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Slides presented in HFM 2022 in Paris.

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- 1. HFM 2022 June 20-25th Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Clarendon Laboratory, University of Oxford Ryutaro Okuma R. Okuma, T. Yajima, T. Fujii, M. Takano & Z. Hiroi J. Phys. Soc. Jpn 87(9), 093702 (2018). R. Okuma, M. Kofu, S. Asai, M. Avdeev, A. Koda, H. Okabe, ... & Z. Hiroi, Nat. Commun. 12(1), 1-7 (2021).
- 2. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Acknowledgement 2 Prof. Z. Hiroi Prof. T. Masuda Dr. S. Asai Dr. M. Kofu Dr. S. O. Kawamura Prof. A. Koda Dr. H. Okabe Dr. M. Hiraishi Prof. K. Kojima Synthesis, XRD@ISSP Neutron diffraction@ANSTO Neutron spectroscopy@J-PARC µSR@J-PARC/TRIUMF Mössbauer@Okayama Prof. M. Takano Prof. T. Fujii Dr. K. Nakajima Prof. R. Kadono Dr. S. Takeshita (TRIUMF) KEK MLF Prof. Maxim Avdeev Okayama University ANSTO JAEA J-PARC MLF University of Tokyo ISSP Facility Dr. T. Yajima
- 3. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Outline 3 • Dimensional reduction by frustration • 3D cluster magnet pharmacosiderite • Spontaneous appearance of 2D magnetism • 1D magnetism due to compass-like interaction + –
- 4. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction by frustration 4 ?
- 5. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction by frustration ? ?
- 6. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction by frustration ? ? ?
- 7. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction by frustration ? ? ? 7
- 8. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction by frustration ? ? ? 8 + –
- 9. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction in geometrically frustrated system 9 Anisotropic Triangular Cs2CuCl4, Ca3ReO5Cl2 2D→1D FCC, BCT BaCuSi2O6, Sr2YRuO6 3D→2D Pyrochlore ZnCr2O4 3D→0D S. E. Sebastian et al Nature (2001) E. Granado et al. PRL (2013) R. Coldea et al. PRL (2001). D. Hirai et al. JPSJ (2019). For overview, C. D. Batista & Z. Nussinov PRB (2005). S. H. Lee et al. Nature (2020)
- 10. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction in geometrically frustrated system 10 Motivation What is the benefit of dimensional reduction? Is there a new route to realise dimensional reduction? Anisotropic Triangular Cs2CuCl4, Ca3ReO5Cl2 2D→1D FCC, BCT BaCuSi2O6, Sr2YRuO6 3D→2D Pyrochlore ZnCr2O4 3D→0D
- 11. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Hint: dimensional reduction by orbital order 11 eg t2g dx2-y2 d3z2-r2 Spin interaction d orbital arrangement K. I. Kugel & D. I. Khomskii JETP (1973); Sov. Phys. Usp. (1982). Perovskite KCuF3 Orbital interaction
