Skip to main content

Thank you for visiting 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.

Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain


One of the simplest quantum many-body systems is the spin-1/2 Heisenberg antiferromagnetic chain, a linear array of interacting magnetic moments. Its exact ground state is a macroscopic singlet entangling all spins in the chain. Its elementary excitations, called spinons, are fractional spin-1/2 quasiparticles created and detected in pairs by neutron scattering. Theoretical predictions show that two-spinon states exhaust only 71% of the spectral weight and higher-order spinon states, yet to be experimentally located, are predicted to participate in the remaining. Here, by accurate absolute normalization of our inelastic neutron scattering data on a spin-1/2 Heisenberg antiferromagnetic chain compound, we account for the full spectral weight to within 99(8)%. Our data thus establish and quantify the existence of higher-order spinon states. The observation that, within error bars, the experimental line shape resembles a rescaled two-spinon one with similar boundaries allows us to develop a simple picture for understanding multi-spinon excitations.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Schematic representation of the magnetic excitations in a spin-1/2 (Heisenberg) antiferromagnetic chain and overview of the neutron scattering results for CuSO4·5D2O.
Figure 2: Excitations in the fully polarized state (μ0H = 5 T).
Figure 3: Excitations in zero magnetic field.
Figure 4: Sum rules.


  1. 1

    Friedrich, W., Knipping, P. & Laue, M. Interferenz-Erscheinungen bei Röntgenstrahlen. Proc. Bavarian Acad. Sci. (Sber. Bayer. Akad. Wiss.) 303–322 (1912).

  2. 2

    Friedrich, W., Knipping, P & Laue, M. Interferenz-Erscheinungen bei Röntgenstrahlen. Naturwissenschaften 39, 361–369 (1952).

    ADS  Article  Google Scholar 

  3. 3

    Morton, T., Charlesworth, J. J. & Lingwood, J. Roger Hiorns: Seizure (Artangel, 2008).

    Google Scholar 

  4. 4

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

    Article  Google Scholar 

  5. 5

    Raghu, S., Qi, X-L., Honerkamp, C. & Zhang, S-C. Topological Mott insulators. Phys. Rev. Lett. 100, 156401 (2008).

    ADS  Article  Google Scholar 

  6. 6

    Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    ADS  Google Scholar 

  7. 7

    Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).

    ADS  Article  Google Scholar 

  8. 8

    Kivelson, S. A., Rokhsar, D. S. & Sethna, J. P. Topology of the resonating valence-bond state: Solitons and high- T c superconductivity. Phys. Rev. B 35, 8865–8868 (1987).

    ADS  Article  Google Scholar 

  9. 9

    Lee, P. A. From high temperature superconductivity to quantum spin liquid: Progress in strong correlation physics. Rep. Prog. Phys. 71, 012501 (2008).

    ADS  Article  Google Scholar 

  10. 10

    Meng, Z. Y. et al. Quantum spin liquid emerging in two-dimensional correlated Dirac fermions. Nature 464, 847–851 (2010).

    ADS  Article  Google Scholar 

  11. 11

    Lieb, E., Schultz, T. & Mattis, D. Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961).

    ADS  MathSciNet  Article  Google Scholar 

  12. 12

    Affleck, I. Quantum spin chains and the Haldane gap. J. Phys. Condens. Matter 1, 3047–3072 (1989).

    ADS  Article  Google Scholar 

  13. 13

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

    ADS  MathSciNet  Article  Google Scholar 

  14. 14

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

    ADS  Article  Google Scholar 

  15. 15

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

    ADS  Article  Google Scholar 

  16. 16

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

    ADS  Article  Google Scholar 

  17. 17

    Kadowaki, H. et al. Observation of magnetic monopoles in spin ice. J. Phys. Soc. Jpn 78, 103706 (2009).

    ADS  Article  Google Scholar 

  18. 18

    Hao, Z. & Tchernyshyov, O. Fermionic spin excitations in two- and three-dimensional antiferromagnets. Phys. Rev. Lett. 103, 187203 (2009).

    ADS  Article  Google Scholar 

  19. 19

    Laughlin, R. B. Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).

    ADS  Article  Google Scholar 

  20. 20

    Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982).

    ADS  MathSciNet  Article  Google Scholar 

  21. 21

    Halperin, B. I. Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Phys. Rev. Lett. 52, 1583–1586 (1984).

    ADS  Article  Google Scholar 

  22. 22

    Kukushkin, I. V. et al. Dispersion of the excitations of fractional quantum Hall states. Science 324, 1044–1047 (2009).

    ADS  Article  Google Scholar 

  23. 23

    Su, W-P. & Schrieffer, J. R. Fractionally charged excitations in charge-density-wave systems with commensurability 3. Phys. Rev. Lett. 46, 738–741 (1981).

    ADS  Article  Google Scholar 

  24. 24

    Heeger, A. J., Kivelson, S., Schrieffer, J. R. & Su, W-P. Solitons in conducting polymers. Rev. Mod. Phys. 60, 781–850 (1988).

    ADS  Article  Google Scholar 

  25. 25

    Greiter, M. & Thomale, R. Non-Abelian statistics in a quantum antiferromagnet. Phys. Rev. Lett. 102, 207203 (2009).

    ADS  Article  Google Scholar 

  26. 26

    Yao, H., Zhang, S-C. & Kivelson, S. A. Algebraic spin liquid in an exactly solvable spin model. Phys. Rev. Lett. 102, 217202 (2009).

