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.

  • Article
  • Published:

Fractional excitations in the square-lattice quantum antiferromagnet

Nature Physics volume 11, pages 62–68 (2015)Cite this article

Subjects

Abstract

Quantum magnets have occupied the fertile ground between many-body theory and low-temperature experiments on real materials since the early days of quantum mechanics. However, our understanding of even deceptively simple systems of interacting spin-1/2 particles is far from complete. The quantum square-lattice Heisenberg antiferromagnet, for example, exhibits a striking anomaly of hitherto unknown origin in its magnetic excitation spectrum. This quantum effect manifests itself for excitations propagating with the specific wavevector (π, 0). We use polarized neutron spectroscopy to fully characterize the magnetic fluctuations in the metal-organic compound Cu(DCOO)2⋅4D2O, a known realization of the quantum square-lattice Heisenberg antiferromagnet model. Our experiments reveal an isotropic excitation continuum at the anomaly, which we analyse theoretically using Gutzwiller-projected trial wavefunctions. The excitation continuum is accounted for by the existence of spatially extended pairs of fractional S = 1/2 quasiparticles, 2D analogues of 1D spinons. Away from the anomalous wavevector, these fractional excitations are bound and form conventional magnons. Our results establish the existence of fractional quasiparticles in the high-energy spectrum of a quasi-two-dimensional antiferromagnet, even in the absence of frustration.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Figure 1: Overview of the magnetic excitation spectrum of CFTD and its interpretation in terms of spin waves or spatially extended fractional excitations.
Figure 2: Summary of the polarized neutron scattering data.
Figure 3: Schematic representation of local spin flip and spatially separated quasiparticle-pair excitations in the Gutzwiller-projected approach.
Figure 4: Variational excitation spectra of the Gutzwiller-projected trial wavefunctions.
Figure 5: Finite-size effects and real-space structure in the |SF〉 state.

Similar content being viewed by others

References

  1. Laughlin, R. B. Nobel lecture: Fractional quantization. Rev. Mod. Phys. 71, 863–874 (1999).

    ADS  MathSciNet  MATH  Google Scholar 

  2. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).

    ADS  Google Scholar 

  3. Hou, C-Y., Chamon, C. & Mudry, C. Electron fractionalization in two-dimensional graphene like structures. Phys. Rev. Lett. 98, 1698–1701 (1979).

    Google Scholar 

  4. Simon, J. et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011).

    ADS  Google Scholar 

  5. Baskaran, G., Zou, Z. & Anderson, P. W. The resonating valence bond state and high-Tc superconductivity—a mean field theory. Solid State Commun. 63, 973–976 (1987).

    ADS  Google Scholar 

  6. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

    Article  ADS  Google Scholar 

  7. Bethe, H. Zur Theorie der Metalle. Z. Phys. A 71, 205–226 (1931).

    MATH  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  9. 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  Google Scholar 

  10. Tennant, D. A., Cowley, R. A., Nagler, S. E. & Tsvelik, A. M. Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering. Phys. Rev. B 52, 13368–13380 (1995).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  12. 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  Google Scholar 

  13. Han, T-H. et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).

    ADS  Google Scholar 

  14. Jeong, M. et al. Field-induced freezing of a quantum spin liquid on the kagome lattice. Phys. Rev. Lett. 107, 237201 (2011).

    ADS  Google Scholar 

  15. Kozlenko, D. P. et al. From quantum disorder to magnetic order in an S = 1/2 kagome lattice: A structural and magnetic study of herbertsmithite at high pressure. Phys. Rev. Lett. 108, 187207 (2012).

    ADS  Google Scholar 

  16. Manousakis, E. The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Rev. Mod. Phys. 63, 1–62 (1991).

    ADS  Google Scholar 

  17. Canali, C. M. & Wallin, M. Spin–spin correlation functions for the square-lattice Heisenberg antiferromagnet at zero temperature. Phys. Rev. B 48, 3264–3280 (1993).

    ADS  Google Scholar 

  18. Reger, J. D. & Young, A. P. Monte Carlo simulations of the spin-1/2 Heisenberg antiferromagnet on a square lattice. Phys. Rev. B 37, 5978–5981 (1988).

