Local dynamics and detection of topology in spin-1 chains

Abstract

Antiferromagnetic spin-1 chains host the celebrated symmetry protected topological Haldane phase, whose spin-1/2 edge states were evidenced in bulk by, e.g., Electron Spin Resonance (ESR). Recent success in assembling effective spin-1 antiferromagnetic chains from nanographene and porphyrin molecules opens the possibility of local, site-by-site, characterization. The nascent technique of combined ESR-STM is able to measure the spin dynamics with atomic real-space resolution, and could fully reveal and manipulate the spin-1/2 degree of freedom. In this work, we combine exact diagonalization and DMRG to investigate the local dynamic spin structure factor of the different phases of the bilinear-biquadratic Hamiltonian with single-ion anisotropy in 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 = 30 its energy should be first tuned by application of 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.

Figures (3)

• Figure 1: Energy properties of the spin $S=1$ bilinear-biquadratic Hamiltonian. (a) The phase diagram. (b) Energy spectrum vs. the total $S_z$ quantum number, for the system with $\theta=\pi/12$ and $N=10$. The energy splitting between the spin singlet and the spin triplet (green circles) is $\Omega=0.004$ while the energy separation between the four lowest energy states and excited ones is $\Delta=0.7$. (c) Dependence of the splitting $\Omega$ on the model, for a fixed chain length. (d) Exponential decay of $\Omega$ with chain length, for various models (defined by $\theta$) in the Haldane phase. The circle vs. triangle symbols indicate the numerical method used (see text).
• Figure 2: Frequency features of the atomically-resolved DSF $\chi_j(\omega)$. (a) Dependence on $\omega$ for a chain of $N=10$ sites in the Haldane phase, with $\Delta$ the Haldane bulkgap (units of $J$). The low-energy peak at the splitting energy $\Omega$ is characteristic of the topological edge states. We apply the usual Lorentzian broadening of the Dirac delta function (Eq. \ref{['eq:Susceptibility']}) with width $\eta=0.02$. (b) A zoom in on the peak at $\Omega$, as panel (a) but with $\eta=0.001$. The inset shows the $\chi_j(\omega)$ on a log scale. (c) As panel (a) but for a chain in the trivial dimerized phase. There are no additional peaks below the one at the bulkgap.
• Figure 3: (a) The signal $\chi_{j=1}^{MAX}$ vs $N$ for four selected models, obtained by DMRG simulations. In DMRG, for the two states with total $S_z=\pm1$ we fix a bond dimension of $50$ and we perform $300$ sweeps, while for finding the two lowest statez with $S_z=0$ we use a bond dimension of $1200$ and we perform $600$ sweeps. (b) Scaling of $\chi_j^{MAX}$ vs $j$ for the same values of $\theta$ as in panel (a), for a chain of $N=18$ sites.