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Observation of fractional edge excitations in nanographene spin chains

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.

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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.

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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.

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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.

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The authors declare no competing interests.

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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.

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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.

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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

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