    ADS  Google Scholar 

  19. Hamer, C. J., Weihong, Z. & Arndt, P. Third-order spin-wave theory for the Heisenberg antiferromagnet. Phys. Rev. B 46, 6276–6292 (1992).

    ADS  Google Scholar 

  20. Anderson, P. W., Baskaran, G., Zou, Z. & Hsu, T. Resonating valence-bond theory of phase transitions and superconductivity in La2CuO4-based compounds. Phys. Rev. Lett. 58, 2790–2793 (1987).

    ADS  Google Scholar 

  21. Auerbach, A. & Arovas, D. P. Spin dynamics in the square-lattice antiferromagnet. Phys. Rev. Lett. 61, 617–620 (1988).

    ADS  Google Scholar 

  22. Hsu, T. C. Spin waves in the flux-phase description of the S = 1/2 Heisenberg antiferromagnet. Phys. Rev. B 41, 11379–11387 (1990).

    ADS  Google Scholar 

  23. Ho, C-M., Muthukumar, V. N., Ogata, M. & Anderson, P. W. Nature of spin excitations in two-dimensional Mott insulators: Undoped cuprates and other materials. Phys. Rev. Lett. 86, 1626–1629 (2001).

    ADS  Google Scholar 

  24. Balents, L., Fisher, M. P. A. & Nayak, C. Dual order parameter for the nodal liquid. Phys. Rev. B 60, 1654–1667 (1999).

    ADS  Google Scholar 

  25. Ghaemi, P. & Senthil, T. Neél order, quantum spin liquids, and quantum criticality in two dimensions. Phys. Rev. B 73, 054415 (2006).

    ADS  Google Scholar 

  26. Greven, M. et al. Neutron scattering study of the two-dimensional spin-S = 1/2 square-lattice Heisenberg antiferromagnet Sr2CuO2Cl2 . Z. Phys. B 96, 465–477 (1995).

    ADS  Google Scholar 

  27. Plumb, K. W., Savici, A. T., Granroth, G. E., Chou, F. C. & Kim, Y-J. High-energy continuum of magnetic excitations in the two-dimensional quantum antiferromagnet Sr2CuO2Cl2 . Phys. Rev. B 89, 180410 (2014).

    ADS  Google Scholar 

  28. Coldea, R. et al. Spin waves and electronic interactions in La2CuO4 . Phys. Rev. Lett. 86, 5377–5380 (2001).

    ADS  Google Scholar 

  29. Headings, N. S., Hayden, S. M., Coldea, R. & Perring, T. G. Anomalous high-energy spin excitations in the high-Tc superconductor-parent antiferromagnet La2CuO4 . Phys. Rev. Lett. 105, 247001 (2010).

    Article  ADS  Google Scholar 

  30. Kim, Y. J. et al. Neutron scattering study of Sr2Cu3O4Cl2 . Phys. Rev. B 64, 024435 (2001).

    ADS  Google Scholar 

  31. Tsyrulin, N. et al. Quantum effects in a weakly frustrated S = 1/2 two-dimensional Heisenberg antiferromagnet in an applied magnetic field. Phys. Rev. Lett. 102, 197201 (2009).

    ADS  Google Scholar 

  32. Tsyrulin, N. et al. Two-dimensional square-lattice S = 1/2 antiferromagnet Cu(Pz)2(ClO4)2 . Phys. Rev. B 81, 134409 (2010).

    ADS  Google Scholar 

  33. Rønnow, H. M. et al. Spin dynamics of the 2D spin-1/2 quantum antiferromagnet copper deuteroformate tetradeuterate (CFTD). Phys. Rev. Lett. 87, 037202 (2001).

    ADS  Google Scholar 

  34. Christensen, N. B. et al. Quantum dynamics and entanglement of spins on a square lattice. Proc. Natl Acad. Sci. USA 104, 15264–15269 (2007).

    ADS  Google Scholar 

  35. Singh, R. R. P. & Gelfand, M. P. Spin-wave excitation spectra and spectral weights in square lattice antiferromagnets. Phys. Rev. B 52, R15695–R15698 (1995).

    ADS  Google Scholar 

  36. Zheng, W., Oitmaa, J. & Hamer, C. J. Series studies of the spin-1/2 Heisenberg antiferromagnet at T = 0: Magnon dispersion and structure factors. Phys. Rev. B 71, 184440 (2005).