- 12. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Orbital-like degeneracy of frustrated clusters Quantum S = 1/2 Classical S = 5/2 Irreps in Td with Stot = 0 2 degenerate states with Stot = 0 12 JSi•Sj J > 0
- 13. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Frustrated cluster in extended lattices Li(In,Ga)Cr4O8 Ba3Yb2Zn5O11 Face Centre (Mn,Fe)4(BeSiO4)3S Co4B6O13 Body Centre Primitive J J’ 3D array of tetrahedra preserving Td symmetry Y. Okamoto et al. PRL (2013). K. Kimura et al. PRB (2014). J. A. Armstron et al. J. Mater. Chem. C (2013). H. Hagiwara et al. (2009). O. Palchik et al. Inorg. Chem. (2003). KFe4(AsO4)3(OH)4•nH2O K10(Mn,Fe,Co)4Sn4S17 “Breathing pyrochlore”
- 14. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Fe3+ cluster in a cubic magnet: pharmacosiderite AsO4 XRD @5 K Neutron @10 K Pharmacosiderite P-43m, a ~ 8Å FeIIIO6 Synthesis by hydrothermal method Largest crystal ~0.3mm A, H2O • AFe4(AsO4)3(OH)4•nH2O • Pharmaco = Toxic, siderite = iron • Not toxic at all • A = Alkali metal, 1/2Ba, NH4, H3O • Ion exchange property
- 15. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Weak ferromagnetic order in tetrahedral clusters J J’ M. Takano et al. JPSJ (1969). R. Okuma et al. JPSJ (2018). Net moment // <100> Fitting by S = 5/2 J - J’ model J ~ 10.6 K, J’ ~ 0.27J TN = 6 K M110 M100
- 16. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Γ5 ground state with reduced ordered moments Magnetic diffraction@ECHIDNA Ordered moment = 1.9µB << 5µB Q = 0 Γ5 order Net moment // <100> Γmag = Γ2 1 + Γ3 2 + Γ4 3 + 2Γ5 3 DM + Ising anisotropy
- 17. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Spin wave-like excitation below TN S(q, ω) @AMATERAS Spin wave simulation J = 0.9 meV, J’ =0.3J, DM = 0.01J, SIA = 0.001J Cryostat from Prof. Iwasa @Ibaraki U. Gapped mode around 0.5 meV Calculation by Spin W
- 18. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Appearance of two-dimensional excitation Bragg peak Asymmetric Broad peak 𝜒1D = 0.036 𝜒2D = 0.023 𝜒3D = 0.031 GoF Γ A Z R X Diffraction pattern (0<E<2meV) suggests 2D order above TN c a b
- 19. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Net moment can align emergent dimension Q. What’s the benefit of dimensional reduction? A. External field can control it B κa κc
- 20. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Potentially lower-dimensional excitation Linear spin wave theory predicts <Si> = 4µB >> 1.9µB Comparable I(Q) Sharper I(E) <Si> = 1.9µB ~ 40% of S 2D (?) S = 5/2 Pharmacosiderite <Si> = 0.5µB ~ 50% of S 2D S = 1/2 without frustration La2CuO4 D. Vaknin et al. PRL (1987). Q~(001) 0 < E < 2 meV
- 21. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters µeV fluctuation detected by Mössbauer&µSR Frequency 1.3(3)µeV Uniform broad peak below TN ↓ Persistent spin fluctuation 57Fe Mössbauer No fluctuation muon spin rotation Exponential decay at high fields ↓ Frequency 1.76(5) µeV T = 2K
