Engineering Biquadratic Interactions in Spin-1 Chains by Spin-1/2 Spacers

TL;DR

The paper addresses how to realize and control unconventional biquadratic interactions in spin-1 chains by inserting pairs of spin-1/2 spacers. Through a hybrid spin-1/$\tfrac{1}{2}$ model and fourth-order perturbation theory, the authors derive an effective BLBQ Hamiltonian with tunable $\beta_{\rm eff}$ and demonstrate, via DMRG, a transition from Heisenberg-like behavior toward AKLT-like topological order while preserving a finite gap and hidden antiferromagnetic order. They identify a quantum phase transition in the large-$J_2/J_3$ regime, detectable via entanglement-spectrum changes without gap closing, between two spin-liquid-like phases. Atomistic modeling of nanographene flakes suggests realistic realizations with $J_2/J_3\approx0.5$, enabling experimental access to AKLT-like states in bottom-up synthesized quantum simulators. Overall, the work offers a blueprint for engineering and tuning unconventional spin interactions in solid-state nanostructures with implications for quantum simulation and topological quantum matter.

Abstract

Low-dimensional quantum systems host a variety of exotic states, such as symmetry-protected topological ground states in spin-1 Haldane chains. Real-world realizations of such states could serve as practical quantum simulators for quantum phases if the interactions can be controlled. However, many proposed models, such as the AKLT state, require unconventional forms of spin interactions beyond standard Heisenberg terms, which do not naturally emerge from microscopic (Coulomb) interactions. Here, we demonstrate a general strategy to induce a biquadratic term between two spin-1 sites and to tune its strength $β$ by placing pairs of spin-1/2 spacers in between them. $β$ is controlled by the ratio between Heisenberg couplings to and in between the spacer spins. Increasing this ratio increases the magnitude of $β$ and decreases the correlation length of edge states, but at a critical value of the ratio, we observe a quantum phase transition between two spin-liquid phases with hidden antiferromagnetic order. Detailed atomistic calculations reveal that chains of nanographene flakes with 22 and 13 atoms, respectively, which could be realized by state-of-the-art bottom-up growth technology, yield precisely the couplings required to approach the AKLT state. These findings deliver a blueprint for engineering unconventional interactions in bottom-up synthesized quantum simulators.

Figures (9)

• Figure 1: (a) Atomistic nanographene chain, where larger red triangles host localized spin-1 moments and smaller blue triangles host spin-$\tfrac{1}{2}$ moments. (b) Effective spin model with antiferromagnetic exchange couplings $J_{2}$ and $J_{3}$. (c) Ground-state representations of the effective spin-1 chain for $J_{2} \ll J_{3}$, $J_{2} \approx J_{3}$, and and $J_{2} \gg J_{3}$. Red and blue balls represent effective spin-$\tfrac{1}{2}$ objects; black circles denote ferromagnetic coupling into spin-1 units; solid blue lines indicate singlet bonds; and dotted blue lines represent mixed singlet--triplet bonds.
• Figure 2: (a), Low-energy spectrum of the hybrid spin-1/spin-$\frac{1}{2}$ chain for a dimer with $N_s=2$, i.e. a spin-1-$\frac{1}{2}$-$\frac{1}{2}$-1 configuration, as a function of $J_2/J_3$, comparing exact diagonalization results (black) with second-order (red dashed) and fourth-order (blue dotted) perturbation theory. The three lowest eigenstates are shown, corresponding to the effective spin-1 manifold. (b), Effective interaction parameter $\beta_{\text{eff}}$ for the hybrid chain, extracted from fitting the spectrum to the 2-site bilinear-biquadratic (BLBQ) from Eq. \ref{['eq:BLBQ']}, plotted as a function of $J_2/J_3$. The fourth-order perturbative result (blue circles) agrees well with the exact result (black) at small $J_2/J_3$, but deviates as spacer triplet excitations enter the low-energy manifold.
• Figure 3: DMRG results for the hybrid spin-1/-$\frac{1}{2}$ model with $N_{\rm s}=2$. (a) Low-energy spectrum as a function of the chain length for the Hamiltonian given by Eq. (\ref{['eq: SpinModel']}) with $J_2/J_3 \sim 0.5$. The ground state is formed by total spin states $S=0$ and $S=1$, and is degenerate in the thermodynamic limit. The inset shows the Haldane gap converging to a finite value in the thermodynamic limit. (b) The spin density of the triplet (with $S_z=1$) for $L=26$. An index $l=3(m-1)+i$, where $m$ is a unit cell index and $i=0,1,2$, with $i=0$ for a spin-1 and $i=1,2$ for spin-$\frac{1}{2}$'s. (c) The correlation length as a function of $J_2/J_3$ for $L=60$. The inset shows the correlation length of the BLBQ Hamiltonian (Eq. \ref{['eq:BLBQ']}) as a function of $\beta$. (d) The effective $\beta_{\mathrm{eff}}$ obtained by matching the correlation length of the hybrid spin-1/-$\frac{1}{2}$ model to that of the BLBQ model.
• Figure 4: Panel (a) shows the evolution of the excitation gap $E_{\mathrm{gap}}$ (blue, left axis) and the SOP $\mathcal{O}^z$ (orange, right axis), as a function of $J_2/J_3$, with the green dotted line denoting the value of the SOP for the AKLT state. The entanglement spectrum as a function of the coupling ratio $J_2/J_3$ is shown for two different reduced matrices: (b) the separation of the two regions is defined by cutting between a spin-1 (red circle) and a spin-1/2 (blue circle) site, while in (c) the separation is defined by cutting between the two spin-1/2 sites. The separation of the system into two subsystems is shown schematically.
• Figure 5: (a) Hybrid spin-1/-$\frac{1}{2}$ dimer made of the atomistic alternating nanographene spin-1's and two spin 1/2's, $N_S=2$. The balls represent carbon atoms, with the line connecting carbon atoms representing nearest-neighbor hopping. We highlight the localization of the effective spin of these nanographenes along the edge and show that each nanographene can be represented by an effective spin. In (b), we show the low-energy many-body spectrum of a dimer labelled by total spin $S$ for different theoretical models with $N_S= 2$. (c) Low-energy many-body spectrum from the C3 Fermionic model for alternating nanographene spin dimers with increasing spin-1/2 buffers $N_s$.