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 ==

==== 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 operators 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

Line 91 ⟶ 90:

=\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 spacial 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"/>.

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 spacial functions.

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>.

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"/>.

==== Formal Definitions ====

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>.

[[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"/> ]]

==== Formal Definitions ====

[[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"/>

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:<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 Description ====

Line 127:

:<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 ==

[[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 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 Multipolar Moments==

[[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 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.

== 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.

== ReferenceReferences ==

{{Reflist}}