- 22. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters µeV fluctuation detected by Mössbauer&µSR Frequency 1.3(3)µeV 57Fe Mössbauer No fluctuation muon spin rotation Exponential decay at high fields ↓ Frequency 1.76(5) µeV T = 2K Energy scale ~ Anisotropy What is the nature of the slow mode? Uniform broad peak below TN ↓ Persistent spin fluctuation
- 23. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters 1D excitation in the spin wave Γ A Z R X 1D mode in the spin wave One dimensional mode connects Γ5&Γ4
- 24. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters “Bond dependent” interaction between clusters 𝑆1 𝑆2 𝑆3 𝑆4 J’(S1+S4)•(S’’2+S’’3) J’(S1+S2)•(S’3+S’4) J’(S1+S3)•(S’’2+S’’4) x y z 𝑆3 ′ 𝑆4 ′ 𝑆1 ′ 𝑆2 ′ 𝑆4 ′′′ 𝑆1 ′′′ 𝑆2 ′′′ 𝑆3 ′′′ 𝑆1 ′′ 𝑆3 ′′ 𝑆4 ′′ 𝑆2 ′′
- 25. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters “Bond dependent” interaction between clusters 𝑆1 𝑆2 𝑆3 𝑆4 –J’(S1+S4)•(S’’’1+S’’’4) –J’(S1+S2)•(S’1+S’2) –J’(S1+S3)•(S’’1+S’’3) Low temperature Stot ~ 0 S1 + S2 ~ –(S3 + S4) S1 + S3 ~ –(S2 + S4) S1 + S4 ~ –(S2 + S3) x y z
- 26. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters “Bond dependent” interaction between clusters 𝑆1 𝑆2 𝑆3 𝑆4 –J’Tz•T’z –J’Tx•T’x –J’Ty•T’y Define cluster spins Tx = (S1 + S2)/2 Ty = (S1 + S3)/2 Tz = (S1 + S4)/2 ~ Γ2 Γ4 ~ Γ5 ~ 𝐻 = −𝐽′ ∑ 𝜇=𝑥,𝑦,𝑧 𝑇𝜇(𝑟) ⋅ 𝑇𝜇(𝑟 + 𝑒𝜇) Tx 2 + Ty 2 + Tz 2 = S(S+1) Tx•Ty= Tz•Tx= Ty•Tz=0 cf. orbital compass model x y z
- 27. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters 1D excitation from “compass-like” interaction 𝐻′ = −𝐽′ ∑ 𝑟 𝑇𝑥(𝑟) ⋅ 𝑇𝑥(𝑟 + 𝑥 ̂ ) + 𝑇𝑦(𝑟) ⋅ 𝑇𝑦(𝑟 + 𝑦 ̂ ) + 𝑇𝑧(𝑟) ⋅ 𝑇𝑧(𝑟 + 𝑧 ̂ ) q = 0 Γ5 x y z
- 28. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters 1D excitation from “compass-like” interaction q = 0 Γ5 with a zero-energy domain line // b x y z 𝐻′ = −𝐽′ ∑ 𝑟 𝑇𝑥(𝑟) ⋅ 𝑇𝑥(𝑟 + 𝑥 ̂ ) + 𝑇𝑦(𝑟) ⋅ 𝑇𝑦(𝑟 + 𝑦 ̂ ) + 𝑇𝑧(𝑟) ⋅ 𝑇𝑧(𝑟 + 𝑧 ̂ )
- 29. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters 1D excitation from “compass-like” interaction q = 0 Γ5 with intersecting domain lines of Γ4 Tangled 1D modes cause the slow fluctuation? x y z 𝐻′ = −𝐽′ ∑ 𝑟 𝑇𝑥(𝑟) ⋅ 𝑇𝑥(𝑟 + 𝑥 ̂ ) + 𝑇𝑦(𝑟) ⋅ 𝑇𝑦(𝑟 + 𝑦 ̂ ) + 𝑇𝑧(𝑟) ⋅ 𝑇𝑧(𝑟 + 𝑧 ̂ )
- 30. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters Dimensional reduction in a frustrated cluster magnet Spontaneous dimensional reduction of q= 0 Γ5 Low energy 1D excitation in q= 0 structure
- 31. Fitting models for µSR spectra Paramagnetic phase (1 − 𝑓 ̂ )𝐺3𝑆(𝑡, 𝜔𝑑 ̂ ) + 𝑓 ̂ 𝐺𝑧 dyn (𝑡, 𝜔𝜇, 𝛿𝜇 ̂ , 𝜈 ̂ ) Ordered phase (1 − 𝑓 ̂ )𝐺3𝑆(𝑡, 𝜔𝑑 ̂ ) ⋅ 𝐺𝑧 stat (𝑡, 𝑥 ̂ ) + 𝑓 ̂ 𝐺𝑧 dyn (𝑡, 𝜔𝜇, 𝛿𝜇 ̂ , 𝜈 ̂ ) ⋅ 𝐺𝑧 stat (𝑡, 𝑥 ̂ ) Definition of formula and parameters 𝐺3𝑆 𝑡, 𝜔𝑑 = 1 6 3 + cos( 3𝜔𝑑𝑡) + 𝛼+cos(𝛽+𝜔𝑑𝑡) +𝛼−cos(𝛽−𝜔𝑑𝑡)) 𝐺𝑧 dyn (𝑡, 𝜔𝜇, 𝛿𝜇, 𝜈) ≈ exp{(−2𝛿𝜇 2 𝜈𝑡)/(𝜔𝜇 2 + 𝜈2 )} 𝐺𝑧 stat(𝑡, 𝑥) ≈ 𝐺∞(𝑥) + 1 − 𝐺∞(𝑥) exp −𝜎𝑠 2𝑡2 , 𝐺∞(𝑥) = 3𝑥2−1 4𝑥2 + (𝑥2−1)2 16𝑥3 log (𝑥+1)2 (𝑥−1)2 , 𝜔𝑑 ̂ = 2𝛾𝜇𝛾𝐼/𝑟 ̂ 3 ,𝛼± = 1 ± 1/ 3 ,𝛽± = (3 ± 3)/2 𝑥1 ̂ = 𝐵0/𝐵loc,1 ̂ ,𝑥2 ̂ = 𝐵0/𝐵loc,2 ̂ ,𝑥 ̂ = 𝐵0/𝐵loc ̂
