Topological phases of matter often share a deep connection between their bulk and edge properties8,9. In the case of the Haldane chain, the bulk exhibits a hidden antiferromagnetic (AFM) order characterized by AFM correlations interlaced with an arbitrary number of Sz = 0 elements, where Sz denotes the spin component in the z-direction. This pattern can only be revealed through non-local string correlations that are sensitive to the local spin states, which, however, require detection of the quantum many-body system with microscopic resolution. Although this was not possible in early experiments on spin-1 chains, evidence for a spin gap, as well as spin-1/2 localized edge states, was found using neutron scattering10,11 or electron resonance experiments12,13 while not directly probing this hidden order or spatially resolving the edge states. Recent developments in quantum simulations enable one to go beyond such solid-state bulk measurements by observing quantum many-body systems with single-site resolution14,15,16,17,18 and in a fully spin- and density-resolved way19,20. This provides a rich diagnostic tool to obtain a direct microscopic picture of the hidden order in experiments21,22. The power of this technique has also been demonstrated recently in a study that was able to reveal a symmetry-protected topological (SPT) phase in the hardcore boson Su–Schrieffer–Heeger (SSH) model using Rydberg atoms23. Here we expand on those results by realizing a finite-temperature version of the Haldane phase in a spin system with tuneable coupling strength, system size and controlled charge fluctuations. We show this by measuring both topological and trivial string order parameters.

An instructive way to engineer the Haldane phase in systems of spin-1/2 fermions is on the basis of the AKLT model4,24, in which a spin-1 particle is formed out of two spin-1/2 particles. Thus, spin-1/2 ladder systems emerge as an experimentally realizable platform for the Haldane phase. Whereas a natural implementation with spin-1 particles on individual rungs requires ferromagnetic rung couplings and antiferromagnetic leg couplings, a much wider variety of couplings in spin-1/2 quantum ladders feature the Haldane phase25,26. This includes the antiferromagnetic Heisenberg case, which we realize here as the strong-interaction limit of the Fermi–Hubbard model.

In our experiment, we prepare such ladders by adiabatically loading a spin-balanced mixture of the two lowest hyperfine states of 6Li into an engineered lattice potential (Methods). As illustrated in Fig. 1a, we realize four isolated two-leg ladders with a variable number (L) of unit cells (where L is therefore also equivalent to length), surrounded by a low-density bath of particles27. The unit cells are chosen to be either along the rungs of the ladders (vertical unit cell, Fig. 1b) or along the diagonals (diagonal unit cells, Fig. 1c). The edges of the ladders are then engineered to match the choice of unit cell: straight edges are chosen for vertical unit cells, whereas one site is blocked on each edge in the case of diagonal unit cells. The atoms in the lowest band of the optical lattice are well described by the Fermi–Hubbard model, with tunnelling energies, ({t}_{parallel }) (chain), ({t}_{perp }) (rung), and on-site interactions U. For half-filling and at strong (U/{t}_{parallel ,perp }approx 13), used throughout most of our experiments (see Methods for details), density fluctuations are suppressed and the spin ladder realizes the Heisenberg model28 with Hamiltonian:

$$hat{H}={J}_{parallel }sum _{begin{array}{c}xin [0,L)\ y=A,Bend{array}}{hat{{bf{S}}}}_{x,y}cdot {hat{{bf{S}}}}_{x+1,y}+{J}_{perp }sum _{xin [0,L)}{hat{{bf{S}}}}_{x,A}cdot {hat{{bf{S}}}}_{x,B}$$

(1)

