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Discrete degeneracies distinguished by the anomalous Hall effect in a metallic kagome ice compound

Nature Physics volume 20, pages 442–449 (2024)Cite this article

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Abstract

In magnetic crystals, despite the explicit breaking of time-reversal symmetry, two equilibrium states related by time reversal are always energetically degenerate. In ferromagnets, this time-reversal degeneracy is reflected in the hysteresis of the magnetic field dependence of the magnetization and, if metallic, in that of the anomalous Hall effect (AHE). Under time-reversal, both these quantities change signs but not their magnitude. Here we show that a time-reversal-like degeneracy appears in the metallic kagome spin ice HoAgGe when magnetic fields are applied parallel to the kagome plane. We find vanishing hysteresis in the field dependence of the magnetization at low temperature, but finite hysteresis in the field-dependent AHE. This suggests the emergence of states with nearly the same energy and net magnetization but different sizes of the AHE and of the longitudinal magnetoresistance. By analysing the experimental data and a minimal tight-binding model, we identify a time-reversal-like operation connecting these near-degenerate states, which is related to the non-trivial distortion of the kagome lattice in HoAgGe. Our work demonstrates the diagnostic power of transport phenomena for identifying hidden symmetries in frustrated spin systems.

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Fig. 1: Time-reversal-like degeneracies in field-induced plateau phases of HoAgGe.
Fig. 2: Magnetic and transport properties of HoAgGe.
Fig. 3: Thermodynamic measurements of HoAgGe in comparison with the Hall data.
Fig. 4: Magnetic and transport properties of HoAgGe under H//c.
Fig. 5: Model calculations for the degenerate plateau states.
Fig. 6: Hall measurements using smaller-range field sweeps.

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Data availability

All data presented in this paper, if not present in the main text or supplementary materials, are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability

The code for the theoretical analyses and model calculations of this study is available from the corresponding authors upon reasonable request.

References

  1. Balents, L. et al. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

    ADS  CAS  PubMed  Google Scholar 

  2. Harris, M. J. et al. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7. Phys. Rev. Lett. 79, 2554–2557 (1997).

    ADS  CAS  Google Scholar 

  3. Ramirez, A. P. et al. Zero-point entropy in ‘spin ice’. Nature 399, 333–335 (1999).

    ADS  CAS  Google Scholar 

  4. Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001).

    ADS  CAS  PubMed  Google Scholar 

  5. Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).

    ADS  CAS  PubMed  Google Scholar 

  6. Morris, D. J. P. et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7. Science 326, 411–414 (2009).

    ADS  CAS  PubMed  Google Scholar 

  7. Fennell, T. et al. Magnetic coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009).

    ADS  CAS  PubMed  Google Scholar 

  8. Hall, E. H. XVIII. On the “rotational coefficient” in nickel and cobalt. Lond. Edinb. Dublin Philos. Mag. J. Sci. 12, 157–172 (1881).

    Google Scholar 

  9. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).

    ADS  Google Scholar 

  10. Machida, Y., Nakatsuji, S., Onoda, S., Tayama, T. & Sakakibara, T. Time-reversal symmetry breaking and spontaneous Hall effect without magnetic dipole order. Nature 463, 210–213 (2010).

    ADS  CAS  PubMed  Google Scholar 

  11. Udagawa, M. & Moessner, R. Anomalous Hall effect from frustration-tuned scalar chirality distribution in Pr2Ir2O7. Phys. Rev. Lett. 111, 036602 (2013).

    ADS  CAS  PubMed  Google Scholar 

  12. Tomizawa, T. & Kontani, H. Anomalous Hall effect in the t2g orbital kagome lattice due to noncollinearity: significance of the orbital Aharonov–Bohm effect. Phys. Rev. B 80, 100401(R) (2009).

    ADS  Google Scholar 

  13. Chen, H., Niu, Q. & MacDonald, A. H. Anomalous Hall effect arising from noncollinear antiferromagnetism. Phys. Rev. Lett. 112, 017205 (2014).

    ADS  PubMed  Google Scholar 

  14. Kübler, J. & Felser, C. Non-collinear antiferromagnets and the anomalous Hall effect. Europhys. Lett. 108, 67001 (2014).

