Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Dispersions of many-body Bethe strings

Abstract

Complex bound states of magnetic excitations, known as Bethe strings, were predicted almost a century ago to exist in one-dimensional quantum magnets1. The dispersions of the string states have so far remained the subject of intense theoretical studies2,3,4,5,6,7. Here, by performing neutron scattering experiments on the one-dimensional Heisenberg–Ising antiferromagnet SrCo2V2O8 in high longitudinal magnetic fields, we reveal in detail the dispersion relations of the string states over the full Brillouin zone, as well as their magnetic field dependencies. Furthermore, the characteristic energy, the scattering intensity and linewidth of the observed string states exhibit excellent agreement with our precise Bethe–ansatz calculations. Our results establish the important role of string states in the quantum spin dynamics of one-dimensional systems, and will invoke studies of their dynamical properties in more general many-body systems.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Crystal structure, magnetic phase diagram and magnetic ground states of SrCo2V2O8.
Fig. 2: Spin configurations and theoretical excitation spectra in the quantum critical regime of the XXZ spin-1/2 antiferromagnetic chain for the Hamiltonian of SrCo2V2O8 (excluding the interchain coupling).
Fig. 3: Dispersion relations of Bethe strings in energy and momentum space.
Fig. 4: Comparison of the scattering intensity at B = 9 T with Bethe–ansatz calculations.
Fig. 5: Magnetic field dependence of Bethe strings.

Similar content being viewed by others

Data availability

The datasets for the inelastic neutron scattering experiment on the time-of-flight LET spectrometer are available from the ISIS facility, Rutherford Appleton Laboratory data portal (10.5286/ISIS.E.RB1510288). The datasets for the inelastic neutron scattering experiment on the cold triple-axis ThALES spectrometer are available from the Institute Laue–Langevin data portal (https://doi.ill.fr/10.5291/ILL-DATA.4-05-700). Additional INS data were taken on the FLEXX spectrometer at HZB, Berlin, Germany. High-field (above 15 T) INS data were taken on the HFM/EXED high magnetic field facility for neutron scattering at HZB, Berlin, Germany. The data represented in Figs. 3, 4 and 5 are available as Source Data Figs. 3, 4 and 5. All other raw and derived data used to support the findings of this study are available from the authors upon request.

Code availability

The code is available upon reasonable request from J.W.

References

  1. Bethe, H. Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205–226 (1931).

    Article  ADS  Google Scholar 

  2. Gaudin, M. Thermodynamics of the Heisenberg–Ising ring for Δ > 1. Phys. Rev. Lett. 26, 1301–1304 (1971).

    Article  ADS  Google Scholar 

  3. Karbach, M. & Muller, G. Introduction to the Bethe ansatz I. Comput. Phys. 11, 36 (1997).

    Article  ADS  Google Scholar 

  4. Kitanine, N., Mailet, J. M. & Terras, V. Form factors of the XXZ Heisenberg spin-1/2 finite chain. Nucl. Phys. B 554, 647–678 (1999).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. Pereira, R. G., White, S. R. & Affleck, I. Exact edge singularities and dynamical correlations in spin-1/2 chains. Phys. Rev. Lett. 100, 027206 (2008).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  10. Haldane, F. D. M. General relation of correlation exponents and spectral properties of one-dimensional Fermi systems: application to the anisotropic S = 1/2 Heisenberg chain. Phys. Rev. Lett. 45, 1358–1362 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  11. Ganahl, M., Rabel, E., Essler, F. H. L. & Evertz, H. G. Observation of complex bound states in the spin-1/2 Heisenberg XXZ chain using local quantum quenches. Phys. Rev. Lett. 108, 077206 (2012).

    Article  ADS  Google Scholar 

  12. Caux, J. S. & Maillet, J. M. Computation of dynamical correlation functions of Heisenberg chains in a magnetic field. Phys. Rev. Lett. 95, 077201 (2005).

    Article  ADS  Google Scholar 

  13. Wang, Z. et al. Experimental observation of Bethe strings. Nature 554, 219–223 (2018).

    Article  ADS  Google Scholar 

  14. Bera, A. K., Lake, B., Stein, W. D. & Zander, S. Magnetic correlations of the quasi-one-dimensional half-integer spin-chain antiferromagnets Sr M 2V2O8 (M =Co, Mn). Phys. Rev. B 89, 094402 (2014).

    Article  ADS  Google Scholar 

  15. Grenier, B. et al. Neutron diffraction investigation of the H-T phase diagram above the longitudinal incommensurate phase of BaCo2V2O8. Phys. Rev. B 92, 134416 (2015).

    Article  ADS  Google Scholar 

  16. Faure, Q. et al. Tomonaga–Luttinger liquid spin dynamics in the quasi-one-dimensional Ising-like antiferromagnet BaCo2V2O8. Phys. Rev. Lett. 123, 027204 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  18. Lake, B., Tennant, D. A., Frost, C. D. & Nagler, S. E. Quantum criticality and universal scaling of a quantum antiferromagnet. Nat. Mater. 4, 329–334 (2005).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. Coldea, R. et al. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. Science 327, 177–180 (2010).

    Article  ADS  Google Scholar 

  23. Wang, Z. Spinon confinement in the one-dimensional Ising-like antiferromagnet SrCo2V2O8. Phys. Rev. B 91, 140404 (R) (2015).