    ADS  Article  Google Scholar 

  27. 27

    Karbach, M. et al. Two-spinon dynamic structure factor of the one-dimensional S = 1/2 Heisenberg antiferromagnet. Phys. Rev. B 55, 12510–12517 (1997).

    ADS  Article  Google Scholar 

  28. 28

    Caux, J-S. & Hagemans, R. The four-spinon dynamical structure factor of the Heisenberg chain. J. Stat. Mech.: Theory Exp. P12013 (2006).

  29. 29

    Heilmann, I. U. et al. Neutron study of the line-shape and field dependence of magnetic excitations in CuCl2·2N(C5D5). Phys. Rev. B 18, 3530–3536 (1978).

    ADS  Article  Google Scholar 

  30. 30

    Nagler, S. E. et al. Spin dynamics in the quantum antiferromagnetic chain compound KCuF3 . Phys. Rev. B 44, 12361–12368 (1991).

    ADS  Article  Google Scholar 

  31. 31

    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).

    ADS  Article  Google Scholar 

  32. 32

    Arai, M. et al. Quantum spin excitations in the spin-Peierls system CuGeO3 . Phys. Rev. Lett. 77, 3649–3652 (1996).

    ADS  Article  Google Scholar 

  33. 33

    Dender, D. C. et al. Magnetic properties of a quasi-one-dimensional S = 1/2 antiferromagnet: Copper benzoate. Phys. Rev. B 53, 2583–2589 (1996).

    ADS  Article  Google Scholar 

  34. 34

    Zheludev, A. et al. Energy separation of single-particle and continuum states in an S = 1/2 weakly coupled chains antiferromagnet. Phys. Rev. Lett. 85, 4799–4802 (2000).

    ADS  Article  Google Scholar 

  35. 35

    Coldea, R., Tennant, D. A., Tsvelik, A. M. & Tylczynski, Z. Experimental realization of a 2D fractional quantum spin liquid. Phys. Rev. Lett. 86, 1335–1338 (2001).

    ADS  Article  Google Scholar 

  36. 36

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

    ADS  Article  Google Scholar 

  37. 37

    Zaliznyak, I. A. et al. Spinons in the strongly correlated copper oxide chains in SrCuO2 . Phys. Rev. Lett. 93, 087202 (2004).

    ADS  Article  Google Scholar 

  38. 38

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

    ADS  Article  Google Scholar 

  39. 39

    Hong, T. et al. Synthesis and structural characterization of 2Dioxane·2H2O·CuCl2: Metal-organic compound with Heisenberg antiferromagnetic S = 1/2 chains. Phys. Rev. B 80, 132404 (2009).

    ADS  Article  Google Scholar 

  40. 40

    Thielemann, B. et al. Direct observation of magnon fractionalization in the quantum spin ladder. Phys. Rev. Lett. 102, 107204 (2009).

    ADS  Article  Google Scholar 

  41. 41

    Walters, A. C. et al. Effect of covalent bonding on magnetism and the missing neutron intensity in copper oxide compounds. Nature Phys. 5, 867–872 (2009).

    ADS  Article  Google Scholar 

  42. 42

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

    ADS  Article  Google Scholar 

  43. 43

    Enderle, M. et al. Two-spinon and four-spinon continuum in a frustrated ferromagnetic spin-1/2 chain. Phys. Rev. Lett. 104, 237207 (2010).

    ADS  Article  Google Scholar 

  44. 44

    Coldea, R. et al. Direct measurement of the spin Hamiltonian and observation of condensation of magnons in the 2D frustrated quantum magnet Cs2CuCl4 . Phys. Rev. Lett. 88, 137203 (2002).

    ADS  Article  Google Scholar 

  45. 45

    Des Cloizeaux, J. & Pearson, J. J. Spin-wave spectrum of the antiferromagnetic linear chain. Phys. Rev. 128, 2131–2135 (1962).

    ADS  Article  Google Scholar 

  46. 46

    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).

    ADS  Article  Google Scholar 

  47. 47

    Van Tol, M. W. & Poulis, N. J. A high-field phase transition in the linear chain compound CuSO4·5D2O. Physica 69, 341–353 (1973).

    ADS  Article  Google Scholar 

  48. 48

    Jimbo, M. & Miwa, T. Algebraic Analysis of Solvable Lattice Models Vol. 85 (CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., 1995).

    MATH  Google Scholar 

Download references


We acknowledge useful discussions with C. Broholm, B. Dalla Piazza, B. Fåk, B. Lake, C. Rüegg and A. Tennant. The work of M.M. was supported in part by US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-FG02-08ER46544. M.E. acknowledges support from the Deutsche Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF), project 03KN5SAA. H.M.R. acknowledges support from the Swiss National Science Foundation (SNF) and the European Research Council (ERC). J-S.C. acknowledges support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organisation for Scientific Research (NWO).

Author information




M.E. and H.M.R. performed the experiment with the help of A.S. on a crystal synthesized by A.K. Data treatment and fits were carried out by M.M., M.E. and H.M.R.; exact theoretical calculations were performed by J-S.C. The physical pictures for multi-spinon excitations were developed through various discussions between J-S.C and M.M., M.E. and H.M.R. M.E., M.M., J-S.C. and H.M.R. wrote the manuscript.

Corresponding author

Correspondence to Martin Mourigal.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 248 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mourigal, M., Enderle, M., Klöpperpieper, A. et al. Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain. Nature Phys 9, 435–441 (2013).

Download citation

Further reading


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