- 32. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters 1D excitation is a candidate of slow fluctuation Γ A Z R X One dimensional mode connects Γ5&Γ4 1D defects can penetrate (not like kagomé) 1D mode in the spin wave
- 33. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters µeV fluctuation detected by Mössbauer&µSR 57Fe Mössbauer No fluctuation T (K) Γ (mm/sec) I. S. (mm/sec) Q. S. (mm/sec) h (T) W (MHz) α (%) θ (°) 6 0.441(5) 0.494(2) 0.1176(6) 0 – – – 4 1.42(9) 0.494 0.1176 61 254(23) 0 49(2) 2.8 0.86(3) 0.494 0.1176 61(5) 313(72) 57(5) 45(1) θ ~ 35˚
- 34. Dimensional reduction by geometrical frustration in a cubic antiferromagnet composed of tetrahedral clusters µeV fluctuation detected by µSR T = 2K H-µ-H model (G3S(t)) f 0.21 ωd / 2π [MHz] 66.45(6) r [nm] 0.1792(5) χ2/N 1.27
- 35. Effects of ion substitution to magnetism r / Å TCW / K TN / K Ground state Cs 1.67 -150 7 Weakferro K(H3O ) 1.38 -150 6 Weakferro Na 1.02 -202 4 Spin Glass? Li 0.67 -255 <2 No LRO NH4 1.54 -280 <2 No LRO

- First, I’d like to thank many collaborators involved in this work. Hiroi sensei who promoted me to present here, he supported me synthesis and writing papers. Yajima san for XRD measurement and analysis, Masuda-sensei and Asai-san for neutron diffraction in ANSTO, actually Max measured it. J-PARC, people in AMATERAS mainly Kofu-san measured this sample and Nakajima-san and Kawamura-san also helped me. For muon measurement, Kadono-sensei and Koda-sensei helped a lot. For Muon measurement Fujii-sensei’s lab and measured with Takano-sensei.
- Simple outline, I will introduce the notion of dimensional reduction in frustrated spin systems. New system frustrated cluster magnet pharmacosiderite is introduced and I will show how 2D magnetism appear below magnetic order and even 1D happen because of unique arrangement of cluster motif
- Dimensional reduction is almost equal to definition of frustration. Ising spin on a triangle. The top spin cannot decide whether up or down.
- Put another triangle aside. Again, the top two spins cannot decide whether up or down. But as you know we put bottom two spins already, we can specify all the spin directions on the bottom.
- Repeat this process. Top spins are always unable to determine the direction.
- This situation describes decoupled two chains even though there are finite coupling between the chains. This is a basic idea of dimensional reduction by frustration. This is robust idea and quantum version is described as destructive interference of phase.
