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.

Observation of fractional edge excitations in nanographene spin chains

Nature volume 598, pages 287–292 (2021)Cite this article

Subjects

A Publisher Correction to this article was published on 03 November 2021

This article has been updated

Abstract

Fractionalization is a phenomenon in which strong interactions in a quantum system drive the emergence of excitations with quantum numbers that are absent in the building blocks. Outstanding examples are excitations with charge e/3 in the fractional quantum Hall effect1,2, solitons in one-dimensional conducting polymers3,4 and Majorana states in topological superconductors5. Fractionalization is also predicted to manifest itself in low-dimensional quantum magnets, such as one-dimensional antiferromagnetic S = 1 chains. The fundamental features of this system are gapped excitations in the bulk6 and, remarkably, S = 1/2 edge states at the chain termini7,8,9, leading to a four-fold degenerate ground state that reflects the underlying symmetry-protected topological order10,11. Here, we use on-surface synthesis12 to fabricate one-dimensional spin chains that contain the S = 1 polycyclic aromatic hydrocarbon triangulene as the building block. Using scanning tunnelling microscopy and spectroscopy at 4.5 K, we probe length-dependent magnetic excitations at the atomic scale in both open-ended and cyclic spin chains, and directly observe gapped spin excitations and fractional edge states therein. Exact diagonalization calculations provide conclusive evidence that the spin chains are described by the S = 1 bilinear-biquadratic Hamiltonian in the Haldane symmetry-protected topological phase. Our results open a bottom-up approach to study strongly correlated phases in purely organic materials, with the potential for the realization of measurement-based quantum computation13.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

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

Fig. 1: On-surface synthesis of triangulene spin chains and observation of zero-energy edge excitations.
Fig. 2: The valence bond solid picture and theoretical calculations of spin excitations in open-ended triangulene spin chains.
Fig. 3: Magnetic excitations in selected open-ended triangulene spin chains and comparison with the bilinear-biquadratic model.
Fig. 4: Magnetic excitations in N = 6 and 13 cyclic triangulene spin chains and comparison with the bilinear-biquadratic model.

Data availability

The data that support the findings of this study are available at the Materials Cloud platform (https://doi.org/10.24435/materialscloud:e8-aq).

Code availability

The custom-designed Python codes that were used for solving the bilinear-biquadratic spin Hamiltonian by exact diagonalization are available on the GitHub repository (https://github.com/GCatarina/ED_BLBQ). All other codes are available from J.F.R. (joaquin.fernandez-rossier@inl.int) upon reasonable request.

Change history

References

  1. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  3. Jackiw, R. & Rebbi, C. Solitons with fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976).

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  5. Kitaev, A. Unpaired Majorana fermions in quantum wires. Phys.-Uspekhi 44, 131–136 (2001).

    Article  ADS  Google Scholar 

  6. Haldane, F. D. M. Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983).

    Article  MathSciNet  Google Scholar 

  7. Affleck, I., Kennedy, T., Lieb, E. H. & Tasaki, H. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799–802 (1987).

    Article  ADS  CAS  PubMed  Google Scholar 

  8. Kennedy, T. Exact diagonalisations of open spin-1 chains. J. Phys. Condens. Matter 2, 5737–5745 (1990).

    Article  ADS  Google Scholar 

  9. White, S. R. & Huse, D. A. Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S=1 Heisenberg chain. Phys. Rev. B 48, 3844–3852 (1993).

    Article  ADS  CAS  Google Scholar 

  10. Gu, Z.-C. & Wen, X.-G. Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B 80, 155131 (2009).

    Article  ADS  Google Scholar 

  11. Pollmann, F., Berg, E., Turner, A. M. & Oshikawa, M. Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B 85, 075125 (2012).

    Article  ADS  Google Scholar 

  12. Clair, S. & de Oteyza, D. G. Controlling a chemical coupling reaction on a surface: tools and strategies for on-surface synthesis. Chem. Rev. 119, 4717–4776 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Wei, T.-C., Affleck, I. & Raussendorf, R. Two-dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation. Phys. Rev. A 86, 032328 (2012).

