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arXiv:2312.07147 · Physical review. B./Physical review. B

Local dynamics and detection of topology in spin-1 chains

Alfonso Maiellaro, Hervé Aubin, Andrej Mesaroš, Pascal Simon

Published Dec 23, 2024• 2 citations
arXivHTMLPDF

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). To fully reveal and manipulate the spin-$1/2$ 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 $N=30$ 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.

DOI
10.1103/physrevb.110.l220410

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We present electron spin resonance experiments in the one-dimensional antiferromagnetic S=1S=1S=1 spin chains NENP and NINO in pulsed magnetic fields up to 50 T. The measured field dependence of the quantum energy gap for B∣∣bB||bB∣∣b is analyzed using the exact diagonalization method and the density matrix renormalization group method. A staggered anisotropy term $(\ensuremath{-}{1)}^{i}{d(S}_{i}^{x}{S}_{i}^{z}{+S}_{i}^{z}{S}_{i}^{x})$ was considered in addition to a staggered field term $(\ensuremath{-}{1)}^{i}{S}_{i}^{x}{B}_{\mathrm{st}}.$ We show that the spin dynamics in high magnetic fields strongly depends on the orthorhombic anisotropy E.
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Abstract The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.
Both the Haldane spin-111 chain and dimerized chains of spin-1/21/21/2 exhibit topologically protected edge states that are robust against specific perturbations. Recently, such spin chains have been specifically assembled on surfaces and we investigate here the robustness of these edge states against coupling to the surface. Since no physical system can be considered perfectly isolated, it is crucial to examine whether topological robustness is maintained in the presence of environmental coupling. We apply exact diagonalization to a Lindblad master equation that couples an alternating Heisenberg spin chain based on spins 1/21/21/2 to a surface via various jump operators. The robustness of topological states is assessed via the time evolution of quantities such as the ground-state degeneracy, correlation function, entropy, and magnetization of edge states. We investigate chains built from dimers with antiferromagnetic and ferromagnetic intra-dimer coupling, which resemble Su-Schrieffer-Heeger and the Haldane models, resp., and assess the impact of zzz-axis anisotropy and longer-ranged couplings. Generally, we find that signatures of topological properties are more robust in Su-Schrieffer-Heeger-like chains than in Haldane-like chains.