Fig. 1: Probing topological phases in spin-1/2 ladders of cold atoms.
figure 1

a, Realization of tailored spin-1/2 ladders in a single plane of a 3D optical lattice with a potential shaped by a DMD. The dilute wings of the potential are well separated from the homogeneous ladder systems. Using quantum gas microscopy, we obtain fully spin- and density-resolved images of the system. The inset shows a single-shot fluorescence image of the prepared ladder without spin resolution. b, c, Connecting spin-1/2 ladders to integer-spin chains by grouping pairs of spins in unit cells. For diagonal unit cells (b) the AFM Heisenberg ladder adiabatically connects to the Haldane spin-1 chain showing spin-1/2 edge states and hidden long-range order (that is, AFM order interspersed with Sz = 0 unit cells). For vertical unit cells (c), the system is in the topologically trivial phase dominated by singlets on the rungs, forming a spin-0 chain. We adapt the edges of the system to match the respective unit cell, that is straight edges for vertical unit cells and tilted edges for diagonal unit cells, which we realize by blocking one site on each end. The energy spectra of the systems grouped by total magnetization Mz display gapped fourfold near-degenerate ground states for the topological configuration and a single ground state for the trivial one. Sketch for L = 7.

with positive leg and rung couplings, ({J}_{parallel ,perp }=4{t}_{parallel ,perp }^{2}/U) and the spin-1/2 operators ({hat{{bf{S}}}}_{x,y}) at site (x, y), with A, B denoting the two legs of the ladder.

The topological properties are most easily explained in the limit ({J}_{perp }gg {J}_{parallel }), where strong spin singlets form along the rungs and the system exhibits an energy gap of ({J}_{perp }). The behaviour on the edges of the ladder then depends on how the system is terminated, and therefore on which unit cells have been chosen. For diagonal unit cells (Fig. 1b), two unpaired spin-1/2 particles remain and the many-body system has a fourfold degeneracy that is only weakly lifted by an edge-to-edge coupling, which vanishes exponentially with system size (Supplementary Information). In the trivial case of vertical unit cells (Fig. 1c), all spins pair into singlets and the ground state is unique. These descriptions remain valid even for weaker ({J}_{perp }/{J}_{parallel }), where the singlet alignment may change between vertical and horizontal, but any line between two rungs cuts an even number of singlets29,30.

To make the analogy between the spin-1/2 ladder and the Haldane integer chain more apparent, we switch to a description in terms of total spin per kth unit cell, ({hat{{bf{S}}}}_{k}={hat{{bf{S}}}}_{k,{rm{A}}}+{hat{{bf{S}}}}_{k,{rm{B}}}), where the indices (A, B) indicate the two spin-1/2 particles in the same unit cell, making an integer spin. In the diagonal unit cell such a system shows a high (≥80%) triplet fraction26 (Supplementary Information). We note that this spin ladder can be adiabatically connected to a spin-1 chain by including ferromagnetic couplings within the unit cell25. However, having a high triplet fraction is not essential for having a well-defined Haldane phase, as both systems share the same universal SPT features26.

The defining property of the Haldane SPT phase is that it is an integer-spin chain with spin-1/2 edge modes: the bulk SO(3) symmetry is said to fractionalize into SU(2) symmetry at the edge. It has no spontaneous symmetry breaking and thus the spin correlation function (langle {hat{S}}_{k}^{z}{hat{S}}_{k+d}^{z}rangle ) is short range. Instead, the aforementioned symmetry fractionalization6,7 can be detected in the bulk using string order parameters3,31:

$${g}_{{mathscr{O}},U}(d)=langle {hat{{mathscr{O}}}}_{k}(mathop{prod }limits_{l=k+1}^{k+d-1}{hat{U}}_{l}){hat{{mathscr{O}}}}_{k+d}rangle $$

(2)

with an on-site symmetry ({hat{U}}_{l}) and endpoint operator ({hat{{mathscr{O}}}}_{k}), where l denotes the unit cell and d the string distance (Fig. 2 and Supplementary Information). This correlator probes the transformation behaviour of the bulk under a symmetry ({hat{U}}_{l}); for example, a spin rotation around the z axis by π, ({hat{R}}_{l}^{z}equiv {rm{exp }}({i}{rm{pi }}{hat{S}}_{l}^{z})). The pure-string correlator ({g}_{{mathbb{1}},{R}^{z}}(d)), where ({hat{{mathscr{O}}}}_{k}=1) and ({hat{U}}_{l}={hat{R}}_{l}^{z}), is non-zero for (dgg 1) if the edge does not have half-integer spins31. This is the case for the topologically trivial configuration but not for the Haldane phase, in which the symmetry is fractionalized. The spin-string operator ({g}_{{S}^{z},{R}^{z}}(d))(ref. 3), ({hat{{mathscr{O}}}}_{k}={hat{S}}_{k}^{z}), exhibits the opposite behaviour and is non-zero only in the Haldane phase (see Supplementary Information for details about the symmetries of the Haldane phase). Thus we can identify the Haldane phase by comparing the two string correlators ({g}_{{S}^{z},{R}^{z}}) and ({g}_{{mathbb{1}},{R}^{z}}), and observe opposite behaviour in the two different regimes.