    Article  ADS  Google Scholar 

  14. Bethe, H. Zur Theorie der Metalle. Z. Physik 71, 205–226 (1931).

    Article  ADS  CAS  MATH  Google Scholar 

  15. Renard, J.-P., Regnault, L.-P. & Verdaguer, M. in Magnetism: Molecules to Materials I: Models and Experiments (eds. Miller, J. S. & Drillon, M.) 49–93 (John Wiley & Sons, 2001).

  16. Soe, W.-H., Manzano, C., De Sarkar, A., Chandrasekhar, N. & Joachim, C. Direct observation of molecular orbitals of pentacene physisorbed on Au(111) by scanning tunneling microscope. Phys. Rev. Lett. 102, 176102 (2009).

    Article  ADS  PubMed  Google Scholar 

  17. Hirjibehedin, C. F., Lutz, C. P. & Heinrich, A. J. Spin coupling in engineered atomic structures. Science 312, 1021–1024 (2006).

    Article  ADS  CAS  PubMed  Google Scholar 

  18. Choi, D.-J. et al. Colloquium: atomic spin chains on surfaces. Rev. Mod. Phys. 91, 041001 (2019).

    Article  ADS  CAS  Google Scholar 

  19. Toskovic, R. et al. Atomic spin-chain realization of a model for quantum criticality. Nat. Phys. 12, 656–660 (2016).

    Article  CAS  Google Scholar 

  20. Yang, K. et al. Probing resonating valence bond states in artificial quantum magnets. Nat. Commun. 12, 993 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  21. Delgado, F., Batista, C. D. & Fernández-Rossier, J. Local probe of fractional edge states of S = 1 Heisenberg spin chains. Phys. Rev. Lett. 111, 167201 (2013).

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Lieb, E. H. Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201–1204 (1989).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  23. Fernández-Rossier, J. & Palacios, J. J. Magnetism in graphene nanoislands. Phys. Rev. Lett. 99, 177204 (2007).

    Article  ADS  PubMed  Google Scholar 

  24. Clar, E. & Stewart, D. G. Aromatic hydrocarbons. LXV. Triangulene derivatives1. J. Am. Chem. Soc. 75, 2667–2672 (1953).

    Article  CAS  Google Scholar 

  25. Goto, K. et al. A stable neutral hydrocarbon radical:  synthesis, crystal structure, and physical properties of 2,5,8-tri-tert-butyl-phenalenyl. J. Am. Chem. Soc. 121, 1619–1620 (1999).

    Article  CAS  Google Scholar 

  26. Inoue, J. et al. The first detection of a Clar’s hydrocarbon, 2,6,10-tri-tert-butyltriangulene: a ground-state triplet of non-Kekulé polynuclear benzenoid hydrocarbon. J. Am. Chem. Soc. 123, 12702–12703 (2001).

    Article  CAS  PubMed  Google Scholar 

  27. Pavliček, N. et al. Synthesis and characterization of triangulene. Nat. Nanotechnol. 12, 308–311 (2017).

    Article  ADS  PubMed  Google Scholar 

  28. Mishra, S. et al. Synthesis and characterization of π-extended triangulene. J. Am. Chem. Soc. 141, 10621–10625 (2019).

    Article  CAS  PubMed  Google Scholar 

  29. Su, J. et al. Atomically precise bottom-up synthesis of π-extended [5]triangulene. Sci. Adv. 5, eaav7717 (2019).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  30. Mishra, S. et al. Synthesis and characterization of [7]triangulene. Nanoscale 13, 1624–1628 (2021).

    Article  CAS  PubMed  Google Scholar 

  31. Mishra, S. et al. Collective all-carbon magnetism in triangulene dimers. Angew. Chem. Int. Ed. 59, 12041–12047 (2020).

    Article  CAS  Google Scholar 

  32. Lado, J. L. & Fernández-Rossier, J. Magnetic edge anisotropy in graphenelike honeycomb crystals. Phys. Rev. Lett. 113, 027203 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  33. Ternes, M., Heinrich, A. J. & Schneider, W.-D. Spectroscopic manifestations of the Kondo effect on single adatoms. J. Phys. Condens. Matter 21, 053001 (2008).