    ADS  Google Scholar 

  37. Syljuåsen, O. F. & Rønnow, H. M. Quantum renormalization of high-energy excitations in the 2D Heisenberg model. J. Phys. Condens. Matter 12, L405–L408 (2000).

    ADS  Google Scholar 

  38. Sandvik, A. W. & Singh, R. R. P. High-energy magnon dispersion and multimagnon continuum in the two-dimensional Heisenberg antiferromagnet. Phys. Rev. Lett. 86, 528–531 (2001).

    ADS  Google Scholar 

  39. Lüscher, A. & Läuchli, A. M. Exact diagonalization study of the antiferromagnetic spin-1/2 Heisenberg model on the square lattice in a magnetic field. Phys. Rev. B 79, 195102 (2009).

    ADS  Google Scholar 

  40. Guarise, M. et al. Measurement of magnetic excitations in the two-dimensional antiferromagnetic Sr2CuO2Cl2 insulator using resonant X-ray scattering: Evidence for extended interactions. Phys. Rev. Lett. 105, 157006 (2010).

    ADS  Google Scholar 

  41. Dalla Piazza, B. et al. Unified one-band Hubbard model for magnetic and electronic spectra of the parent compounds of cuprate superconductors. Phys. Rev. B 85, 100508 (2012).

    ADS  Google Scholar 

  42. Ishii, K. et al. High-energy spin and charge excitations in electron-doped copper oxide superconductors. Nature Commun. 5, 3714 (2014).

    ADS  Google Scholar 

  43. Braicovich, L. et al. Magnetic excitations and phase separation in the underdoped La2−xSrxCuO4 superconductor measured by resonant inelastic X-ray scattering. Phys. Rev. Lett. 104, 077002 (2010).

    ADS  Google Scholar 

  44. Zhitomirsky, M. E. & Chernyshev, A. L. Colloquium: Spontaneous magnon decays. Rev. Mod. Phys. 85, 219–242 (2013).

    ADS  Google Scholar 

  45. Tang, Y. & Sandvik, A. W. Confinement and deconfinement of spinons in two dimensions. Phys. Rev. Lett. 110, 217213 (2013).

    ADS  Google Scholar 

  46. Kohno, M., Starykh, O. A. & Balents, L. Spinons and triplons in spatially anisotropic frustrated antiferromagnets. Nature Phys. 3, 790–795 (2007).

    ADS  Google Scholar 

  47. 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  Google Scholar 

  48. Gros, C. Superconductivity in correlated wave functions. Phys. Rev. B 38, 931–934 (1988).

    ADS  Google Scholar 

  49. Gros, C. Physics of projected wavefunctions. Ann. Phys. 189, 53–88 (1989).

    ADS  Google Scholar 

  50. Li, T. & Yang, F. Variational study of the neutron resonance mode in the cuprate superconductors. Phys. Rev. B 81, 214509 (2010).

    ADS  Google Scholar 

  51. Dmitriev, D. V., Krivnov, V. Y., Likhachev, V. N. & Ovchinnikov, A. A. Variation function with vortexes in the Heisenberg 2-dimensional antiferromagnetic model. Phys. Solid State 38, 397 (1996) [Translated: Fiz. Tverd. Tela 38, 397 (1996)]

    Google Scholar 

  52. Wen, X-G. & Lee, P. A. Theory of underdoped cuprates. Phys. Rev. Lett. 76, 503–506 (1996).

    ADS  Google Scholar 

  53. Nayak, C. Density-wave states of nonzero angular momentum. Phys. Rev. B 62, 4880–4889 (2000).

    ADS  Google Scholar 

  54. Lee, T. K. & Feng, S. Doping dependence of antiferromagnetism in La2CuO4: A numerical study based on a resonating-valence-bond state. Phys. Rev. B 38, 11809–11812 (1988).

    ADS  Google Scholar 

  55. Trivedi, N. & Ceperley, D. M. Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study. Phys. Rev. B 41, 4552–4569 (1990).

    ADS  Google Scholar 

  56. Calandra Buonaura, M. & Sorella, S. Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers. Phys. Rev. B 57, 11446–11456 (1998).

