## An interacting topological phase

Most topologically nontrivial systems discovered to date consist of noninteracting particles. Their properties can therefore be explained within a single-particle picture. De Léséleuc *et al.* engineered a topological phase of bosonic atoms in which interactions play a crucial role. The atoms, which were in highly excited Rydberg states, were held in an array of optical tweezers. Depending on the spatial arrangement of the tweezers, the dipole-dipole interactions between the atoms gave rise to two configurations with different topological properties.

*Science*, this issue p. 775

## Abstract

The concept of topological phases is a powerful framework for characterizing ground states of quantum many-body systems that goes beyond the paradigm of symmetry breaking. Topological phases can appear in condensed-matter systems naturally, whereas the implementation and study of such quantum many-body ground states in artificial matter require careful engineering. Here, we report the experimental realization of a symmetry-protected topological phase of interacting bosons in a one-dimensional lattice and demonstrate a robust ground state degeneracy attributed to protected zero-energy edge states. The experimental setup is based on atoms trapped in an array of optical tweezers and excited into Rydberg levels, which gives rise to hard-core bosons with an effective hopping generated by dipolar exchange interaction.

The concept of symmetry breaking can be used to characterize a variety of quantum phases. Topological quantum phases, however, do not fit into this paradigm. The most prominent example of a topological phase is the integer quantum Hall state with its robust edge states, giving rise to a quantized Hall conductance (*1*). The prediction of topological insulators (*2*) revealed another class of topological states of matter, denoted as symmetry-protected topological phases (SPTs). They occur in systems that display an excitation gap in the bulk (i.e., bulk insulators) and are invariant under a global symmetry. Their defining property is that the ground state at zero temperature cannot be transformed into a conventional insulating state upon deformations of the system that do not close the excitation gap or violate the symmetry. In particular, the zero-energy edge states are robust to any perturbation commuting with the symmetry operators.

SPT phases were first predicted and observed in materials in which the interaction between electrons can be effectively neglected (*3*, *4*). In this specific case of noninteracting fermions, SPT phases can be classified based on the action of the Hamiltonian on a single particle (*5*, *6*). Thus, the appearance of robust edge states is fully understood from the single-particle eigenstates. This notable simplification motivated experimental studies of topological phenomena at the single-particle level in classical systems such as coupled mechanical oscillators (*7*, *8*), radiofrequency circuits (*9*), optical devices (*10*–*13*), and plasmonic systems (*14*, *15*). Similar studies were conducted also in quantum systems such as ultracold atoms (*16*–*22*), where atomic interactions were either negligible (*16*–*21*) or used only to fill up Bloch bands (*22*), but the observed physics could be explained by a noninteracting model.

By contrast, the ground state of noninteracting bosons is a Bose-Einstein condensate. Therefore, bosonic SPT phases only appear in systems with interactions between the particles. Their classification requires the analysis of the quantum many-body ground state and is based on group cohomology (*23*, *24*). A notable example is the Haldane phase of the antiferromagnetic spin-1 chain (*25*), which has been experimentally observed in some solid-state materials (*26*, *27*). In the context of artificial matter, interactions have been introduced between bosonic particles in a system with a topological band structure (*28*, *29*), but studies were restricted to the two-body limit, still far from the many-body regime.

Here, we report the realization of a many-body SPT phase of interacting bosonic particles in an artificial system. Our setup is based on a staggered one-dimensional chain of Rydberg atoms, each restricted to a two-level system, resonantly coupled together by the dipolar interaction (*30*). We use this system to encode hard-core bosons—i.e., bosonic particles with infinite on-site interaction energy–coherently hopping along the chain. The system then realizes a bosonic version of the Su-Schrieffer-Heeger (SSH) model (*31*), which was originally formulated to describe fermionic particles hopping on a dimerized lattice. Our bosonic setup gives rise to two distinct phases of a bulk–half-filled chain: a trivial one with a single ground state and an excitation gap; and an SPT phase, with a fourfold ground state degeneracy due to edge states, and a bulk excitation gap. After an adiabatic preparation of a bulk–half-filled chain, we detect the ground state degeneracy in the topological phase and probe the zero-energy edge states. Furthermore, we experimentally demonstrate the robustness of the SPT phase under a perturbation respecting the protecting symmetry and show that this robustness cannot be explained at the single-particle level, a feature that distinguishes our system from noninteracting SPT phases.

## SSH model for hard-core bosons

The SSH model is formulated on a one-dimensional lattice with an even number of sites *N* and staggered hopping of particles (Fig. 1A). It is convenient to divide the lattice into two sublattices: *i*. In our realization, the nearest-neighbor hoppings are dominating—with J denoting the strong link and

At the single-particle level, the spectrum of the Hamiltonian in Eq. 1, shown in Fig. 2A, is obtained by diagonalizing the coupling matrix *5*, *6*), which notably constrains the hopping matrix *32*, *33*), polaritons in an array of micropillars (*14*), or mechanical granular chains (*34*).

We now turn to the properties of the quantum many-body ground state corresponding to a bulk–half-filled chain. For noninteracting fermions, the properties of the SSH chain follow from the Fermi sea picture based on the single-particle eigenstates: In the trivial configuration one obtains a single insulating ground state, whereas in the topological configuration, (i) an excitation gap appears in the bulk, and (ii) the ground state is fourfold degenerate as the two zero-energy edge modes can be either empty or occupied. For interacting bosons, the description of the many-body ground state is much more challenging. In the special case with only nearest-neighbor hoppings J and *35*). These properties are robust against the addition of extra terms to the Hamiltonian as long as the bulk gap remains finite and the protecting symmetry is respected, as required to describe a state in the SPT phase. Here, two symmetries have to be respected: the particle number conservation and the symmetry represented by the anti-unitary operator*23*, *24*) allows for a topological and a trivial phase for the symmetry described by Eq. 2. Moreover, in contrast to the chiral symmetry (protecting the single-particle properties and the fermionic SPT phase), next–nearest-neighbor hoppings *36*).

