Letter | Published:

Experimental observation of Bethe strings

Nature volume 554, pages 219223 (08 February 2018) | Download Citation

Abstract

Almost a century ago, string states—complex bound states of magnetic excitations—were predicted to exist in one-dimensional quantum magnets1. However, despite many theoretical studies2,3,4,5,6,7,8,9,10,11, the experimental realization and identification of string states in a condensed-matter system have yet to be achieved. Here we use high-resolution terahertz spectroscopy to resolve string states in the antiferromagnetic Heisenberg–Ising chain SrCo2V2O8 in strong longitudinal magnetic fields. In the field-induced quantum-critical regime, we identify strings and fractional magnetic excitations that are accurately described by the Bethe ansatz1,3,4. Close to quantum criticality, the string excitations govern the quantum spin dynamics, whereas the fractional excitations, which are dominant at low energies, reflect the antiferromagnetic quantum fluctuations. Today, Bethe’s result1 is important not only in the field of quantum magnetism but also more broadly, including in the study of cold atoms and in string theory; hence, we anticipate that our work will shed light on the study of complex many-body systems in general.

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References

  1. 1.

    Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205–226 (1931)

  2. 2.

    & One-dimensional chain of anisotropic spin-spin interactions. II. Properties of the ground-state energy per lattice site for an infinite system. Phys. Rev. 150, 327–339 (1966)

  3. 3.

    Thermodynamics of the Heisenberg-Ising ring for Δ ≥ 1. Phys. Rev. Lett. 26, 1301–1304 (1971)

  4. 4.

    & One-dimensional anisotropic Heisenberg model at finite temperatures. Prog. Theor. Phys. 48, 2187–2209 (1972)

  5. 5.

    , , & Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field. Phys. Rev. B 24, 1429–1467 (1981)

  6. 6.

    , & Form factors of the XXZ Heisenberg spin-1/2 finite chain. Nucl. Phys. B 554, 647–678 (1999)

  7. 7.

    & Line-shape predictions via Bethe ansatz for the one-dimensional spin-1/2 Heisenberg antiferromagnet in a magnetic field. Phys. Rev. B 62, 14871–14879 (2000)

  8. 8.

    , & Evaluation of dynamic spin structure factor for the spin-1/2 XXZ chain in a magnetic field. J. Phys. Soc. Jpn 73, 3008–3014 (2004)

  9. 9.

    , & Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime. J. Stat. Mech. 2005, P09003 (2005)

  10. 10.

    Dynamically dominant excitations of string solutions in the spin-1/2 antiferromagnetic Heisenberg chain in a magnetic field. Phys. Rev. Lett. 102, 037203 (2009)

  11. 11.

    , , & Observation of complex bound states in the spin-1/2 Heisenberg XXZ chain using local quantum quenches. Phys. Rev. Lett. 108, 077206 (2012)

  12. 12.

    Bound states of two spin waves in the Heisenberg ferromagnet. Phys. Rev. 132, 85–97 (1963)

  13. 13.

    The spectrum of the continuous isotropic quantum Heisenberg chain: quantum solitons as magnon bound states. J. Phys. Chem. 13, L195–L200 (1980)

  14. 14.

    Entanglement dynamics and quantum-state transport in spin chains. Phys. Rev. A 69, 034304 (2004)

  15. 15.

    The Bethe ansatz after 75 years. Phys. Today 60, 36–40 (2007)

  16. 16.

    , , & Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3. Phys. Rev. Lett. 70, 4003–4006 (1993)

  17. 17.

    et al. Confinement of fractional quantum number particles in a condensed-matter system. Nat. Phys. 6, 50–55 (2010)

  18. 18.

    et al. Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain. Nat. Phys. 9, 435–441 (2013)

  19. 19.

    et al. Orbital-exchange and fractional quantum number excitations in an f-electron metal, Yb2Pt2Pb. Science 352, 1206–1210 (2016)

  20. 20.

    & What is the spin of a spin wave? Phys. Lett. A 85, 375–377 (1981)

  21. 21.

    Magnetic properties of CoCl2 and NiCl2. Phys. Rev. 131, 546–555 (1963)

  22. 22.

