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arXiv:2402.13590

Tunable topological phases in nanographene-based spin-1/2 alternating-exchange Heisenberg chains

Chenxiao Zhao, Gonçalo Catarina, Jinjiang Zhang, J. C. G. Henriques, Lin Yan Yang, Ji Ma, Xinliang Feng, Oliver Gröning, Pascal Ruffieux, Joaquín Fernández-Rossier, Román Fasel

Published Feb 21, 2024• 7 citations
arXivHTMLPDF

Abstract

Unlocking the potential of topological order within many-body spin systems has long been a central pursuit in the realm of quantum materials. Despite extensive efforts, the quest for a versatile platform enabling site-selective spin manipulation, essential for tuning and probing diverse topological phases, has persisted. Here, we utilize on-surface synthesis to construct spin-1/2 alternating-exchange Heisenberg (AH) chains[1] with antiferromagnetic couplings $J_1$ and $J_2$ by covalently linking Clar's goblets -- nanographenes each hosting two antiferromagnetically-coupled unpaired electrons[2]. Utilizing scanning tunneling microscopy, we exert atomic-scale control over the spin chain lengths, parities and exchange-coupling terminations, and probe their magnetic response by means of inelastic tunneling spectroscopy. Our investigation confirms the gapped nature of bulk excitations in the chains, known as triplons[3]. Besides, the triplon dispersion relation is successfully extracted from the spatial variation of tunneling spectral amplitudes. Furthermore, depending on the parity and termination of chains, we observe varying numbers of in-gap $S=1/2$ edge spins, enabling the determination of the degeneracy of distinct topological ground states in the thermodynamic limit-either 1, 2, or 4. By monitoring interactions between these edge spins, we identify the exponential decay of spin correlations. Our experimental findings, corroborated by theoretical calculations, present a phase-controlled many-body platform, opening promising avenues toward the development of spin-based quantum devices.