    ADS  Google Scholar 

  15. Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212–215 (2015).

    ADS  CAS  PubMed  Google Scholar 

  16. Nayak, A. K. et al. Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3Ge. Sci. Adv. 2, e1501870 (2016).

    ADS  Google Scholar 

  17. Sürgers, C., Fischer, G., Winkel, P. & Löhneysen, H. V. Large topological Hall effect in the non-collinear phase of an antiferromagnet. Nat. Commun. 5, 3400 (2014).

    ADS  PubMed  Google Scholar 

  18. Liu, Z. Q. et al. Electrical switching of the topological anomalous Hall effect in a non-collinear antiferromagnet above room temperature. Nat. Electron. 1, 172–177 (2018).

    ADS  CAS  Google Scholar 

  19. Zhao, K. et al. Anomalous Hall effect in the noncollinear antiferromagnetic antiperovskite Mn3Ni1−xCuxN. Phys. Rev. B 100, 045109 (2019).

    ADS  CAS  Google Scholar 

  20. Zhou, X. et al. Spin-order dependent anomalous Hall effect and magneto-optical effect in the noncollinear antiferromagnets Mn3XN with X = Ga, Zn, Ag, or Ni. Phys. Rev. B 99, 104428 (2019).

    ADS  CAS  Google Scholar 

  21. Gurung, G., Shao, D.-F., Paudel, T. R. & Tsymbal, E. Y. Anomalous Hall conductivity of noncollinear magnetic antiperovskites. Phys. Rev. Mater. 3, 044409 (2019).

    CAS  Google Scholar 

  22. Šmejkal, L., González-Hernández, R., Jungwirth, T. & Sinova, J. Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets. Sci. Adv. 6, eaaz8809 (2020).

    ADS  PubMed  PubMed Central  Google Scholar 

  23. Li, X., MacDonald, A. H. & Chen, H. Quantum anomalous Hall effect through canted antiferromagnetism. Preprint at https://doi.org/10.48550/arXiv.1902.10650 (2019).

  24. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    ADS  MathSciNet  CAS  Google Scholar 

  25. Chen, H. et al. Manipulating anomalous Hall antiferromagnets with magnetic fields. Phys. Rev. B 101, 104418 (2020).

    ADS  CAS  Google Scholar 

  26. Onsager, L. Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265 (1931).

    ADS  CAS  Google Scholar 

  27. Xiao, C. et al. Linear magnetoresistance induced by intra-scattering semiclassics of Bloch electrons. Phys. Rev. B 101, 201410(R) (2020).

    ADS  Google Scholar 

  28. Wang, Y. et al. Antisymmetric linear magnetoresistance and the planar Hall effect. Nat. Commun. 11, 216 (2020).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  29. Wolf, M. & Schotte, K. D. Ising model with competing next-nearest-neighbour interactions on the Kagome lattice. J. Phys. A: Math. Gen. 21, 2195–2209 (1988).

    ADS  MathSciNet  Google Scholar 

  30. Takagi, T. & Mekata, M. Magnetic ordering of Ising spins on Kagomé lattice with the 1st and the 2nd neighbor interactions. J. Phys. Soc. Jpn 62, 3943–3953 (1993).

    ADS  CAS  Google Scholar 

  31. Wills, A. S., Ballou, R. & Lacroix, C. Model of localized highly frustrated ferromagnetism: the kagomé spin ice. Phys. Rev. B 66, 144407 (2002).

    ADS  Google Scholar 

  32. Matsuhira, K., Hiroi, Z., Tayama, T., Takagi, S. & Sakakibara, T. A new macroscopically degenerate ground state in the spin ice compound Dy2Ti2O7 under a magnetic field. J. Phys. Condens. Matter 14, L559 (2002).

    ADS  CAS  Google Scholar 

  33. Tabata, Y. et al. Kagomé ice state in the dipolar spin ice Dy2Ti2O7. Phys. Rev. Lett. 97, 257205 (2006).

    ADS  CAS  PubMed  Google Scholar 

  34. Möller, G. & Moessner, R. Magnetic multipole analysis of kagome and artificial spin-ice dipolar arrays. Phys. Rev. B 80, 140409(R) (2009).

