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Observation of spin-quadrupolar excitations in Sr2CoGe2O7 by high-field electron spin resonance. — When we think of a spin, usually we imagine an arrow pointing somewhere (representing the expectation values of the components of the spin operator), and with the arrow we associate a magnetic moment. Upon time reversal, the arrow reverses its direction.
This is a reasonable picture for the spin 1/2 of the electron, but for larger spins this does not exhaust all the possibilities. For example, the dimension of the Hilbert space is 3 for the spin 1, and we can construct spin states for which the expectation values of all the three spin operators vanish — the state does not point anywhere, it cannot be represented by an arrow.
The simplest example is the 0 eigenstate of the S2 operator. In fact, there are three linearly independent such states (the zero eigenstate of the Sx, Sy and Sz operators), spanning the Hilbert space. Though they cannot be represented by an arrow, they still break the rotational symmetry, since quadratic forms of spin operators differentiate among them. Instead of arrows, we can use directors (like in the case of liquid crystals), as the rotation by π around an axis perpendicular to the director returns the same state (up to a phase factor). These states are called spin-quadrupoles. Furthermore, these states do not break the time-reversal symmetry.
Similarly, the long-range-ordered states of interacting spins are usually time-reversal-breaking states, with a configuration of “arrows” that repeats itself on the lattice. However, under favorable conditions, interacting spins can produce ordered states where the order parameter is of spin-quadrupolar character which does not break the time reversal symmetry.
Theoretically, such phases have been established in spin-one Heisenberg models extended with higher-order spin interactions. Even more interestingly, time-reversal invariant ordered states can also be realized in spin-1/2 systems, where the quadrupole-like order parameter is distributed between two spins on a bond, leading to a so-called nematic ordering.
These theoretical developments have inspired the quest to nematic and quadrupole phases in real materials. However, when relying on standard experimental methods, such phases usually remain hidden. Most of the experimental probes detect spin-dipolar (ΔS=1) transitions, and they do not interact with the spin-quadrupoles, as their detection requires ΔS=2 transitions.
In a collaboration with experimental researchers from Osaka University, we found an unambiguous experimental observation of spin-quadrupolar excitations in the layered Sr2CoGe2O7 multiferroic compound. In this compound, the Co ions are in the centers of tetrahedra formed by the four surrounding O ions (Fig. 1). Since the inversion symmetry is absent, the relativistic spin-orbit coupling allows the coupling of the electric polarizations to the spin-quadrupolar operators. Due to this magnetoelectric coupling present in the Sr2CoGe2O7, the non-magnetic, purely spin-quadrupolar excitation becomes electrically active and detectable by electromagnetic waves, like the electron spin resonance spectroscopy.
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Figure 1. The schematic crystal structure of the Sr2CoGe2O7 multiferroic compound projected onto the ab plane. The green spheres represent the magnetic Co2+ ions with S = 3/2 surrounded by four O2− ions (red) in an alternating tetrahedral environment.
In the electron spin resonance spectra of Sr2CoGe2O7 above the saturation field of 20T, a mode with twice the g-factor of the usual modes is observed (Fig. 2). This indicates the absorption of two magnons, just what is needed for the creation of a quadrupole wave. Indeed, we could explain the features of the experimental spectra taken in different geometries by a simple theoretical model of the spin-quadrupolar wave providing not only a qualitative description, but also a quantitative agreement.
Figure 2. Frequency-field diagrams of the ESR resonance fields of Sr2CoGe2O7 for magnetic fields parallel to the [100] direction of the external magnetic field. The solid lines represent the dipolar resonance modes from the multiboson spin-wave theory. The red dashed line indicates a resonance mode with a slope twice larger than the others, corresponding to a two-magnon excitation — the quadrupolar mode.
The most significant point of our finding is the first observation of non-magnetic spin-quadrupolar excitation in an antiferromagnetic material (Fig. 3). Such spin-quadrupolar degrees of freedom become inherent in systems with larger than S=1/2 magnetic moments, regardless of the presence of magneto-electric coupling. Upon condensing such multipolar excitations, magnetically disordered exotic quantum phases may arise. The experimental identification of quadrupole excitations with vanishing gap gives us a possibility to identify long-sought nematic phases, which stand without any usual magnetic fingerprint and are almost impossible to tell apart from other non-magnetic phases. Furthermore, our work will stimulate the application of the magnetoelectric effect as a spectroscopy tool.
Figure 3. Schematic plot of (a) the Q1 quadrupolar mode for H∥[100] and (b) the dipolar modes for H∥[110], as seen from the direction of the magnetic field. In both cases the oscillating component of the uniform electric polarization P (shown by orange ellipse) is perpendicular to the external magnetic field H, therefore they are active in the Faraday configuration. The green ellipse represents the rotating quadrupolar moments, while the green arrows the precessing dipolar spins on the two sublattices. The red arrows show the electric polarization vectors which are excited by the oscillating electric field.
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Grants and international cooperation
OTKA K-115632: Magnetism and superconductivity in intermetallic nanocomposites (B.
Újfalussy, 2015-2019) in consortium with BME3
OTKA K-106047: Correlated states and excitations in d- and f-electron systems and ultracold Fermi gases (K. Penc, 2013-2017)
NKFI K-124176: Magnetism, topology, and entanglement in quantum insulators (K. Penc, 2017-2021)
OTKA K-109570: Fundamentals of complex, multi-component metallic alloys (L. Vitos, 2013-2018)
HAS NKM-44/2017: Optical magnetoelectric effect and spin dynamics in multiferroics (K.
Penc, 2017-2019)
Publications
Articles
1. Akaki M, Yoshizawa D, Okutani A, Kida T, Romhányi J, Penc K, Hagiwara M: Direct observation of spin-quadrupolar excitations in Sr2CoGe2O7 by high-field electron spin resonance. PHYS REV B 96:(21) 214406/1-16 (2017)
2. Al-Zoubi N, Schönecker S, Johansson B, Vitos L: Assessing the Exact Muffin-Tin Orbitals method for the Bain path of metals. PHILOS MAG 97:(15) 1243-1264 (2017)
3. Barmina E, Kosogor A, Khovaylo V, Gorshenkov M, Lyange M, Kuchin D, Dilmieva E, Koledov V, Shavrov V, Taskaev S, Chatterjee R, Varga LK: Thermomechanical properties and two-way shape memory effect in melt spun Ni57Mn21Al21Si1 ribbons. J ALLOY COMPD 696: 310-314 (2017)
4. Li C-M, Johansson B, Vitos L: Physical mechanism of δ-δ′-ϵ phase stability in plutonium.
SCI REP-UK 7:(1) 5632/1-7 (2017)
5. Dong Z, Li W, Chen D, Schönecker S, Long M, Vitos L: Longitudinal spin fluctuation contribution to thermal lattice expansion of paramagnetic Fe. PHYS REV B 95:(5) 054426/1-9 (2017)
6. Dong ZH, Schonecker S, Chen DF, Li W, Long MJ, Vitos L: Elastic properties of paramagnetic austenitic steel at finite temperature: Longitudinal spin fluctuations in multicomponent alloys. PHYS REV B 96:(17) 174415/1-12 (2017)
7. Finco A, Rózsa L, Hsu PJ, Kubetzka A, Vedmedenko E, von Bergmann K, Wiesendanger R: Temperature-induced increase of spin spiral periods. PHYS REV LETT 119:(3) 037202/1-5 (2017)
8. Huang S, Vida Á, Li W, Molnár D, Kwon SK, Holmström E, Varga B, Varga LK, Vitos L:
Thermal expansion in FeCrCoNiGa high-entropy alloy from theory and experiment.
APPL PHYS LETT 110:(24) 241902-1-5 (2017)
9. Huang S, Vida Á, Heczel A, Holmström E, Vitos L: Thermal expansion, elastic and magnetic properties of FeCoNiCu-based high-entropy alloys using first-principle theory. JOM-J MIN MET MAT S 69:(11) 2107-2112 (2017)
10. Kim FH, Penc K, Nataf P, Mila F: Linear flavor-wave theory for fully antisymmetric SU(N) irreducible representations. PHYS REV B 96:(20) 205142/1-11 (2017)
3 BME: Budapest University of Technology and Economics
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11. Kovac J, Novak L, Varga LK: Study of the magnetization processes in amorphous and nanocrystalline FINEMET by the numerical decomposition of the hysteresis loops.