DOI
10.48550/arxiv.2402.13590

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Unlocking the potential of topological order within many-body spin systems has long been a central pursuit in the realm of quantum materials. Despite extensive efforts, the quest for a versatile platform enabling site-selective spin manipulation, essential for tuning and probing diverse topological phases, has persisted. Here, we utilize on-surface synthesis to construct spin-1/2 alternating-exchange Heisenberg (AH) chains[1] with antiferromagnetic couplings J1J_1J1​ and J2J_2J2​ by covalently linking Clar's goblets -- nanographenes each hosting two antiferromagnetically-coupled unpaired electrons[2]. Utilizing scanning tunneling microscopy, we exert atomic-scale control over the spin chain lengths, parities and exchange-coupling terminations, and probe their magnetic response by means of inelastic tunneling spectroscopy. Our investigation confirms the gapped nature of bulk excitations in the chains, known as triplons[3]. Besides, the triplon dispersion relation is successfully extracted from the spatial variation of tunneling spectral amplitudes. Furthermore, depending on the parity and termination of chains, we observe varying numbers of in-gap S=1/2S=1/2S=1/2 edge spins, enabling the determination of the degeneracy of distinct topological ground states in the thermodynamic limit-either 1, 2, or 4. By monitoring interactions between these edge spins, we identify the exponential decay of spin correlations. Our experimental findings, corroborated by theoretical calculations, present a phase-controlled many-body platform, opening promising avenues toward the development of spin-based quantum devices.
Magnetic nanographenes are emerging as versatile building blocks for artificial spin lattices, enabling the exploration of flagship one-dimensional quantum-magnetism models with unprecedented control. The spin-1 Heisenberg model, including bilinear and biquadratic exchange, was first realized using [3]-triangulenes, revealing the Haldane phase. More recently, Clar’s goblets enabled the spin-1/2 Heisenberg model with exchange alternation, uncovering additional topological phases. Here we show that spin-1 nanographenes can be used to explore bond-alternating chains both in the Haldane phase and beyond it, in a dimerized phase with emergent edge spin-1. We use density matrix renormalization group (DMRG) to analyze how biquadratic exchange, which is known to be large in spin-1 nanographenes, determines the phase transition boundary. Combining multiconfigurational and first-principles calculations, we identify two realistic candidates to realize these two different phases: the recently synthesized extended Clar’s goblet and a passivated [4]-triangulene. We demonstrate how to distinguish these phases experimentally using inelastic electron tunneling spectroscopy, paving the way for their observation.
Inelastic electron tunneling spectra (IETS) are evaluated for spin-1/2 Heisenberg chains showing different phases of their spin ordering. The spin ordering is controlled by the value of the two different Heisenberg couplings on the two sides of each of the chain's atoms (bond-alternating chains). The perfect anti-ferromagnetic phase, i.e. a unique exchange coupling, marks a topological quantum phase transition (TQPT) of the bond-alternating chain. Our calculations show that the TQPT is recognizable in the excited states of the chain and hence that IETS is in principle capable of discriminating the phases. We show that perfectly symmetric chains, such as closed rings mimicking infinite chains, yield the same spectra on both sides of the TQPT and IETS cannot reveal the nature of the spin phase. However, for finite size open chains, both sides of the TQPT are associated with different IETS spectra, especially on the edge atoms, thus outlining the transition.
Inferring microscopic Hamiltonians from experimental data is a central challenge in quantum materials and quantum simulation. In low-dimensional spin systems, exchange interactions are often assumed to be spatially uniform, despite structural and environmental inhomogeneities that can locally modify the coupling. Here, we leverage a local, length-independent machine learning methodology to reconstruct spatially modulated exchange interactions directly from inelastic scanning tunneling spectroscopy maps. We demonstrate this approach with nanographene spin chains, identifying both near-uniform and inhomogeneous regimes across the synthesized magnets. The reconstructed models quantitatively reproduce the experimental spectra and recover the correct scaling of the excitation gap with system size. Our results establish a general strategy to bridge local spectroscopic measurements with effective many-body Hamiltonians.
Antiferromagnetic spin-1 chains host the celebrated symmetry-protected topological Haldane phase, whose spin-1/21/21/2 edge states were evidenced in bulk by, e.g., electron spin resonance (ESR). To fully reveal and manipulate the spin-1/21/21/2 degree of freedom, one may build on the recent success in assembling effective spin-1 antiferromagnetic chains from nanographene and porphyrin molecules, and on the nascent technique of combined ESR and scanning tunneling microscopy, which is able to measure the spin dynamics site by site with atomic real-space resolution. In this Letter, we combine exact diagonalization and density matrix renormalization group to investigate the local dynamic spin structure factor of the different phases of the bilinear-biquadratic Hamiltonian with single-ion anisotropy in the presence of an external magnetic field. We find that the signature of the Haldane phase is a low-energy peak created by singlet-triplet transitions in the edge-state manifold. We predict that the signature peak is experimentally observable, although for chains of length above N=30N=30N=30 its energy should be first tuned by application of an external magnetic field. We fully characterize the peak in real space and energy, and further show its robustness to weak anisotropy and a relevant range of temperatures.
Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an even-Haldane phase of a dimerized S = 1 2 Heisenberg chain and a Haldane phase of an S = 1 chain realized when Hund coupling locks two S = 1 2 spins per monomer into S = 1. Density-functional theory places the active manifold deep in the Mott regime ( U / t > 100), justifying a spin-only Heisenberg description; a compact ( t, U ) → J mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon-Haldane features in S +– ( q,ω). Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.
We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from S=32S=\frac{3}{2}S=23​ to S=112S=\frac{11}{2}S=211​ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and 2S+12\frac{2S+1}{2}22S+1​ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with SSS and converge to a single point at J2J1=14\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}J1​J2​​=41​ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for S=12S=\frac{1}{2}S=21​, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ values.