Fig. 2: Trivial versus topological configurations.
figure 2

a, The atomic density distribution (hat{n}) of ladders with diagonal and vertical unit cells. b, The amplitudes of the spin-string correlator ({g}_{{S}^{z},{R}^{z}}) (green circles) and the string-only correlator ({g}_{{mathbb{1}},{R}^{z}}) (grey squares) observed as a function of the spin distance over d unit cells. The cartoon illustrates the unit cells, the total spin Sz per unit cell and the string correlators for a subsystem with d = 3. In the trivial configuration (rung unit cells), (|{g}_{{mathbb{1}},{R}^{z}}(d)|) is well above zero, whereas (|,{g}_{{S}^{z},{R}^{z}}(d)|) is rapidly vanishing at d > 1. By contrast, for the topological configuration (diagonal unit cells), (|,{g}_{{S}^{z},{R}^{z}}(d)|), shows a long-range correlation, whereas (|{g}_{{mathbb{1}},{R}^{z}}(d)|) is close to zero. In both cases, the two-point spin–spin correlation C(d) decays rapidly to zero as a function of the distance d (insets). The correlators ({g}_{{mathbb{1}},{R}^{z}},{g}_{{S}^{z},{R}^{z}}) and C(d) are evaluated for fixed total magnetization ({m}^{z})  = 0. c, Amplitudes of the rung- and inversion-averaged local magnetizations (|{m}^{z}(x)|) plotted as a function of position x along the chains for different ({m}^{z}). In the unbalanced spin sector of the topological configuration (({m}^{z}) = ±1), the result displays a localization of the excess spins at the edges, signalling the presence of edge states. All data were taken with ({J}_{perp }/{J}_{parallel }=1.3(2)). Error bars denote one standard error of the mean (s.e.m.) and are smaller than their marker size if not visible.

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Another perspective on ({g}_{{S}^{z},{R}^{z}}) can be gained by recognizing it as a normal two-point correlator at distance d, which ignores all spin-0 contributions along the way (‘squeezed space’22,32). In the underlying spin-1/2 system, this order stems from N − 1 consecutive rungs dominantly consisting of N − 1 singlets and two spin-1/2 states, which have a combined total spin of +1, 0 or −1.

To observe the characteristics of the SPT phase, we prepare a two-leg ladder of length L = 7 and ({J}_{perp }/{J}_{parallel }=1.3(2)) in both the topological and the trivial configuration. The tailored potential yields a homogeneous filling of the system with sharp boundaries (Fig. 2a), which is characterized by a remaining density variance over the system of 2 × 10−4. To focus on the spin physics, we select realizations with ({N}_{uparrow }+{N}_{downarrow }=2L) per ladder. Additionally, we exclude ladders with an excessive number of doublon–hole fluctuations and do not consider strings with odd atom numbers in the string or the endpoints of the correlator (Methods). We characterize the spin-balanced ladder systems (({M}^{z}equiv ({N}_{uparrow }-{N}_{downarrow })/2=0)) by evaluation of the string order parameters, as defined in equation (2). In the topological configuration, we observe fast decay of ({g}_{{S}^{z},{R}^{z}}) over a distance of approximately one site and a long-range correlation up to d = 6, with a final value of ({g}_{{S}^{z},{R}^{z}}simeq 0.1) (Fig. 2b). In contrast, for the trivial configuration, the correlation decays rapidly to zero as a function of the string correlator length. The opposite behaviour is seen for ({g}_{{mathbb{1}},{R}^{z}}(d)), demonstrating the hidden correlations expected for both phases.

