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Dispersions of many-body Bethe strings


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.

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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.

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 ( 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.


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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




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.

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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.

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Bera, A.K., Wu, J., Yang, W. et al. Dispersions of many-body Bethe strings. Nat. Phys. 16, 625–630 (2020).

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