    ADS  Google Scholar 

  35. Chern, G.-W., Mellado, P. & Tchernyshyov, O. Two-stage ordering of spins in dipolar spin ice on the kagome lattice. Phys. Rev. Lett. 106, 207202 (2011).

    ADS  PubMed  Google Scholar 

  36. Chern, G.-W. & Tchernyshyov, O. Magnetic charge and ordering in kagome spin ice. Phil. Trans. R. Soc. A 370, 5718–5737 (2012).

    ADS  MathSciNet  PubMed  Google Scholar 

  37. Qi, Y., Brintlinger, T. & Cumings, J. Direct observation of the ice rule in an artificial kagome spin ice. Phys. Rev. B 77, 094418 (2008).

    ADS  Google Scholar 

  38. Ladak, S., Read, D. E., Perkins, G. K., Cohen, L. F. & Branford, W. R. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359–363 (2010).

    CAS  Google Scholar 

  39. Dun, Z. L. et al. Magnetic ground states of the rare-earth tripod kagome lattice Mg2RE3Sb3O14 (RE = Gd, Dy, Er). Phys. Rev. Lett. 116, 157201 (2016).

    ADS  CAS  PubMed  Google Scholar 

  40. Xiao, D. et al. Berry phase correction to electron density of states in solids. Phys. Rev. Lett. 95, 137204 (2005).

    ADS  PubMed  Google Scholar 

  41. Thonhauser, T. et al. Orbital magnetization in periodic insulators. Phys. Rev. Lett. 95, 137205 (2005).

    ADS  CAS  PubMed  Google Scholar 

  42. Shi, J. et al. Quantum theory of orbital magnetization and its generalization to interacting systems. Phys. Rev. Lett. 99, 197202 (2007).

    ADS  PubMed  Google Scholar 

  43. Zhao, K. et al. Realization of the kagome spin ice state in a frustrated intermetallic compound. Science 367, 1218–1223 (2020).

    ADS  CAS  PubMed  Google Scholar 

  44. Gopalan, V. & Litvin, D. B. Rotation-reversal symmetries in crystals and handed structures. Nat. Mater. 10, 376–381 (2011).

    ADS  CAS  PubMed  Google Scholar 

  45. VanLeeuwen, B. K. & Gopalan, V. The antisymmetry of distortions. Nat. Commun. 6, 8818 (2015).

    ADS  CAS  PubMed  Google Scholar 

  46. Li, N. et al. Low-temperature transport properties of the intermetallic compound HoAgGe with a kagome spin-ice state. Phys. Rev. B 106, 014416 (2022).

    ADS  MathSciNet  CAS  Google Scholar 

  47. Dubovik, V. M. & Tugushev, V. V. Toroid moments in electrodynamics and solid-state physics. Phys. Rep. 187, 145–202 (1990).

    ADS  Google Scholar 

  48. Chen, H. Electronic chiralization as an indicator of the anomalous Hall effect in unconventional magnetic systems. Phys. Rev. B 106, 024421 (2022).

    ADS  CAS  Google Scholar 

  49. Ye, L. et al. Electronic transport on the Shastry–Sutherland lattice in Ising-type rare-earth tetraborides. Phys. Rev. B 95, 174405 (2017).

    ADS  Google Scholar 

  50. Morosan, E. et al. Thermodynamic and transport properties of RAgGe (R=Tb–Lu) single crystals. J. Magn. Magn. Mater. 277, 298–321 (2004).

    ADS  CAS  Google Scholar 

  51. Prozorov, R. & Kogan, V. G. Effective demagnetizing factors of diamagnetic samples of various shapes. Phys. Rev. Appl. 10, 014030 (2018).

    ADS  CAS  Google Scholar 

  52. Tokiwa, Y. & Gegenwart, P. High-resolution alternating-field technique to determine the magnetocaloric effect of metals down to very low temperatures. Rev. Sci. Instrum. 82, 013905 (2011).

    ADS  CAS  PubMed  Google Scholar 

  53. Petříček, V., Dušek, M. & Palatinus, L. Crystallographic Computing System JANA2006: general features. Z. Kristallogr. Cryst. Mater. 229, 345–352 (2014).