## Experimental realization with Rydberg atoms

Our realization of the bosonic SSH model is performed on an artificial structure with *N* = 14 sites of individually trapped ^{87}Rb atoms (*37*–*39*) (Fig. 1, C and D). The motion of the atoms is frozen during our experiment, occurring in a few microseconds. Coupling between the atoms is achieved, despite the large interatomic distance (~10 µm), by preparing the atoms in Rydberg states for which the dipole-dipole coupling is enhanced to a few megahertz (*30*, *40*).

We first prepare each atom in a Rydberg s-level, *35*). From there, the atom can be coherently transferred to a Rydberg p-level, *30*) (Fig. 1B); we use this to engineer the hopping matrix *35*). For each experimental run, we thus obtain the occupancy of each site, which is then averaged by repeating the experiment every ~0.3 s.

To implement the sub-lattice symmetry, we use the angular dependence of the dipolar coupling *41*). The dipolar interaction also gives rise to longer-range hopping on the order of ~0.2 MHz to third neighbors, which do not qualitatively change the properties of our system but are fully taken into account for quantitative comparison between theory and experiment.

## Single-particle spectrum

As a benchmark of our system, we first study the properties of a single particle in the chain. The single-particle spectrum is probed by microwave spectroscopy (Fig. 2A). Initializing the vacuum state *35*). In Fig. 2C, we quantitatively show the localization of edge modes at the boundary by postselecting experimental runs where at most one particle was created. We observe that the particle populates significantly only the leftmost and rightmost sites, and their second neighbors, as expected from the sublattice symmetry (edge states have support on one of the two subchains only) and in good agreement with a parameter-free calculation (black crosses) (*35*).

For any finite chain, the left and right edge modes hybridize to form symmetric and antisymmetric states with an energy difference *42*), and then let the system evolve freely. We show in Fig. 2E the experimental results for three chains of four, six, and eight sites. The energy *35*).

## Many-body ground state

We now turn to the study of the many-body system. In Fig. 3, B and C, we show the full energy spectrum in the trivial and topological configurations, calculated using exact diagonalization and ordered by increasing number of particles (*35*). In the trivial case, there is a single ground state at half-filling. By contrast, the topological configuration exhibits four degenerate ground states corresponding to the bulk half-filled, and which are characterized by zero, one, or two additional particles mainly residing at the edges. To prepare the ground state, we perform a microwave adiabatic sweep (Fig. 3A), where the final detuning *35*).

We present in Fig. 3, D and E, the dependence of the local density of particles on *35*)]. The local bulk properties are independent of the topology of the setup, but the situation is drastically different for the edge occupancy: In the trivial configuration, the edge sites behave as the bulk sites, whereas for the topological chain, the boundaries remain depleted for *35*).

We gain more insight into the many-body state by analyzing the correlations between particles, which we can measure because our detection scheme provides the full site-resolved particle distribution. In the strongly dimerized regime *23*, *43*) with

We now demonstrate the degeneracy of the many-body ground state in the topological phase and the bulk excitation gap. We first prepare a six-particle many-body ground state with the bulk at half-filling but empty edge states by an adiabatic sweep ending at

## Probing the protecting symmetry

We now probe the robustness of the fourfold ground state degeneracy to small perturbations that respect the protecting symmetry represented by *35*) by using simulations that include van der Waals interactions. We also checked that when we prepare the ground state with a half-filled bulk, i.e., whenever

The above experiment illustrates that, in contrast to a noninteracting SPT phase, the robustness of the bosonic many-body ground state at half-filling cannot be understood at the single-particle level. To gain an intuition for the differences between the SPT phase of noninteracting fermions and of hard-core bosons, we use the following simple picture. Considering only the three rightmost sites (the edge and a dimer), and taking the perturbative limit (*35*). This simplified model captures why the fermionic degeneracy is broken by the

## Discussion and outlook

We prepared quantum many-body states in two topologically different phases and observed four signatures of an SPT phase for interacting bosons: (i) a ground state degeneracy characterized by zero-energy edge states, (ii) an excitation gap in the bulk, (iii) a nonvanishing string order, and (iv) a robustness of these properties against perturbations respecting the protecting symmetry

The symmetry protecting our SPT phase defines the symmetry group *25*). We show numerically in (*35*) that our quantum many-body ground state is in the same SPT phase as the Haldane state, as they can be smoothly connected by adding hopping terms with complex amplitudes as well as additional Ising-type interactions

Our work demonstrates that Rydberg platforms, which combine flexible geometries and a natural access to the strongly correlated regime via the hard-core constraint, can explore unconventional quantum many-body states of matter. In combination with schemes for obtaining topological band structures in higher dimensions (*44*), it opens the prospect for the realization of long-range entangled states such as fractional Chern insulators with anyonic excitations.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**

**Funding:**This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no. 817842 (PASQuanS), from “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM, projects QUANTICA and XYLOS) and from the IXCORE-Fondation pour la Recherche. H.P.B. is supported by the European Union under the ERC consolidator grant SIRPOL (grant no. 681208).

**Author contributions:**S.d.L., V.L., P.S., and D.B. carried out the experimental measurements and data analysis; S.W. and N.L. performed the theoretical analysis. All work was supervised by H.P.B., T.L., and A.B. All authors contributed to the writing of the manuscript.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**The experimental data and theory code of this study are available at (

*45*).