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:

Experimental observation of Bethe strings

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.

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

Figure 1: Quantum spin chain in SrCo2V2O8, psinon–(anti)psinon pairs and strings.
Figure 2: Softening of spinons and emergent magnetic excitations at the quantum phase transition in SrCo2V2O8.
Figure 3: Absorption spectra of psinon–psinon, psinon–antipsinon, two-string and three-string excitations for Bc < B < Bs and of magnons for B > Bs in SrCo2V2O8.
Figure 4: Magnetic excitations in the longitudinal-field Heisenberg–Ising chain SrCo2V2O8.

Similar content being viewed by others

References

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

    Article  CAS  ADS  Google Scholar 

  2. Yang, C. N. & Yang, C. P. 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)

    CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. Müller, G., Thomas, H., Beck, H. & Bonner, J. C. Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field. Phys. Rev. B 24, 1429–1467 (1981)

    Article  ADS  Google Scholar 

  6. 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 

  7. Karbach, M. & Müller, G. 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)

    Article  CAS  ADS  Google Scholar 

  8. Sato, J., Shiroishi, M. & Takahashi, M. 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)

    Article  CAS  ADS  Google Scholar 

  9. 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 

  10. 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 

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

    Article  CAS  ADS  MathSciNet  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  16. Tennant, D. A., Perring, T. G., Cowley, R. A. & Nagler, S. E. Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3 . Phys. Rev. Lett. 70, 4003–4006 (1993)

    Article  CAS  ADS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  20. 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 

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

    Article  CAS  ADS  Google Scholar 

  22. Bera, A. K., Lake, B., Stein, W.-D. & Zander, S. Magnetic correlations of the quasi-one-dimensional half-integer spin-chain antiferromagnets SrM2V2O8 (M = Co, Mn). Phys. Rev. B 89, 094402 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  24. Wang, Z. 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)

    Article  ADS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  26. Yang, W ., Wu, J ., Xu, S ., Wang, Z. & Wu, C. Quantum spin dynamics of the axial antiferromagnetic spin-1/2 XXZ chain in a longitudinal magnetic field. Preprint at https://arxiv.org/abs/1702.01854 (2017)

  27. Essler, F. H. L. & Konik, R. M. in From Fields to Strings: Circumnavigating Theoretical Physics (eds Shifman, M. et al.) Vol. 1, 684–830 (World Scientific, 2005)

    Book  Google Scholar 

  28. Nishida, Y., Kato, Y. & Batista, C. D. Efimov effect in quantum magnets. Nat. Phys. 9, 93–97 (2013)

    Article  CAS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  30. Lieb, E. H. & Wu, F. Y. 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)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. Shiba, H., Ueda, Y., Okunishi, K., Kimura, S. & Kindo, K. Exchange interaction via crystal-field excited states and its importance in CsCoCl3 . J. Phys. Soc. Jpn 72, 2326–2333 (2003)

    Article  CAS  ADS  Google Scholar 

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

  36. 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 

  37. 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 

  38. He, F., Jiang, Y.-Z., Yu, Y.-C., Lin, H.-Q. & Guan, X.-W. Quantum criticality of spinons. Phys. Rev. B 96, 220401(R) (2017)

    Article  ADS  Google Scholar 

Download references

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

Authors and Affiliations

Authors

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.

Corresponding author

Correspondence to Zhe Wang.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

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.

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

Extended data figures and tables

Extended Data Figure 1 Crystal and magnetic structure of SrCo2V2O8.

a, The screw-chain structure consists of edge-shared CoO6 octahedra. Each chain has screw-axis symmetry with a period of four Co2+ ions (as numbered by the integers 1, 2, 3 and 4), corresponding to the lattice constant along the c axis. The Néel-ordered phase is illustrated by antiparallel arrows representing magnetic moments at the Co2+ sites. Intra-chain nearest-neighbour interaction is denoted by J. b, Viewing from the c axis, each unit cell contains four screw chains with left- or right-handed screw axes. The leading inter-chain coupling J is indicated, which is between the Co2+ ions in the same layer (denoted by the same integer as the Co site) and from chains with the same chirality. It is very small compared to the intra-chain interaction, J/J < 10−2 (refs 32, 33).

Extended Data Figure 2 High-field magnetization and magnetic susceptibility of SrCo2V2O8.

a, Magnetization M as a function of an applied longitudinal magnetic field B along the Ising axis (B c), measured at 1.7 K (circles). Theoretical magnetization of the Heisenberg–Ising chain model is shown by the dashed line. b, Magnetic susceptibility dM/dH as a function of the applied longitudinal field B. A quantum phase transition from the Néel-ordered phase to the critical phase is revealed by the onset of magnetization and the peak in the susceptibility curve at the critical field Bc = 4 T. Saturated magnetization is observed above the field Bs = 28.7 T and indicated by the sharp peak in the susceptibility. The small anomaly at Bhs = 25 T seen in the susceptibility is close to the field of half-saturated magnetization.

Extended Data Figure 3 Low-energy phonon spectrum of SrCo2V2O8.

The phonon spectra of SrCo2V2O8 measured for the polarization Eω a at 5 K. Strong reflectivity due to phonon excitations is observed in the spectral range 8–13.5 meV.

Extended Data Figure 4 Schematics of patterns of Bethe quantum numbers.

a, The ground state. b, One-pair psinon–psinon state 1ψψ. c, One-pair psinon–antipsinon state 1ψψ*. d, Length-two string state 1χ(2)R. The system size is taken as N = 32 and the magnetization is .

Extended Data Figure 5 DSFs.

a, b, S+−(q, ω) and S−+(q, ω), respectively, as functions of energy ħω/J (vertical axis) and momentum q/π (horizontal axis) for 2m = 0.4 and N = 200. The gapless continua are formed by real Bethe eigenstates (psinon–antipsinon pairs in S+− and psinon–psinon pairs in S−+). For S+− (a), the higher-energy continua correspond to excitations of two-string (ħω > 3J) and three-string (ħω > 5J) states.

Extended Data Figure 6 The momentum-integrated ratios.

a, b, ν+− for S+− and ν−+ for S−+, respectively, as functions of magnetization 2m. In a, the green line is the 1ψψ* contribution. The blue, red and the black lines are augmented by progressively taking into account the 2ψψ*, two-string and three-string contributions, respectively. In b, the blue and black lines represent the 1ψψ and 1ψψ + 2ψψ contributions, respectively.

Extended Data Figure 7 DSF of psinon–psinon pairs as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

Extended Data Figure 8 DSF of psinon–antipsinon pairs as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

Extended Data Figure 9 DSF factor of two-string states as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

Extended Data Figure 10 DSF of three-string states as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

PowerPoint slides

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Wu, J., Yang, W. et al. Experimental observation of Bethe strings. Nature 554, 219–223 (2018). https://doi.org/10.1038/nature25466

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature25466

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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