Furthermore, the two-point spin correlation, (C(d)equiv {g}_{{S}^{z},{mathbb{1}}}(d)=)(langle {hat{S}}_{k}^{z}{hat{S}}_{k+d}^{z}rangle ,,) yields only the short-range AFM correlation characteristic for a gapped phase (see insets in Fig. 2b). It is interesting to note that at the largest distance in the topological case, C(d = 6) displays a clear (negative) correlation between the two edge spins, despite small correlations at shorter distances. This (classical) correlation confirms the existence of a non-magnetized bulk, such that spins on the edges of the system must be of opposite direction at global Mz = 0.

We probe the edges explicitly by measuring the amplitude of the local rung-averaged magnetization mz(x) as a function of rung position x for different sectors of the ladder magnetization Mz (Fig. 2c). In the case of an imbalanced spin mixture with Mz = ±1, we see that the two end sites exhibit on average a higher magnetization than the bulk rungs in the topological configuration. This is consistent with the bulk of the ground states of both phases forming a global singlet, and only the edges of the topological phase carrying an excess spin-1/2 without energy cost. The measured bulk magnetization can be attributed to finite-temperature effects (Supplementary Information).

The SPT phase is expected to be robust26 on variation of the ratio ({J}_{perp }/{J}_{parallel }), but maintains a finite gap in the system. We realize both the trivial and topological configuration with different ({t}_{perp }/{t}_{parallel }) at almost fixed U and study the string correlators at maximal distance (L − 1) versus ({J}_{perp }/{J}_{parallel }) (Fig. 3a). For the topological configuration with diagonal unit cells, we observe ({g}_{{mathbb{1}},{R}^{z}}(L-1)simeq 0) and (|{g}_{{S}^{z},{R}^{z}}| > 0) for all ({J}_{perp }/{J}_{parallel }) with a maximum around ({J}_{perp }/{J}_{parallel }simeq 1.3(2)) (that is, ({J}_{perp }/({J}_{perp }+{J}_{parallel })simeq 0.56(4))), whereas for the trivial case the role of the correlators is reversed. Both phases continuously connect in the limit of two disconnected chains at ({J}_{perp }=0). These observations demonstrate qualitatively all the key predictions of the antiferromagnetic spin-1/2 ladder at temperature T = 0 (ref. 26) and the strengths of the measured correlations are consistent with exact diagonalization (ED) calculations using an entropy per particle (S/N=(0.3-0.45),{k}_{{rm{B}}}) (shaded lines in Fig. 3a).

Fig. 3: Influence of spin-coupling strength on the string order parameters and the edge states.
figure 3

a, The two string order parameters, ({g}_{{S}^{z},{R}^{z}}) (green) and ({g}_{{mathbb{1}},{R}^{z}}) (grey), for both trivial (left) and topological (right) configurations measured as a function of ({J}_{perp }/({J}_{perp }+{J}_{parallel })). Both ({g}_{{S}^{z},{R}^{z}}) and ({g}_{{mathbb{1}},{R}^{z}}) stay finite in their respective phases and are largely consistent with zero in the other phase. The data were taken at a chain length of L = 5 except for one data point marked by a triangle at L = 7. Shaded curves are the ED results of the two order parameters at finite entropy per particle, S/N = (0.3−0.45) kB and L = 5. The inset shows the measured ({g}_{{S}^{z},{R}^{z}}) as a function of the chain length L at ({J}_{perp }/{J}_{parallel }=1.3(2)) (that is, ({J}_{perp }/({J}_{perp }+{J}_{parallel })=0.56(4))). The decay in the magnitude of the string order parameter with length is expected at finite temperatures in quantitative agreement with ED results (lines) at S/N ≈ 0.4 kB. b, Edge state localization at ({J}_{perp }/{J}_{parallel }=1.3(2)). In the ({m}^{z}) = ±1 spin sectors of the topological configuration, the unit cell local magnetization (|{m}^{z}(k)|) at chain position k shows excess magnetization localized at the edges for different lengths. The black line is a fit to our inversion-averaged data. c, The localization length ξ of the edge states increases with the leg coupling ({J}_{parallel }) but saturates at a value set by temperature and system size L = 5. Lines are ED results at S/N = 0.3 kB and 0 kB. The inset shows the independence of ξ with respect to L extracted from the plots in b, as well as ED results for S/N = 0.3 kB. Error bars denote one standard error of the mean (s.e.m) and are smaller than their marker size if not visible.