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Acknowledgements

We thank O. Tchernyshyov, M. Udagawa, H. Deng, I. Kezsmarki, J. Shen, H. Ren, S. Su, T. Dong and N. Wang for helpful discussions and experimental support. The work in Augsburg (K.Z., Y.T. and P.G.) was supported by the German Research Foundation (DFG) through SPP1666 (project no. 220179758), TRR80 (project no. 107745057) and TRR360 (project no. 492547816) and via the Sino-German Cooperation Group on Emergent Correlated Matter. K.Z. acknowledges the support by the National Key R&D Program of China (Grant No. 2023YFA1406003), National Natural Science Foundation of China (Grants No. 12274015), the Beijing Nova Program (Grant No. Z211100002121095), and the Fundamental Research Funds for the Central Universities. H.C. acknowledges the support by NSF CAREER grant DMR-1945023. A portion of this work was conducted at the Synergetic Extreme Condition User Facility (SECUF).

Author information

Authors and Affiliations

  1. Experimentalphysik VI, Centre for Electronic Correlations and Magnetism, University of Augsburg, Augsburg, Germany

    K. Zhao, Y. Tokiwa & P. Gegenwart

  2. School of Physics, Beihang University, Beijing, China

    K. Zhao

  3. Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan

    Y. Tokiwa

  4. Department of Physics, Colorado State University, Fort Collins, CO, USA

    H. Chen

  5. School of Advanced Materials Discovery, Colorado State University, Fort Collins, CO, USA

    H. Chen

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Contributions

K.Z. and P.G. proposed the experiments. K.Z. synthesized single crystals and measured magnetic properties, specific heat and electronic transport properties. Y.T. conducted magnetic Grüneisen parameter measurements. H.C. proposed the theoretical analysis and provided model calculations. K.Z., H.C. and P.G. wrote the manuscript with input from all authors.

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Correspondence to K. Zhao, H. Chen or P. Gegenwart.

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Nature Physics thanks Hiroaki Ishizuka and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information

Supplementary Notes 1–15, Figs. 1–34 and Tables 1 and 2.

Source data

Source Data Fig. 1

The anomalous part \({\rho }_{{ca}}^{\rm{AH}}\) (constant-in-field) of the Hall resistivity \({\rho }_{{ca}}\), together with corrected magnetization M(H) data (Fig. 1b); summary of magnetic susceptibility, magnetization M(H), magnetic specific heat Cmag and magnetic Grüneisen parameter Γmag of HoAgGe under H//b (Fig. 1c).

Source Data Fig. 2

The magnetization M, a- and c-axis MR ρa and ρc, as well as the Hall resistivity ρca and conductivity σac of HoAgGe under H//b at various temperatures.

Source Data Fig. 3

The magnetic specific heat Cmag, magnetic Grüneisen parameter Γmag, magnetic entropy Smag and Hall resistivity ρca of HoAgGe at 8 K (Fig. 3a) and 3 K (Fig. 3b).

Source Data Fig. 4

The magnetic susceptibility of HoAgGe under H along c and b axes (Fig. 4a); the magnetization (M) under H//c at 2 K, with dM/dH curves (Fig. 4b); the a-axis MR ρa and the Hall resistivity ρab at 2 K under H//c (Fig. 4c).

Source Data Fig. 5

Band structure and Berry-curvature-related quantities orbital magnetization and linear magnetoconductivity intrinsic AHC (Fig. 5c); intrinsic AHC, Berry-curvature-contributed linear magnetoconductivity and orbital magnetization (Fig. 5d–f).

Source Data Fig. 6

The Hall resistivity ρca (Fig. 6a) and symmetric part of the raw Rca signal (Fig. 6b) of HoAgGe obtained using different field-sweep protocols under H//b at 2 K.

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Zhao, K., Tokiwa, Y., Chen, H. et al. Discrete degeneracies distinguished by the anomalous Hall effect in a metallic kagome ice compound. Nat. Phys. 20, 442–449 (2024). https://doi.org/10.1038/s41567-023-02307-w

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