Multipolar exchange interaction: Difference between revisions

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Magnetic materials with strong [[spin-orbit interaction]], such as: LaFeAsO,<ref name="LaFeAsO">{{cite journal | last=Cricchio | first=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 | last=Gonnelli | first=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 }}</ref> PrFe₄P₁₂,<ref name="PrFe4P12">{{cite journal | last=Kiss | first=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 }}</ref><ref>{{cite journal | last=Sato | first=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 | last=Takimoto | first=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 }}</ref> UO₂,<ref name="UO2">{{cite journal | last=Pi | first=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 }}</ref><ref>{{cite journal | last=Giannozzi | first=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 | last=Mironov | first=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 | last=Carretta | first=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 | last=Caciuffo | first=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 }}</ref> NpO₂,<ref name="NpO2">{{cite journal | last=Santini | first=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 | last=Santini | first=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 | last=Kubo | first=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 }}</ref> Ce_1−xLa_xB₆,<ref name="Ce1−xLaxB6">{{cite journal | last=Mannix | first=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 | last=Chandra | first=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 }}</ref><ref>{{cite journal | last=Cricchio | first=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 }}</ref><ref>{{cite journal | last=Ikeda | first=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 }}</ref><ref>{{cite journal | last=Kiss | first=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 }}</ref><ref>{{cite journal | last=Rau | first=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 }}</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 | last=Santini | first=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}}</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 Operatorsoperator Expansionexpansion ==

=== Basic Conceptsconcepts ===

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 operator]]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

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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 | last=Pi | first=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 }}</ref><ref>{{cite journal | last=Pi | first=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 }}</ref>

=== Formal Definitionsdefinitions ===

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

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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 theorem of tensor operators, the 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 harmonic tensor operators of <math> J=1 </math> case, we have

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If so, a quadrupole-quadrupole interaction (see next section) can be considered as a two steps dipole-dipole interaction. For example, <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 terms are not necessarily smaller than low rank terms. In many systems, high rank terms are more important than low rank terms.<ref name="Review"/>

== Multipolar Exchangeexchange Interactionsinteractions ==

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

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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 between 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|right|Flipping the phases of multipoles <ref name="multipolar exchange"/>]]

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

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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.

== ComputeComputing Couplingcoupling Constantsconstants ==

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 | last=Siemann | first=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 | last=Wills | first=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.