    Article  ADS  PubMed  Google Scholar 

  34. Li, J. et al. Single spin localization and manipulation in graphene open-shell nanostructures. Nat. Commun. 10, 200 (2019).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  35. Mishra, S. et al. Topological frustration induces unconventional magnetism in a nanographene. Nat. Nanotechnol. 15, 22–28 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  36. Ortiz, R. & Fernández-Rossier, J. Probing local moments in nanographenes with electron tunneling spectroscopy. Progr. Surf. Sci. 95, 100595 (2020).

    Article  CAS  Google Scholar 

  37. Oberg, J. C. et al. Control of single-spin magnetic anisotropy by exchange coupling. Nat. Nanotechnol. 9, 64–68 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  38. Jacob, D., Ortiz, R. & Fernández-Rossier, J. Renormalization of spin excitations and Kondo effect in open-shell nanographenes. Phys. Rev. B 104, 075404 (2021).

    Article  ADS  CAS  Google Scholar 

  39. Li, J. et al. Uncovering the triplet ground state of triangular graphene nanoflakes engineered with atomic precision on a metal surface. Phys. Rev. Lett. 124, 177201 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  40. Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2016).

    Article  ADS  PubMed  Google Scholar 

  41. Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).

    Article  ADS  Google Scholar 

  42. Hieulle, J. et al. On-surface synthesis and collective spin excitations of a triangulene-based nanostar. Angew. Chem. Int. Ed. https://doi.org/10.1002/anie.202108301 (2021).

  43. Giannozzi, P. et al. Quantum ESPRESSO toward the exascale. J. Chem. Phys. 152, 154105 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  44. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  ADS  CAS  PubMed  Google Scholar 

  45. Lejaeghere, K. et al. Reproducibility in density functional theory calculations of solids. Science 351, aad3000 (2016).

    Article  PubMed  Google Scholar 

  46. Hutter, J., Iannuzzi, M., Schiffmann, F. & VandeVondele, J. CP2K: atomistic simulations of condensed matter systems. Wiley Interdiscip. Rev. Comput. Mol. Sci. 4, 15–25 (2014).

    Article  CAS  Google Scholar 

  47. Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010).

    Article  ADS  PubMed  Google Scholar 

  48. Goedecker, S., Teter, M. & Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B 54, 1703–1710 (1996).

    Article  ADS  CAS  Google Scholar 

  49. VandeVondele, J. & Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys. 127, 114105 (2007).

    Article  ADS  PubMed  Google Scholar 

  50. Wilhelm, J., Del Ben, M. & Hutter, J. GW in the Gaussian and plane waves scheme with application to linear acenes. J. Chem. Theory Comput. 12, 3623–3635 (2016).

    Article  CAS  PubMed  Google Scholar 

  51. Neaton, J. B., Hybertsen, M. S. & Louie, S. G. Renormalization of molecular electronic levels at metal-molecule interfaces. Phys. Rev. Lett. 97, 216405 (2006).

    Article  ADS  CAS  PubMed  Google Scholar 

  52. Kharche, N. & Meunier, V. Width and crystal orientation dependent band gap renormalization in substrate-supported graphene nanoribbons. J. Phys. Chem. Lett. 7, 1526–1533 (2016).

    Article  CAS  PubMed  Google Scholar 

  53. Yakutovich, A. V. et al. AiiDAlab – an ecosystem for developing, executing, and sharing scientific workflows. Comput. Mater. Sci. 188, 110165 (2021).

    Article  CAS  Google Scholar 

  54. Ortiz, R. et al. Exchange rules for diradical π-conjugated hydrocarbons. Nano Lett. 19, 5991–5997 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  55. Tran, V.-T., Saint-Martin, J., Dollfus, P. & Volz, S. Third nearest neighbor parameterized tight binding model for graphene nano-ribbons. AIP Adv. 7, 075212 (2017).

    Article  ADS  Google Scholar 

  56. Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor software library for tensor network calculations. Preprint at https://arxiv.org/abs/2007.14822 (2020).

  57. Weinberg, P. & Bukov, M. QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems part I: spin chains. SciPost Phys. 2, 003 (2017).