    ADS  Google Scholar 

  57. Liu, Z. & Manousakis, E. Variational calculations for the square-lattice quantum antiferromagnet. Phys. Rev. B 40, 11437–11440 (1989).

    ADS  Google Scholar 

  58. Franjic, F. & Sorella, S. Spin-wave wave function for quantum spin models. Prog. Theor. Phys. 97, 399–406 (1997).

    ADS  Google Scholar 

  59. Paramekanti, A., Randeria, M. & Trivedi, N. High-T c superconductors: A variational theory of the superconducting state. Phys. Rev. B 70, 054504 (2004).

    ADS  Google Scholar 

  60. Ivanov, D. A. Resonating-valence-bond structure of Gutzwiller-projected superconducting wave functions. Phys. Rev. B 74, 024525 (2006).

    ADS  Google Scholar 

  61. Syromyatnikov, A. V. Spectrum of short-wavelength magnons in a two-dimensional quantum Heisenberg antiferromagnet on a square lattice: Third-order expansion in 1/S. J. Phys. Condens. Matter 22, 216003 (2010).

    ADS  Google Scholar 

  62. Kim, C. et al. Observation of spin-charge separation in one-dimensional SrCuO2 . Phys. Rev. Lett. 77, 4054–4057 (1996).

    ADS  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge fruitful discussions with C. Broholm, L. P. Regnault, S. Sachdev and M. Zhitomirsky. Work in EPFL was supported by the Swiss National Science Foundation, the MPBH network, and European Research Council grant CONQUEST. The work of D.A.I. was supported by the Swiss National Foundation through the NCCR QSIT. Computational work was supported by the Swiss National Supercomputing Center (CSCS) under project ID s347. Work at Johns Hopkins University was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Material Sciences and Engineering under grant DE-FG02-08ER46544. N.B.C. was supported by the Danish Agency for Science, Technology and Innovation under DANSCATT.

Author information

Authors and Affiliations

  1. Laboratory for Quantum Magnetism, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Switzerland

    B. Dalla Piazza, M. Mourigal, G. J. Nilsen & H. M. Rønnow

  2. Institut Laue-Langevin, BP 156, F-38042 Grenoble Cedex 9, France

    M. Mourigal & M. Enderle

  3. Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA

    M. Mourigal

  4. Department of Physics, Technical University of Denmark (DTU), DK-2800 Kgs. Lyngby, Denmark

    N. B. Christensen

  5. Laboratory for Neutron Scattering, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

    N. B. Christensen & P. Tregenna-Piggott

  6. Department of Chemistry, University of Edinburgh, Edinburgh EH9 3JJ, UK

    G. J. Nilsen

  7. ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 OQX, UK

    T. G. Perring

  8. London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, London WC1H 0AH, UK

    D. F. McMorrow

  9. Institute for Theoretical Physics, ETH Zürich, CH-8093 Zürich, Switzerland

    D. A. Ivanov

  10. Institute for Theoretical Physics, University of Zürich, CH-8057 Zürich, Switzerland

    D. A. Ivanov

  11. RIKEN Centre for Emergent Matter Science (CEMS), Wako 351-0198, Japan

    H. M. Rønnow

Authors
  1. B. Dalla Piazza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mourigal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. B. Christensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. J. Nilsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. P. Tregenna-Piggott
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. T. G. Perring
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. M. Enderle
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. D. F. McMorrow
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. D. A. Ivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. H. M. Rønnow
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

B.D.P. and D.A.I. performed the theoretical work. B.D.P. wrote and ran the numerical calculations. M.M., N.B.C., M.E. and T.G.P. performed the experiments. G.J.N., P.T-P. and N.B.C. grew the samples. M.M. analysed the data guided by M.E., N.B.C. and H.M.R. B.D.P., M.M., D.A.I. and H.M.R. wrote the paper with contributions from all co-authors. D.F.M., D.A.I. and H.M.R. supervised the project.

Corresponding authors

Correspondence to B. Dalla Piazza or M. Mourigal.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 1051 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dalla Piazza, B., Mourigal, M., Christensen, N. et al. Fractional excitations in the square-lattice quantum antiferromagnet. Nature Phys 11, 62–68 (2015). https://doi.org/10.1038/nphys3172

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys3172

This article is cited by

Search

Advanced 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