- Because this is robust you can see this effect in many frustrated magnets. Anisotropic triangular lattice is the closest realization of the previous idea, and even though the coupling between the chain is very large it behaves as a one-dimensional system and shows 1D spinon excitation along the chain. Face-centred cubic lattice is another classic example, where antiferromagnetic interaction between the plane is cancelled out with each other and becomes 2D system. In spinel chromium oxide magnon excitation is localized inside the hexagon because the hopping is cancelled out by local antiparallel arrangement.
- Based on these numerous examples motivated me to ask some questions. What do you get from dimensional reduction. Is there anything you cannot get in simple low dimensional magnets? How are they different from p]ure 1D or 2D magnet? Because I am a crystal grower, I would like to look for the answer by investigatin]g new types of materials
- Actually dimensional reduction does not happen only in geometrically frustrated systems. Text-book example is orbital order. KCuF3 is a distorted perovskite and is a very three dimensional structure. But because Jahn teller distortion stabilis]es dx2-y2 orbital, connectivity of dx2-y2 orbital is one-dimensional and the compound is known as one of the most one-dimensional spin system. Ultimately one dimensional orbital pattern results from its directional character and the effective very anisotropic interaction between them, which is called compass model. Now know]n as Kitaev type model.
- Anyway, spin only system can have orbital like anisotropic degree of freedom if they form cluster. For example spins arranged in a antiferromagnetically coupled tetrahedron, the ground state is two-fold degenerate, and the state is expressed as the way of putting singlets on the two bond. This is kind of orbital like anistropic motif, at least it breaks the symmetry of tetrahedra. Classical counterpart has a richer structure, if we classify the spin arrangement that satisfies the sum is zero based on tetrahedron’s symmetry, five distinct arrangement appears, all of them can be a source of anisotropically interacting units if placed in an extended lattice
- I would like to put clusters in lattice for them to interact. To make the most of its anisotropic feature highest symmetry compatible with the cluster is considered. There are some materials known in this context. First one is Face cetered cubic lattice, which is known as breathing pyrochlore lattice, the compounds know are like ordered spinel LiInGa oxide, and Ytterbium oxide. Nex one is body centred cubic this is found in mineral Helvite, where Mn and Fe have spins and Cobalt borate compound is also known. Finally the simplest case pharmacosiderite is the target in this study.
- Pharmacosiderite consists of tetrahedral cluster of FeO6 which is bridged by a tetrahedral arsenate. As it contains arsenic, the mineral name means toxic buy actually not. Because of its open framework, it can incorporate several ions, from alkali metals to ammnonium hydronium, and balium. Here we study the simplest H3O pharmacosiderite, synthesized by hydrothermal method. The crystal size is up to 0.3mm so most of the measurement were done in powder form. Neutron and XRD at low temperature confirmed the structure is cubic P-43m down to 5 K.
- Powder and single crsyal measurement revealed weak ferromagnetic order and comparable cluster interaction. Broad peak is fitted to spin-5/2 J and J ‘ model which gives 10.6K for intra cluster and roughly 1/3 of J as J’ inter cluster interaction. Below 6 K weak ferromagnetic order was observed. Single crystal study revealed that the net moment points in <100> direction of the cubic axes.
- To pin down the magnetic structure, neutron diffraction was performed ECHDINA ANSTO. All the magnetic peaks at 1.6 K were assigned as q = 0, In the irreps of q = 0 structure only gamma4 and gamm5 can account for finite 100 peak and gamma5 results in better agreement. Gamma5 is stabilised by Dzyaloshinskii Moriya and smaller ising anisotropy and basiccaly coplanar but the ising anisotropy cants the structure towards <100> axis slightly. Interesting thing here is that the ordered moment is very reduced and suggests fluctuation in the ground state.