    Article  ADS  Google Scholar 

  58. Weinberg, P. & Bukov, M. QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: bosons, fermions and higher spins. SciPost Phys. 7, 020 (2019).

    Article  ADS  CAS  Google Scholar 

  59. Fernández-Rossier, J. Theory of single-spin inelastic tunneling spectroscopy. Phys. Rev. Lett. 102, 256802 (2009).

    Article  ADS  PubMed  Google Scholar 

  60. Spinelli, A., Bryant, B., Delgado, F., Fernández-Rossier, J. & Otte, A. F. Imaging of spin waves in atomically designed nanomagnets. Nat. Mater. 13, 782–785 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  61. Coleman, P. New approach to the mixed-valence problem. Phys. Rev. B 29, 3035–3044 (1984).

    Article  ADS  CAS  Google Scholar 

  62. Jacob, D. & Kurth, S. Many-body spectral functions from steady state density functional theory. Nano Lett. 18, 2086–2090 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  63. Jacob, D. Simulation of inelastic spin flip excitations and Kondo effect in STM spectroscopy of magnetic molecules on metal substrates. J. Phys. Condens. Matter 30, 354003 (2018).

    Article  PubMed  Google Scholar 

  64. Jacob, D. & Fernández-Rossier, J. Competition between quantum spin tunneling and Kondo effect. Eur. Phys. J. B 89, 210 (2016).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank O. Gröning and J. C. Sancho-García for fruitful discussions. This work was supported by the Swiss National Science Foundation (grant numbers 200020-182015 and IZLCZ2-170184), the NCCR MARVEL funded by the Swiss National Science Foundation (grant number 51NF40-182892), the European Union’s Horizon 2020 research and innovation program (grant number 881603, Graphene Flagship Core 3), the Office of Naval Research (N00014-18-1-2708), ERC Consolidator grant (T2DCP, grant number 819698), the German Research Foundation within the Cluster of Excellence Center for Advancing Electronics Dresden (cfaed) and EnhanceNano (grant number 391979941), the Basque Government (grant number IT1249-19), the Generalitat Valenciana (Prometeo2017/139), the Spanish Government (grant number PID2019-109539GB-C41), and the Portuguese FCT (grant number SFRH/BD/138806/2018). Computational support from the Swiss Supercomputing Center (CSCS) under project ID s904 is gratefully acknowledged.

Author information

Author notes

  1. Shantanu Mishra

    Present address: IBM Research—Zurich, Rüschlikon, Switzerland

  2. These authors contributed equally: Shantanu Mishra and Gonçalo Catarina

Authors and Affiliations

  1. Empa—Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

    Shantanu Mishra, Kristjan Eimre, Carlo A. Pignedoli, Pascal Ruffieux & Roman Fasel

  2. International Iberian Nanotechnology Laboratory, Braga, Portugal

    Gonçalo Catarina & Joaquín Fernández-Rossier

  3. University of Alicante, Sant Vicent del Raspeig, Spain

    Gonçalo Catarina & Ricardo Ortiz

  4. Technical University of Dresden, Dresden, Germany

    Fupeng Wu, Ji Ma & Xinliang Feng

  5. University of the Basque Country, San Sebastián, Spain

    David Jacob

  6. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

    David Jacob

  7. Max Planck Institute of Microstructure Physics, Halle, Germany

    Xinliang Feng

  8. University of Bern, Bern, Switzerland

    Roman Fasel

Authors
  1. Shantanu Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gonçalo Catarina
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fupeng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ricardo Ortiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. David Jacob
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Kristjan Eimre
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ji Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Carlo A. Pignedoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Xinliang Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Pascal Ruffieux
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Joaquín Fernández-Rossier
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. Roman Fasel
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

X.F., P.R. and R.F. conceived the project. F.W. and J.M. synthesized and characterized the precursor molecules. S.M. performed the on-surface synthesis, and STM and STS measurements. G.C., R.O. and J.F.R. performed the tight-binding, CAS, ED and DMRG calculations. D.J. performed the MOAM-NCA calculations. K.E. and C.A.P. performed the DFT and GW calculations. All authors contributed toward writing the manuscript.

