref><ref>{{cite journal | lastlast1=Cricchio | firstfirst1=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| arxiv=0904.3883 | s2cid=20622071 }}</ref><ref>{{cite journal | lastlast1=Ikeda | firstfirst1=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 | publisher=Springer Science and Business Media LLC | volume=8 | issue=7 | date=3 June 2012 | issn=1745-2473 | doi=10.1038/nphys2330 | pages=528–533| arxiv=1204.4016 | s2cid=119108102 }}</ref><ref>{{cite journal | lastlast1=Kiss | firstfirst1=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 | s2cid=118892596 }}</ref><ref>{{cite journal | lastlast1=Rau | firstfirst1=Jeffrey G. | last2=Kee | first2=Hae-Young | title=Hidden and antiferromagnetic order as a rank-5 superspin in URu₂Si₂ | journal=Physical Review B | publisher=American Physical Society (APS) | volume=85 | issue=24 | date=13 June 2012 | issn=1098-0121 | doi=10.1103/physrevb.85.245112 | page=245112| arxiv=1203.1047 | s2cid=118313829 }}</ref> and many other compounds, are found to have magnetic ordering constituted by high rank multipoles, e.g. quadruple, octople, etc.<ref name="Review">{{cite journal | lastlast1=Santini | firstfirst1=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| 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" />

== Tensor operator 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">{{cite journal | lastlast1=Pi | firstfirst1=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 | publisher=American Physical Society (APS) | volume=112 | issue=7 | date=20 February 2014 | issn=0031-9007 | doi=10.1103/physrevlett.112.077203 | page=077203| arxiv=1308.1488 | s2cid=42262386 }}</ref><ref>{{cite journal | lastlast1=Pi | firstfirst1=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 | publisher=American Physical Society (APS) | volume=90 | issue=4 | date=31 July 2014 | issn=1098-0121 | doi=10.1103/physrevb.90.045148 | page=045148| arxiv=1406.0221 | s2cid=118960388 }}</ref>

=== Formal definitions ===

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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 | lastlast1=Siemann | firstfirst1=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}}</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 | lastlast1=Wills | firstfirst1=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}}</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 ==