- To investigate the origin of the fluctuation, we have performed inelastic neutron experiments in AMATERAS@J-PARC. The spectra are well dispersing up to 4 meV and spin-wave starts from 100 and 200. Only at very low temperatures, opening of gap-like structure was observed around 0.5 meV. These features are basically reprodcuced by the same parameters used for fitting of magnetic susceptibility. DM interaction mainly open up a gap around 0.5 meV
- To understand the nature of the excitation single crystal spin-wave dispersion is plotted. As you can see, gamma-z line isvery flat and other directions are is dispersive. This direction is perpendicular to coplanar plane spins span. This is typical of dimensional reduction by frustration. The green bonds directed along c-axis interaction is cancelled out because of the antiparallel spin arrangement of the two spins. Instead for the blue bonds running in the plane, Interaction is active because there are no cancellation. To see this effect if we look at the diffraction pattern which was obtained by integrating low energy region, we can clearly asymmetric broad peak in addition to the sharp Bragg peaks. Asymmetric peak is signature of low dimensional order and persists above TN as it describes short range order. If we fit the broad peak by 1d and 2d, 3d order, 2D order can describes the best among them. So 2D fluctuation seems to describe the spin fluctuation of pharmacosiderite.
- 2D fluctuation is meaningful if it is combined with the weak ferromagnetic order. The interacting planes are always normal to net component, It should be aligned by magnetic field easily. For example, thermal conductivity will show a clear anisotropy depending on the field direction. We are trying to measure this effect by growing a larger single crystal. But this is not the end of the story.
- So regarding the reduced ordered moment, purely two dimensional order is unlikely to explain everything. This is because for example, 2d quantum antiferromagnet compound without frustration, parent phase of superconductor has just 50% of reduction although pharmacosiderite is more than 60%. Comparison of pharmacosiderite data to the linear spin wave theory predicts 4 mub as the ordered moment much bigger than the experimental value. For momentum dependence of the calculationshows good agreement and the energy dependence is greatly sharpened. So I suspect there are still spin wave-like zero energy excitation near the ground state.
- Indeed we’ve detected persistent very low energy excitation in the ground state. In iron Mossbauer spectroscopy, spectrum in the ordered state has six peaks but they are not as sharp as the perfectly ordered state. We can fit this by Blume model, which is kind of spins are constantly changing their direction up and down in an Ising way and estimate the fluctuation energy. The estimated fluctuation is 1.3 microeV . Muon spectroscopy has also showed evidence of microeV fluctuation which is shown in the relaxing component of the spectrum at high fields. The value is 1.76 microeV which is consistent with Mossbauer. So in addition to 2D fluctuation detected by neutron there is a very slow mode persisting in the ground state.
- We don’t have clear experimental evidence of the nature of the slow mode, but we believe that this is coming from enhanced one-dimensional spin-wave like excitation.
- In the calculated spin-wave spectrum, There is one-dimensional mode. This corresponds to a mode that connects gamma5 ground state and gamma4, which is degenerate in the absence of anisotropy. Let me explain why this happens in the language of cluster spin.
- This is the interaction around one spin of tetrahedron. This looks complex enough.
- But at low temperatures sum of the cluster spins are expected to satisfy these relations and the cluster interactions become much simpler
- If we define cluster spins as a set of orthogonal vectors, then the interaction becomes a bond dependent interaction like a Kitaev model. More similar situation happens in the model of orbital order in the perovskite. In this notation noncoplanar spin configuration like all out is a set of three vectors and coplanar noncollinear structure like gamma4 and 5 is a set of two vectors.
- Let’s consider the elementary excitation from q = 0 gamma 5 state. This is one of the ground state you can easily show it. Because Red spins connected by red lines are aligned. Blue spins are aligned. No z component shown by green above means that there are no coupling between this plane.
- The situation doesn’t change if we rotate an array of blue spins about the red spins like this. This doesn’t cost energy if no anisotropy.
- This domain can intersect each other so I expect these excitations remain mobile even at low temperature, which was seen in musr and mossbauer spectroscopy.