Corresponding authors

Correspondence to Xinliang Feng, Pascal Ruffieux or Joaquín Fernández-Rossier.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Berthold Jäck, Yi Zhou and the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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 Fig. 1 Scanning tunnelling spectroscopy measurements of the frontier bands of triangulene spin chains.

a, b, dI/dV spectroscopy on TSCs with cis (a) and trans (b) intertriangulene bonding configurations (open feedback parameters: V = −1.5 V, I = 250 pA; Vrms = 16 mV). Acquisition positions are marked with filled circles in c, d. Irrespective of the bonding configuration, TSCs exhibit an electronic band gap of 1.6 eV. c, d, High-resolution STM images (top panels), and constant-current dI/dV maps of the valence (middle panels) and conduction (bottom panels) bands of cis (c) and trans (d) TSCs. Scanning parameters: V = −0.4 V, I = 250 pA (top and middle panels, c, d) and V = 1.1 V, I = 280 pA (bottom panels, c, d); Vrms = 30 mV. All measurements were performed with a CO functionalized tip.

Extended Data Fig. 2 Gas-phase density functional theory calculations on triangulene spin chains.

a, e, DFT band structure and density of states (DOS) plots of TSCs with cis (a) and trans (e) intertriangulene bonding configurations in their antiferromagnetic ground state. Energies E are given with respect to the vacuum level. A Gaussian broadening of 100 meV has been applied to the DOS plots. Note that spin up and spin down bands are energetically degenerate. b, f, Corresponding band structure plots around the frontier bands. k denotes the reciprocal lattice vector. The unit cells for the band structure calculations contain four and two triangulene units for cis and trans TSCs, respectively, with the lattice periodicities a = 30.0 Å (cis TSC) and 17.4 Å (trans TSC). The dashed lines indicate the middle of the band gap. The calculations reveal nearly dispersionless frontier bands due to a weak intertriangulene electronic hybridization. In addition, TSCs exhibit a band gap of 0.68 eV irrespective of the intertriangulene bonding configuration. c, g, Ground state spin density distributions for cis (c) and trans (g) TSCs. Spin up and spin down densities are denoted in blue and red, respectively. d, h, Local DOS maps of the valence (VB) and conduction (CB) bands of cis (d) and trans (h) TSCs. Spin density distributions and local DOS maps were calculated at a height of 3 Å above the TSCs.

Extended Data Fig. 3 Derivation of the bilinear-biquadratic model.

a, b, Schematic energy level diagram of N = 2 (a) and 3 (b) oTSCs for the Heisenberg, Hubbard and BLBQ models. Analytical expressions for the spin models are provided in the Supplementary Information (Supplementary Note 2). The Hubbard model is defined such that each triangulene unit is represented by a four-site lattice (c) and the many-body energy levels are computed with DMRG, taking t = −1.11 eV, t′ = −0.20 eV and U = 1.45|t|. The parameters of the BLBQ model (\(J\) = 18 meV and \(\beta \) = 0.09) are obtained by matching its excitation energies to those of the Hubbard model for the N = 2 TSC. c, Description of the four-site toy model with the intra- and intertriangulene hopping, t and t′, respectively, indicated. The coloured filled circles denote the two sublattices. d, e, Comparison of the excitation energies for an N = 3 oTSC computed with CAS(6,6) for the complete Hubbard model with t1 = −2.70 eV, t2 = 0 eV and t3 = −0.35 eV (d), and with DMRG for the four-site Hubbard model (e), as the atomic Hubbard U is varied. Dashed lines indicate the experimental spin excitation energies of 14 meV for N = 2 TSC (a) and, 11 and 35 meV for N = 3 oTSC (b, d, e). Note that the Heisenberg model fails to capture both the experimental spin excitation energies for the N = 3 oTSC (b), and the Hubbard model results for the N = 2 (a) and N = 3 (b) oTSCs.

Extended Data Fig. 4 Experimental and theoretical spectroscopic signatures of spin excitations in an N = 4 open-ended triangulene spin chain.