    , , & Magnetic correlations of the quasi-one-dimensional half-integer spin-chain antiferromagnets SrM2V2O8 (M = Co, Mn). Phys. Rev. B 89, 094402 (2014)

  23. 23.

    et al. Spinon confinement in the one-dimensional Ising-like antiferromagnet SrCo2V2O8. Phys. Rev. B 91, 140404 (2015)

  24. 24.

    et al. From confined spinons to emergent fermions: observation of elementary magnetic excitations in a transverse-field Ising chain. Phys. Rev. B 94, 125130 (2016)

  25. 25.

    et al. Extended quantum critical phase in a magnetized spin-1/2 antiferromagnetic chain. Phys. Rev. Lett. 91, 037205 (2003)

  26. 26.

    ., ., ., & Quantum spin dynamics of the axial antiferromagnetic spin-1/2 XXZ chain in a longitudinal magnetic field. Preprint at (2017)

  27. 27.

    & in From Fields to Strings: Circumnavigating Theoretical Physics (eds et al.) Vol. 1, 684–830 (World Scientific, 2005)

  28. 28.

    , & Efimov effect in quantum magnets. Nat. Phys. 9, 93–97 (2013)

  29. 29.

    et al. Microscopic observation of magnon bound states and their dynamics. Nature 502, 76–79 (2013)

  30. 30.

    & Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension. Phys. Rev. Lett. 20, 1445–1448 (1968)

  31. 31.

    & The Bethe-ansatz for N  = 4 super Yang-Mills. J. High Energy Phys. 3, 13 (2003)

  32. 32.

    et al. Longitudinal and transverse Zeeman ladders in the Ising-like chain antiferromagnet BaCo2V2O8. Phys. Rev. Lett. 114, 017201 (2015)

  33. 33.

    et al. Spinon confinement in a quasi-one dimensional anisotropic Heisenberg magnet. Phys. Rev. B 96, 054423 (2017)

  34. 34.

    , , , & Exchange interaction via crystal-field excited states and its importance in CsCoCl3. J. Phys. Soc. Jpn 72, 2326–2333 (2003)

  35. 35.

    Thermodynamics of One-Dimensional Solvable Models (Cambridge Univ. Press, 2005)

  36. 36.

    & Computation of dynamical correlation functions of Heisenberg chains in a magnetic field. Phys. Rev. Lett. 95, 077201 (2005)

  37. 37.

    , & Exact edge singularities and dynamical correlations in spin-1/2 chains. Phys. Rev. Lett. 100, 027206 (2008)

  38. 38.

    , , , & Quantum criticality of spinons. Phys. Rev. B 96, 220401(R) (2017)

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Acknowledgements

We thank I. Bloch, M. Karbach, T. Lorenz and X. Zotos for discussions. We acknowledge partial support by the DFG via the Transregional Collaborative Research Center TRR 80, and by the HFML-RU/FOM and the HLD-HZDR, members of the European Magnetic Field Laboratory (EMFL). J.W., W.Y., S.X. and C.W. are supported by NSF grant number DMR-1410375 and AFOSR grant number FA9550-14-1-0168. C.W. also acknowledges partial support from the National Natural Science Foundation of China (grant number 11729402).

Author information

Author notes

    • Jianda Wu

    Present address: Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany.

Affiliations

  1. Experimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany

    • Zhe Wang
    •  & Alois Loidl
  2. Institute of Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany

    • Zhe Wang
  3. Department of Physics, University of California, San Diego, California 92093, USA

    • Jianda Wu
    • , Wang Yang
    • , Shenglong Xu
    •  & Congjun Wu
  4. Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany

    • Anup Kumar Bera
    • , A. T. M. Nazmul Islam
    •  & Bella Lake
  5. Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India

    • Anup Kumar Bera
  6. High Field Magnet Laboratory, Radboud University, 6525 ED Nijmegen, The Netherlands

    • Dmytro Kamenskyi
  7. Hochfeld Magnetlab Dresden, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany

    • Joseph Matthew Law
  8. Institut für Festkörperphysik, Technische Universität Berlin, 10623 Berlin, Germany

    • Bella Lake

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Contributions

Z.W. conceived and performed the optical experiments, analysed the data and coordinated the project. J.W., W.Y. and S.X. carried out the Bethe-ansatz calculations. A.K.B. and A.T.M.N.I. prepared and characterized the high-quality single crystals. A.K.B. and J.M.L. performed the high-field magnetization measurements. D.K. assisted with the high-field optical experiments. B.L., C.W. and A.L. supervised the project. Z.W., J.W., W.Y., C.W. and A.L. wrote the manuscript with input from all authors. All authors discussed the results.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Zhe Wang.

Reviewer Information Nature thanks M. Batchelor, J. van den Brink and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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DOI

https://doi.org/10.1038/nature25466

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