Source data

We reveal these features despite the finite temperature in our system, which would destroy the long-range hidden order in an infinite system. The total entropy in our system is, however, still low enough to yield a large fraction of realizations of the topological ground state. In larger systems, the total number of thermal excitations grows (at fixed entropy per particle) and the non-local correlator (|,{g}_{{S}^{z},{R}^{z}}(L-1)|) decreases (see inset of Fig. 3a), consistent with vanishing correlations in the thermodynamic limit, thus yielding a restriction on our system size at our level of experimental precision and entropy per particle (Supplementary Information). Finite size effects are explored in detail in the Supplementary Information. We note that the difference between the SPT phase and the trivial phase is here clearly shown by considering both ({g}_{{S}^{z},{R}^{z}}) and ({g}_{{mathbb{1}},{R}^{z}}).

To investigate the localization length of the edge states, we evaluate our data for ({m}^{z}) = ±1 and plot the local magnetization per unit cell ({m}^{z})(k) for different system sizes (Fig. 3b). Because of the singlets in the bulk, the excess spin is most likely to be found at the edges of the system. This spin partly polarizes the neighbouring sites antiferromagnetically, leading to an exponentially localized net magnetization with AFM substructure33. The data are well described by the fit function ({m}^{z}(k)={m}_{{rm{B}}}+{m}_{{rm{E}}}left({(-1)}^{k}{{rm{e}}}^{-k/xi }+{(-1)}^{L-k-1}{{rm{e}}}^{-(L-k-1)/xi }right)) with free bulk magnetization mB, edge magnetization mE and decay length ξ. In Fig. 3c, we show how this localization length ξ decreases as we approach the limit of rung singlets, ({J}_{perp }gg {J}_{parallel }). Comparison with ED lets us identify two regimes: at ({J}_{perp }gtrsim {J}_{parallel }), the measured ξ drops with larger ({J}_{perp }) and coincides with theory independent of temperature, whereas at low ({J}_{parallel }) thermal effects dominate, limiting the increase of ξ to three sites for our system (Supplementary Information).

Thus far, we have worked in the Mott limit where density fluctuations can be ignored, such that the spin Hamiltonian, equation (1), is a good effective description of the Fermi–Hubbard ladder. However, it is known that the Haldane SPT phase can be unstable to density fluctuations34,35,36. By reducing (U/{t}_{parallel }), the symmetry in the unit cell in the bulk changes from SO(3) to SU(2), as it now may contain both half-integer and integer total spin. This effectively removes the distinction between bulk and edge (Supplementary Information). This means that the edge mode and string order parameter are no longer topologically non-trivial, which is also manifested in the fact that the two phases can be adiabatically connected by tuning through a low-(U/{t}_{parallel }) regime if one breaks additional symmetries but preserves spin-rotation symmetry34,35,36. In particular, the above string orders lose their distinguishing power: ({g}_{{S}^{z},{R}^{z}}) and ({g}_{{mathbb{1}},{R}^{z}}) will both generically have long-range order away from the Mott limit34.

Intriguingly, despite the breakdown of the above symmetry argument and string order parameter, the Hubbard ladder (with diagonal unit cell) remains a non-trivial SPT phase due to its sublattice symmetry. This symmetry is a direct consequence of the ladder being bipartite (see Supplementary Information for details). It is simplest to see that this protects the SPT phase in the limit U = 0, where the two spin species decouple, such that our model reduces to two copies of the SSH chain37. It is known that such a stack remains in a non-trivial SPT phase in the presence of interactions, namely U ≠ 0 (ref. 38). Moreover, together with the parity symmetry of spin-down particles, ({hat{P}}_{l}^{downarrow }equiv {rm{exp }}left[{i}{rm{pi }}left({hat{n}}_{l,{rm{A}}}^{downarrow }+{hat{n}}_{l,{rm{B}}}^{downarrow }right)right]), it then gives rise to a different string order parameter: the topological phase is characterized by long-range order in ({g}_{{S}^{z},{P}^{downarrow }}), whereas it has vanishing correlations for ({g}_{{mathbb{1}},{P}^{downarrow }}), with the roles being reversed in the trivial phase. This novel string order is derived in the Supplementary Information. Remarkably, in the Heisenberg limit, it coincides with the conventional string order parameter used before.

