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arXiv:2410.21424 · PRX Quantum

Spin-1 Haldane Phase in a Chain of Rydberg Atoms

Johannes Mögerle, Katharina Brechtelsbauer, A. T. Gea-Caballero, Javier Prior, Gabriel Emperauger, Guillaume Bornet, Cheng Chen, Thierry Lahaye, Antoine Browaeys, Hans Peter Büchler

Published May 19, 2025• 16 citations
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Abstract

We present a protocol to implement a spin-1 chain in Rydberg systems using three Rydberg states close to a Förster resonance. In addition to dipole-dipole interactions, strong van der Waals interactions naturally appear due to the presence of the Förster resonance and give rise to a highly tunable Hamiltonian. The resulting phase diagram is studied using the infinite density-matrix renormalization group and reveals a highly robust Haldane phase—a prime example of a symmetry-protected topological phase. We find experimentally accessible parameters to probe the Haldane phase in current Rydberg systems, and demonstrate an efficient adiabatic preparation scheme. This paves the way to probe the remarkable properties of spin fractionalization in the Haldane phase.

DOI
10.1103/prxquantum.6.020332

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We present a protocol to implement a spin-1 chain in Rydberg systems using three Rydberg states close to a Förster resonance. In addition to dipole-dipole interactions, strong van der Waals interactions naturally appear due to the presence of the Förster resonance and give rise to a highly tunable Hamiltonian. The resulting phase diagram is studied using the infinite density-matrix renormalization group and reveals a highly robust Haldane phase—a prime example of a symmetry-protected topological phase. We find experimentally accessible parameters to probe the Haldane phase in current Rydberg systems, and demonstrate an efficient adiabatic preparation scheme. This paves the way to probe the remarkable properties of spin fractionalization in the Haldane phase.
We provide the phase diagram of a two-parameter spin-1 chain that has a symmetry-protected topological (SPT) Haldane phase using computational algorithms along with tensor-network tools. We improve previous results, showing the existence of a new phase and new triple points. New striking features are the triple end of the Haldane phase and the complexity of phases bordering the Haldane phase in proximity---allowing moving to nearby non-SPT phases via small perturbations. These characteristics make the system, which appears in Rydberg excitons, e.g., in Cu2O{\mathrm{Cu}}_{2}\mathrm{O}Cu2​O, a prime candidate for applications.
Motzkin spin chain is a well-known mathematical model with connections to symmetry-protected topological phases, such as the Haldane phase, as well as to concepts in the AdS/CFT correspondence. They exhibit highly entangled ground states that violate the area law and are exceptionally difficult to simulate with conventional numerical methods. Numerical simulations of the Motzkin ground state become further challenging at large system sizes due to their high-dimensional spin structure, rendering it a natural test bed for quantum simulation with ultra-cold systems. Here, we propose a Rydberg-atom based quantum simulation scheme that effectively realizes Motzkin spins using an experimentally accessible set of parameters. We show that the resulting effective Motzkin ground state reproduces the characteristic entanglement scaling and the block-structure properties of the reduced density matrix associated with the ideal Motzkin state. Our results establish a pathway toward a concrete experimental realization of Motzkin spins beyond purely mathematical constructions, opening avenues for exploring other similar exotic non-area-law entangled phases in programmable Rydberg simulators.
The Motzkin spin chain is a well-known mathematical model with connections to symmetry-protected topological phases, such as the Haldane phase, as well as to concepts in the AdS/CFT correspondence. They exhibit highly entangled ground states that violate the area law and are exceptionally difficult to simulate with conventional numerical methods. Numerical simulations of the Motzkin ground state become further challenging at large system sizes due to their high-dimensional spin structure, rendering it a natural test bed for quantum simulation with ultracold systems. Here, we propose a Rydberg-atom based quantum simulation scheme that effectively realizes Motzkin spins using an experimentally accessible set of parameters. We show that the resulting effective Motzkin ground state reproduces the characteristic entanglement scaling and the block-structure properties of the reduced density matrix associated with the ideal Motzkin state. Our results establish a pathway toward a concrete experimental realization of Motzkin spins beyond purely mathematical constructions, opening avenues for exploring other similar exotic non-area-law entangled phases in programmable Rydberg simulators.
We study the ground-state phase diagram of the quantum spin-2 XXZ chain in the presence of on-site anisotropy using a matrix-product state based infinite-system density matrix renormalization group (iDMRG) algorithm. One of the interests in this system is in connecting the highly quantum-mechanical spin-1 phase diagram with the classical $S=\ensuremath{\infty}$ phase diagram. Several of the recent advances within DMRG make it possible to perform a detailed analysis of the whole phase diagram. We consider different types of on-site anisotropies, which allows us to establish the validity of the following statements: (1) the spin-2 model can be tuned into a phase, which is equivalent to the ``topologically nontrivial'' spin-1 Haldane phase, and (2) the spin-2 Haldane phase at the isotropic Heisenberg point is adiabatically connected to the ``trivial'' large-DDD phase, with a continuous change of the Hamiltonian parameters. Furthermore, we study the spin-3 XXZ chain to help explain the development of the classical phase diagram. We present details on how to use the iDMRG method to map out the phase diagram and include an extensive discussion of the numerical methods.
Dysprosium atoms have proven to be a promising platform for quantum simulation due to their strong magnetic moment and high tunability of interactions. In this work, we propose dysprosium atoms for simulating the one-dimensional spin-1 XXZ Heisenberg model, which is known to have a rich phase diagram including the famous Haldane phase. To realize the model, we make use of the strong dipolar exchange interactions that naturally occur in the ground state of dysprosium due to its large electron angular momentum of JJJ = 8. To implement spin-1 particles, we encode the spin degree of freedom into three Zeeman sublevels, which are energetically isolated by applying a magnetic field. Using the density-matrix renormalization group, we analyze the ground-state properties of the resulting effective model. We find that a chain of fermionic dysprosium atoms in a suitable magnetic field can form a Haldane state with the characteristic spin-1/2 edge modes. Furthermore, we discuss the use of AC Stark shifts and Raman-type schemes for bosonic dysprosium to isolate effective spin-1 systems and to increase the tunability of model parameters.
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.