Comparison between experimental and theoretical (using the four-site Hubbard and BLBQ models) d2I/dV2 spectra of an N = 4 oTSC shows a good agreement in both the energies and the modulation of the spin spectral weight across the different units in the TSC. Numerals along the abscissa denote the unit number of the TSC. BLBQ model calculations are performed with two different Teff values for the tunnelling quasiparticle, which determine the linewidth of the d2I/dV2 profile. Model parameters are the same as in Extended Data Fig. 3.

Extended Data Fig. 5 Average magnetization for the first three Sz = +1 states of an N = 16 open-ended triangulene spin chain, obtained with the bilinear-biquadratic model.

Calculations were performed with \(J\) = 18 meV and \(\beta \) = 0.09. Orange filled circles denote the magnetization profile of the state with the lowest excitation energy E = 0.4 meV, much smaller than the theoretical Haldane gap (9 meV), and \(|S,{S}_{z}\rangle =|1,+1\rangle \). The average magnetization is clearly the largest at the terminal units, and is strongly depleted at the central units, as expected for an edge state. Blue and green filled circles denote spin excitations with energies larger than the theoretical Haldane gap. Blue filled circles correspond to a state with E = 12.1 meV and \(|S,{S}_{z}\rangle =|1,+1\rangle \), where the magnetization profile forms a nodeless standing wave with maximum average magnetization at the central units. This can be identified as a spin wave state, except for the minor upturn at the terminal units. Green filled circles are associated to a state with E = 11.6 meV and \(|S,{S}_{z}\rangle =|2,+1\rangle \), where the average magnetization shares similarities with both the edge and nodeless spin wave states.

Extended Data Fig. 6 Theoretical and experimental spin excitation energies of open-ended and cyclic triangulene spin chains.

a, Spin excitation energies calculated by ED of the BLBQ model (\(J\) = 18 meV and \(\beta \) = 0.09) for oTSCs with N = 2–16 (circles) and cTSCs with N = 5, 6, 12, 13, 14, 15 and 16 (crosses) up to 50 meV. The size of the symbols accounts for the spin spectral weight of the corresponding spin excitation. The lowest energy bulk excitation, as indicated for the N = 16 cTSC, converges to the Haldane gap (9 meV) with increasing N. b, Experimental spin excitation energies up to 50 meV for seventeen oTSCs with N between 2 and 20, and eight cTSCs with N = 5, 6, 12, 13, 14, 15, 16 and 47. The lowest energy bulk excitation, indicated for the N = 47 cTSC, converges to the Haldane gap (14 meV) with increasing N. Experimentally, starting from both N = 16 oTSC and cTSC, convergence to the Haldane gap is observed. Note the odd–even effect observed for the lowest energy excitation of cTSCs, seen both in theory and experiments.

Extended Data Fig. 7 Non-crossing approximation results for the multi-orbital Anderson model of an N = 3 open-ended triangulene spin chain (t1 = −2.70 eV, t2 = 0 eV, t3 = −0.35 eV and U = 1.90|t1|) coupled to the surface (Γ/π = 13 meV).

a, Total spectral function of CAS(6,6) at different temperatures T for the case of particle–hole symmetry. b, Orbital-resolved spectral function of CAS(6,6) for T = 4.64 K and for the particle–hole symmetric case. c, Detuning from particle–hole symmetry: total spectral function of CAS(6,6) for different values of δε and T = 4.64 K. d, Local spectral functions at T = 4.64 K for carbon sites of one of the outer triangulene units and the central triangulene unit (δε = 200 meV). The inset shows a sketch of the N = 3 oTSC with the two carbon sites marked with the corresponding coloured filled circles. The spectral functions in individual panels are offset vertically for visual clarity.

Supplementary information

Supplementary Information

Supplementary Figs. 1–49 and Supplementary Notes 1 and 2: additional STM and STS data, effect of extrinsic spin-orbit coupling on triangulenes, analytical solutions of the Heisenberg and BLBQ models, materials and methods in solution synthesis and characterization, solution synthetic procedures, and NMR spectroscopy and high-resolution mass spectrometry of chemical compounds.

Peer Review File

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mishra, S., Catarina, G., Wu, F. et al. Observation of fractional edge excitations in nanographene spin chains. Nature 598, 287–292 (2021). https://doi.org/10.1038/s41586-021-03842-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-021-03842-3

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

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