In the topological phase it is meaningful to normalize ({g}_{{S}^{z},{P}^{downarrow }}) to ({tilde{g}}_{{S}^{z},{P}^{downarrow }}=eta {g}_{{S}^{z},{P}^{downarrow }}) with ({eta }^{-1}=langle |{hat{S}}_{k}^{z}||{hat{S}}_{k+d}^{z}|rangle ), which effectively excludes endpoints with spin Sz = 0. Indeed, we find unchanged string correlations ({tilde{g}}_{{S}^{z},{P}^{downarrow }}) and ({g}_{{mathbb{1}},{P}^{downarrow }}) down to the lowest experimentally explored value (U/{t}_{parallel }=2.5(2)) (Fig. 4a, b) and edge state signals down to (U/{t}_{parallel }=5.0(3)) (Fig. 4c). Density matrix renormalization group (DMRG) calculations for (Lto infty ) confirm non-zero ({tilde{g}}_{{S}^{z},{P}^{downarrow }}) (L−1) at T = 0 and for all rung-coupling strengths (Fig. 4d), while ({g}_{{mathbb{1}},{P}^{downarrow }}(L-1)) is strictly zero. Owing to the normalization ({tilde{g}}_{{S}^{z},{P}^{downarrow }}(L-1)) goes to 1 for ({J}_{perp }gg {J}_{parallel }).

Fig. 4: Robustness of the Haldane phase to density fluctuations.
figure 4

a, b, The hidden SPT order is preserved even at low Hubbard interactions, as revealed by the novel string correlators (|,{tilde{g}}_{{S}^{z},{P}^{downarrow }}(d),|) (green circles) and (|{g}_{{mathbb{1}},{P}^{downarrow }}(d)|) (grey squares) on the basis of the spin-down parity ({hat{P}}^{downarrow }). (|,{tilde{g}}_{{S}^{z},{P}^{downarrow }},|) stays non-zero, whereas (|{g}_{{mathbb{1}},{P}^{downarrow }}|) is consistent with zero for d = L − 1 over the measured interaction range. The same qualitative behaviour is seen in zero-temperature DMRG calculations (shaded line) with (Lto infty ). c, Spatial distribution of excess magnetization (({m}^{z}) = ±1) for decreasing (U/{t}_{parallel }). Even far away from the Heisenberg regime, the edge state signal remains strong and only diminishes for very weak (U/{t}_{parallel }). d, Map of zero-temperature DMRG ((Lto infty )) results for the spin-string correlator in the entire parameter space of the topological phase. It shows a strictly non-zero ({hat{g}}_{{S}^{z},{P}^{downarrow }}) while ({g}_{{mathbb{1}},{P}^{downarrow }}(L-1)=0) everywhere in this phase. The black circles indicate the parameters of the measurements. All experimental data were taken at ({J}_{perp }/{J}_{parallel }=1.3(2)) and L = 5 in the tilted geometry. ({m}^{z}) = 0 in a, b and d. Error bars denote one standard error of the mean (s.e.m.) and are smaller than their marker size if not visible.

Source data

In our work, we realized a finite-temperature version of the topological Haldane SPT phase using the full spin and density resolution of our Fermi quantum gas microscope. We demonstrated the robustness of the edge states and the hidden order of this SPT phase in both the Heisenberg and the Fermi–Hubbard regime. In the future, studies may extend the two-leg ladder to a varying number of legs, in which one would expect clear differences between even and odd numbers of legs39 and topological effects away from half-filling40, or may investigate topological phases in higher dimensions41. Furthermore, the ladder geometry holds the potential to reveal hole–hole pairing42 at temperatures more favourable than in a full two-dimensional system.



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