Prediction of a topological phase transition in exchange alternating spin-1 nanographene chains

TL;DR

The paper addresses realizing and detecting a topological phase transition between the Haldane and dimerized phases in bond-alternating $S=1$ spin chains built from nanographene molecules. It extends the standard $S=1$ BAH/BLBQ model to include biquadratic exchange with relative strengths $\beta_1=B_1/J_1$ and $\beta_2=B_2/J_2$, mapping the critical dimerization $d_c$ as a function of $(\beta_1,\beta_2)$ via DMRG. By proposing two molecular realizations—the extended Clar's goblet and passivated $[4]$-triangulene—and deriving their exchange parameters with CI-CAS and DFT, it predicts which side of the transition each chain occupies. IETS simulations reveal clear, edge-sensitive signatures distinguishing the Haldane and dimerized phases, and the results show robustness to disorder, providing a practical route for experimental exploration of $1$-D topological magnetism in nanographene systems. The work thus connects microscopic molecular design to topological phase control and detection in artificial spin chains.

Abstract

The use of magnetic nanographenes as building blocks for artificial spin lattices is enabling the exploration of flagship model Hamiltonians of one-dimensional quantum magnetism with an unprecedented degree of control. The spin-1 Heisenberg model, incorporating both linear and quadratic exchange interactions, was first realized using [3]-triangulenes, where the hallmark Haldane phase with spin fractionalization was observed. Later, the spin-1/2 Heisenberg Hamiltonian with exchange alternation was realized with Clar's goblets, where two additional topological phases were identified. Here we show that spin-1 nanographenes can also be used to explore the topological phase transition between the Haldane phase and a dimerized phase predicted for spin-1 chains with bond-alternation. We first study how the boundary of the phase transition is modified by non-linear exchange, known to be present in spin-1 nanographenes, using density matrix renormalization group (DMRG). Combining multiconfigurational with first-principles calculations, we propose two candidates to realize different topological phases of the model: a recently synthesized extended Clar's goblet, and a passivated [4]-triangulene. Moreover, we show how these two phases can be identified experimentally using inelastic electron tunneling spectroscopy (IETS). This work paves the way for the experimental realization of these topological phases, which can be locally probed with scanning tunneling microscopy.

Figures (5)

• Figure 1: Nanographene spin chains.a, Chain of [3]-triangulenes which realizes the BLBQ Hamiltonian. A depiction of the ground state wave function in the Haldane phase is also shown mishra2021observation; b, c, $S=1/2$ BAH Hamiltonian, with $J_1 < J_2$ and $J_1 > J_2$ respectively, realized with Olympicenes zhao2024tunable, and the respective depictions of the ground state wave function. d, Phase diagram of the $S=1$ BAH Hamiltonian with bilinear ($J_1$, $J_2$) and biquadratic ($B_1$, $B_2$) exchange couplings. The nanographene chains which realize each phase are depicted, as well as their ground state wave functions.
• Figure 2: Critical dimerization map. Colormap of the critical dimerization value at which a topological phase transition occurs in the Hamiltonian of equation (\ref{['Eq: S=1 BAH']}) as a function of the relative strength of the biquadratic and bilinear exchanges, $\beta_{1,2} = B_{1,2} / J_{1,2}$ obtained with DMRG for chains with $N = 40$$S=1$ spins. The reference value is $d_c(\beta_{1,2}=0) \approx 0.277$.
• Figure 3: Molecular building blocks.a, $S=1$ nanographenes which can be used to realize chains described by the Hamiltonian of equation (\ref{['Eq: S=1 BAH']}), with the wave function of its zero modes (ZM) depicted below each molecule. Full (empty) circles mark the majority (minority) sublattice. Possible dimers obtained from these monomers are depicted on the right, where arrows indicate the passivated site. b, c, Single particle spectrum for the dimers of panel a. The dashed boxes highlight the active space used in the CI-CAS calculation for each system. d, e, Energies obtained by solving the Hubbard model in the CI-CAS approximation. The energy differences between the singlet and the triplet and quintuplet in the BLBQ model are also shown. The parameters $t = -2.7$eV, $t_3=t/10$ and $U = |t|$ were used.
• Figure 4: Simulation of IETS.a, b, c, d, Simulation of the $dI/dV$ maps for chains with 10 sites described with equation (\ref{['Eq: S=1 BAH']}), made from extended-Oly. and passivated [4]-triangulenes, using the exchanges of Table I. e, f, Pictorial representation of the spectrum of the spin Hamiltonian for the two types of chains with different terminations, i.e. chains starting with a strong or weak exchange. g, h, $dI/dV$ signal at zero bias across the chain obtained from panels a, b, c, d.
• Figure 5: Disorder dependence. Gap to the first excited state for a chain starting on a strong exchange as a function of the disorder parameter $p$ defined in the text. For each $p$ a chain with $N = 40$ spins is diagonalized $100$ times, with each gap being represented by a semi-transparent black circle. The blue line represents the average gap for each $p$. The dashed line shows the gap at the onset of the Haldane phase without disorder (changing $J_2$ such that $d = d_c = 0.44$) which is only finite due to finite-size effects.