ACTA PHYS POL A 131:(4) 732-734 (2017) (Proc. 16th Czech and Slovak Conference on Magnetism (CSMAG), Košice, Slovakia: 13-17 June 2016)
12. Lakatos-Varsányi M, Murányi R, Hajdu F, Berényi R, Varga LK: Nanostructured pulsed current metal coatings of Fe and Fe–Ni for microelectronic applications. T I MET FINISH 95:(1) 6-8 (2017)
13. Lee J-Y, Koo YM, Lu S, Vitos L, Kwon SK: The behaviour of stacking fault energy upon interstitial alloying. SCI REP-UK 7:(1) 11074/1-6 (2017)
14. Li X, Schönecker S, Zhao J, Vitos L, Johansson B: Elastic anharmonicity of bcc Fe and Fe-based random alloys from first-principles calculations. PHYS REV B 95:(2) 024203/1-11 (2017)
15. Lizárraga R, Pan F, Bergqvist L, Holmström E, Gercsi Z, Vitos L: First principles theory of the hcp-fcc phase transition in cobalt. SCI REP-UK 7:(1) 3778/1-8 (2017)
16. Lu S , Li RH, Kádas K , Zhang HL, Tian YZ, Kwon SK, Kokko K , Hu Q-M, Hertzman S , Vitos L: Stacking fault energy of C-alloyed steels: The effect of magnetism. ACTA MATER 122: 72-81 (2017)
17. Östlin A, Vitos L, Chioncel L: Analytic continuation-free Green's function approach to correlated electronic structure calculations. PHYS REV B 96:(12) 125156/1-13 (2017) 18. Palotás K, Rózsa L, Simon E, Udvardi L, Szunyogh L: Spin-polarized scanning tunneling
microscopy characteristics of skyrmionic spin structures exhibiting various topologies.
PHYS REV B 96:(2) 024410/1-9 (2017)
19. Rózsa L, Palotás K, Deák A, Simon E, Yanes R, Udvardi L, Szunyogh L, Nowak U:
Formation and stability of metastable skyrmionic spin structures with various topologies in an ultrathin film. PHYS REV B 95:(9) 094423/1-9 (2017)
20. Rózsa L, Atxitia U, Nowak U: Temperature scaling of the Dzyaloshinsky-Moriya interaction in the spin wave spectrum. PHYS REV B 96:(9) 094436/1-11 (2017)
21. Song HQ, Tian FY, Hu QM, Vitos L, Wang YD, Shen J, Chen NX: Local lattice distortion in high-entropy alloys. PHYS REV MATER 1:(2) 023404/1-8 (2017)
22. Sun X, Zhang H, Lu S, Ding X, Wang Y, Vitos L: Phase selection rule for Al-doped CrMnFeCoNi high-entropy alloys from first-principles. ACTA MATER 140: 366-374 (2017)
23. Szolnoki L, Kiss A, Dora B, Simon F: Spin-relaxation time in materials with broken inversion symmetry and large spin-orbit coupling. SCI REP UK 7:(1) 9949/1-10 (2017) 24. Szolnoki L, Dora B, Kiss A, Fabian J, Simon F: An intuitive approach to the unified theory
of spin relaxation. PHYS REV B 96:(24) 245123/1-9 (2017)
25. Tian F, Wang Y, Vitos L: Impact of aluminum doping on the thermo-physical properties of refractory medium-entropy alloys. J APPL PHYS 121:(1) 015105/1-9 (2017)
26. Tian FY, Varga LK, Vitos L: Predicting single phase CrMoWX high entropy alloys from empirical relations in combination with first-principles calculations. INTERMETALLICS 83: 9-16 (2017)
27. Tian L-Y, Ye L-H, Hu Q-M, Lu S , Zhao JJ, Vitos L: CPA descriptions of random Cu-Au alloys in comparison with SQS approach. COMP MATER SCI 128: 302-309 (2017) 28. Tian L-Y, Wang G, Harris JS, Irving DL, Zhao J, Vitos L: Alloying effect on the elastic
properties of refractory high-entropy alloys. MATER DESIGN 114: 243-252 (2017)
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29. Tian L-Y, Lizárraga R, Larsson H, Holmström E, Vitos L: A first principles study of the stacking fault energies for fcc Co-based binary alloys. ACTA MATER 136: 215-223 (2017)
30. Vida Á, Chinh NQ, Lendvai J, Heczel A, Varga LK: Microstructures and transition from brittle to ductile behavior of NiFeCrMoW high entropy alloys. MATER LETT 195: 14-17 (2017)
31. Zhao W, Li W, Sun Z, Gong S, Vitos L: Tuning the plasticity of Ni-Mo solid solution in Ni-based superalloys by ab initio calculations. MATER DESIGN 124: 100-107 (2017)
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