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

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. Access through your institution Buy or subscribe This is a preview of

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* Log in * Learn about institutional subscriptions * Read our FAQs * Contact customer support SIMILAR CONTENT BEING VIEWED BY OTHERS TUNABLE TOPOLOGICAL PHASES IN NANOGRAPHENE-BASED SPIN-1/2

ALTERNATING-EXCHANGE HEISENBERG CHAINS Article 28 October 2024 SPIN EXCITATIONS IN NANOGRAPHENE-BASED ANTIFERROMAGNETIC SPIN-1/2 HEISENBERG CHAINS Article Open access 14 March 2025 HIGHLY

ENTANGLED POLYRADICAL NANOGRAPHENE WITH COEXISTING STRONG CORRELATION AND TOPOLOGICAL FRUSTRATION Article 19 February 2024 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.

([email protected]) upon reasonable request. CHANGE HISTORY * _ 03 NOVEMBER 2021 A Correction to this paper has been published: https://doi.org/10.1038/s41586-021-04150-6 _

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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 * Shantanu Mishra Present address: IBM Research—Zurich, Rüschlikon,

Switzerland * These authors contributed equally: Shantanu Mishra and Gonçalo Catarina AUTHORS AND AFFILIATIONS * Empa—Swiss Federal Laboratories for Materials Science and Technology,

Dübendorf, Switzerland Shantanu Mishra, Kristjan Eimre, Carlo A. Pignedoli, Pascal Ruffieux & Roman Fasel * International Iberian Nanotechnology Laboratory, Braga, Portugal Gonçalo

Catarina & Joaquín Fernández-Rossier * University of Alicante, Sant Vicent del Raspeig, Spain Gonçalo Catarina & Ricardo Ortiz * Technical University of Dresden, Dresden, Germany

Fupeng Wu, Ji Ma & Xinliang Feng * University of the Basque Country, San Sebastián, Spain David Jacob * IKERBASQUE, Basque Foundation for Science, Bilbao, Spain David Jacob * Max Planck

Institute of Microstructure Physics, Halle, Germany Xinliang Feng * University of Bern, Bern, Switzerland Roman Fasel Authors * Shantanu Mishra View author publications You can also search

for this author inPubMed Google Scholar * Gonçalo Catarina View author publications You can also search for this author inPubMed Google Scholar * Fupeng Wu View author publications You can

also search for this author inPubMed Google Scholar * Ricardo Ortiz View author publications You can also search for this author inPubMed Google Scholar * David Jacob View author

publications You can also search for this author inPubMed Google Scholar * Kristjan Eimre View author publications You can also search for this author inPubMed Google Scholar * Ji Ma View

author publications You can also search for this author inPubMed Google Scholar * Carlo A. Pignedoli View author publications You can also search for this author inPubMed Google Scholar *

Xinliang Feng View author publications You can also search for this author inPubMed Google Scholar * Pascal Ruffieux View author publications You can also search for this author inPubMed 

Google Scholar * Joaquín Fernández-Rossier View author publications You can also search for this author inPubMed Google Scholar * Roman Fasel View author publications You can also search for

this author inPubMed 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, d_I_/d_V_ spectroscopy on TSCs with _cis_ (A) and _trans_ (B) intertriangulene bonding configurations (open feedback parameters: _V_ = −1.5 V, _I_ = 

250 pA; _V_rms = 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 d_I_/d_V_ 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); _V_rms = 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 _t_1 = −2.70 eV, _t_2 = 0 eV and _t_3 = −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) d2_I_/d_V_2

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 _T_eff values for the tunnelling quasiparticle, which determine the linewidth of the d2_I_/d_V_2

profile. Model parameters are the same as in Extended Data Fig. 3. EXTENDED DATA FIG. 5 AVERAGE MAGNETIZATION FOR THE FIRST THREE _S__Z_ = +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 (_T_1 = −2.70 EV, _T_2 = 0 EV, _T_3 = −0.35 EV AND _U_ = 1.90|_T_1|) 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 δ_ε_ 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 (δ_ε_ = 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 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: 30 April 2021 * Accepted: 20 July 2021 *

Published: 13 October 2021 * Issue Date: 14 October 2021 * DOI: https://doi.org/10.1038/s41586-021-03842-3 SHARE THIS ARTICLE Anyone you share the following link with will be able to read

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