    Article  ADS  Google Scholar 

  24. Polyakov, A. M. Quark confinement and topology of gauge theories. Nucl. Phys. B 120, 429–458 (1977).

    Article  ADS  Google Scholar 

  25. Shelton, D. G., Nersesyan, A. A. & Tsvelik, A. M. Antiferromagnetic spin ladders: crossover between spin S = 1/2 and S = 1 chains. Phys. Rev. B 53, 8521–8532 (1996).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  27. Yang, W., Wu, J., Xu, S., Wang, Z. & Wu, C. One-dimensional quantum spin dynamics of Bethe string states. Phys. Rev. B 100, 184406 (2019).

    Article  ADS  Google Scholar 

  28. Korepin, V. E., Bogoliubov, N. M. & Izergin, A. G. Quantum Inverse Scattering Method and Correlation Functions (Cambridge Univ. Press, 1997).

  29. Yang, C. N. & Yang, C. P. One-dimensional chain of anisotropic spin–spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system. Phys. Rev. 150, 321–327 (1966).

    Article  ADS  Google Scholar 

  30. Lejay, P. et al. Crystal growth and magnetic property of MCo2V2O8 (M = Sr and Ba). J. Cryst. Growth 317, 128–131 (2011).

    Article  ADS  Google Scholar 

  31. Skourski, Y., Kuz’min, M. D., Skokov, K. P., Andreev, A. V. & Wosnitza, J. High-field magnetization of Ho2Fe17. Phys. Rev. B 83, 214420 (2011).

    Article  ADS  Google Scholar 

  32. Russina, M. & Mezei, F. First implementation of repetition rate multiplication in neutron spectroscopy. Nucl. Instrum. Meth. A 604, 624–631 (2009).

    Article  ADS  Google Scholar 

  33. Nakamura, M. et al. First demonstration of novel method for inelastic neutron scattering measurement utilizing multiple incident energies. J. Phys. Soc. Jpn 78, 093002 (2009).

    Article  ADS  Google Scholar 

  34. Arnold, O. et al. Mantid—Data analysis and visualization package for neutron scattering and μ-SR experiments. Nucl. Instrum. Meth. A 764, 156–166 (2014).

    Article  ADS  Google Scholar 

  35. Le, M. D. et al. Gains from the upgrade of the cold neutron triple-axis spectrometer FLEXX at the BER-II reactor. Nucl. Instrum. Meth. A 729, 220–226 (2013).

    Article  ADS  Google Scholar 

  36. Prokhnenko, O., Smeibidl, P., Stein, W.-D., Bartkowiak, M. & Stüsser, N. HFM/EXED: the high magnetic field facility for neutron scattering at BER II. J. Large Scale Res. Facilities 3, A115 (2017).

    Article  Google Scholar 

  37. Bartkowiak, M., Stuesser, N. & Prokhnenko, O. The design of the inelastic neutron scattering mode for the extreme environment diffractometer with the 26 T high field magnet. Nucl. Instrum. Meth. A 797, 121–129 (2015).

    Article  ADS  Google Scholar 

  38. Korepin, V. E. Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  39. Slavnov, N. A. Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz. Theor. Math. Phys. 79, 502–508 (1989).

    Article  MathSciNet  Google Scholar 

  40. Caux, J. S., Hagemans, R. & Maillet, J. M. Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime. J. Stat. Mech. 2005, P09003 (2005).

    Article  Google Scholar 

Download references

Acknowledgements

We thank the HFM/EXED team and P. Smeibidl, R. Wahle and S. Gerischer for their technical support during the measurements. J.W. acknowledges additional support from a Shanghai talent programme. The high-field experiments at Dresden were supported by Hochfeld Magnetlabor Dresden at HZDR, a member of the European Magnetic Field Laboratory (EMFL).

Author information

Authors and Affiliations

Authors

Contributions

A.K.B. and B.L. conceived the experiments. A.K.B., J.W. and B.L. coordinated the project. A.K.B. and A.T.M.N.I. prepared and characterized the high-quality single crystals. A.K.B., B.K. and J.M.L. performed the bulk measurements. A.K.B., B.L. and R.B. performed the LET experiments. B.L. and M.Boehm. performed ThALES experiments. B.L., M.Bartkowiak. and O.P. performed the HFM/EXED measurements. B.L. and J.X. performed the FLEXX measurements. J.W. and W.Y. carried out the Bethe–ansatz calculations. A.K.B. and B.L. analysed the experimental data. J.W. and W.Y. analysed the Bethe–ansatz calculations. The comparison between experimental and theoretical results was made by A.K.B., J.W., W.Y. and B.L. A.K.B. wrote the manuscript, with contributions from B.L., J.W., W.Y. and Z.W. All authors discussed the data and its interpretation.

Corresponding authors

Correspondence to Anup Kumar Bera, Jianda Wu, Wang Yang or Bella Lake.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–6, Discussion and Table 1.

Source data

Source Data Fig. 3

Source Data for Fig. 3.

Source Data Fig. 4

Source Data for Fig. 4.

Source Data Fig. 5

Source Data for Fig. 5.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bera, A.K., Wu, J., Yang, W. et al. Dispersions of many-body Bethe strings. Nat. Phys. 16, 625–630 (2020). https://doi.org/10.1038/s41567-020-0835-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-020-0835-7

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing