## Looking for patterns in an optical lattice

One of the simplest models of interacting fermions on a two-dimensional (2D) lattice—the Hubbard model—becomes too tricky to simulate on classical computers as the density of empty lattice sites (holes) increases. Chiu *et al.* used a quantum microscope to take snapshots of thousands of realizations of the 2D Hubbard model in an optical lattice filled with fermionic lithium atoms at varying hole densities (see the Perspective by Schauss). The authors used pattern recognition algorithms to analyze the images, in which each lattice site was individually resolved. Comparing these patterns to the predictions of several theoretical models, they found the most consistency with the so-called geometric string model.

## Abstract

Understanding strongly correlated quantum many-body states is one of the most difficult challenges in modern physics. For example, there remain fundamental open questions on the phase diagram of the Hubbard model, which describes strongly correlated electrons in solids. In this work, we realize the Hubbard Hamiltonian and search for specific patterns within the individual images of many realizations of strongly correlated ultracold fermions in an optical lattice. Upon doping a cold-atom antiferromagnet, we find consistency with geometric strings, entities that may explain the relationship between hole motion and spin order, in both pattern-based and conventional observables. Our results demonstrate the potential for pattern recognition to provide key insights into cold-atom quantum many-body systems.

Quantum superposition describes quantum systems as simultaneously realizing different configurations. Such behavior is believed to be at the heart of phenomena in strongly correlated quantum many-body systems, which cannot be described by single-particle or mean-field theories. An intriguing consequence of the superposition principle is the existence of hidden order in correlated quantum systems: Although every individual configuration is characterized by a particular pattern, the average over these configurations leads to an apparent loss of order. By contrast, instantaneous projective measurements have the potential to reveal these underlying patterns.

One notable example of a system with hidden order is the one-dimensional (1D) Fermi-Hubbard model at strong coupling (*1*, *2*). Although 1D chains with additional holes or particles beyond an average of one particle per site (doped) yield average two-point spin correlations that decay more rapidly with distance than chains with an average of one particle per site (half-filled), this magnetic ordering can be revealed by accounting for the fluctuating positions across individual configurations of the additional dopants within each chain. The apparent loss of magnetic order is in fact hidden order, hidden by the dopants and their varying positions (*3*, *4*). Although direct detection of this hidden string order remains inaccessible in solids, experiments with ultracold atoms enable projective measurements, or “snapshots,” and generally can provide access to such structures (*5*). In particular, quantum gas microscopy (*6*) enables site-resolved imaging and access to correlators that have been constructed to reveal the hidden order (*7*).

The hidden order in 1D is well understood, but the physics of the 2D Hubbard model is fundamentally more complex because of an intricate interplay between spin and charge degrees of freedom; as a result, formulating an appropriate correlation function to search for hidden order becomes considerably more challenging. The 2D Hubbard model is believed to capture the rich physics of high-temperature superconductivity and other phases (*8*–*10*) such as the strange metal, stripe, antiferromagnet (AFM), or pseudogap phase, but a unified understanding of these phenomena is still lacking. For example, the behavior of individual dopants in an AFM is not agreed upon, including whether hidden string order is present and dopants hide magnetic correlations by shifting the positions of a string of spins. Quantum gas microscopy, however, provides a perspective that goes beyond the framework of two- or multipoint correlations. Hidden string order can be searched for directly within individual snapshots of the quantum mechanical wave function, where quantum fluctuations are resolved.

Here we perform a microscopic study of the hole-doped Fermi-Hubbard model and report indications of string patterns in 2D over a wide doping range. Our measurements use ultracold fermions in an optical lattice down to the lowest currently achievable temperatures, where at low doping AFM correlations extend across the system size (*11*). We identify string patterns in individual projective measurements and compare them with predictions from microscopic theoretical approaches.

## Candidate theories for the doped Hubbard model

We study the Fermi-Hubbard model, which is defined by the Hamiltonian*t* of spin-½ fermions

The Fermi-Hubbard model is well understood when the band is half-filled at an average of one particle per site (Fig. 1B). For temperatures *11*).

Much less is known about the doped Fermi-Hubbard model. However, it is understood that dopant delocalization for kinetic energy minimization competes with spin interactions in the background AFM. Experiments on the cuprates have also shown that at temperatures *9*). These two metallic phases (pseudogap and strange metal) defy a description in terms of conventional quasiparticles and still lack a unified theoretical understanding.

Although phenomenological, numeric, and mean-field (MF) approaches have provided key insights in the past, quantum gas microscopy is naturally suited to assess microscopic theoretical approaches. One such theory is Anderson’s resonating valence bond (RVB) picture (*12*), which considers trial wave functions of free holes moving through a spin liquid composed of singlet coverings. We consider one particular class of RVB wave functions that have been studied extensively, called π-flux states (*13*). They stem from an MF density matrix *k*_{B} is Boltzmann’s constant, *14*) for details]. Snapshots of the trial state in the Fock basis can be obtained by Monte-Carlo sampling, with temperature *T* as a free fit parameter (*15*).

A second microscopic approach that we examine is the geometric-string theory (*16*), where AFM order at half-filling is hidden in doped states via hole motion. This theory extends earlier work (*17*–*19*) and establishes a relationship between the AFM parent state at half-filling and the strongly correlated quantum states at finite doping. Here, holes move through the parent AFM by displacing each spin along its trajectory by one lattice site, whereas the AFM quantum state remains otherwise unmodified; this is the frozen-spin approximation (*20*). The delocalization of each hole can then be described as a superposition state of hole trajectories, or geometric strings, whose lengths ℓ depend on the strength of AFM correlations and the ratio of the kinetic energy *t* to the superexchange *J*. For any given temperature, a distribution function

We directly assess these microscopic theoretical approaches with a quantum gas microscope, which provides projective measurements of the quantum mechanical wave function for the doped Hubbard model in the parity-projected Fock basis. Our experimental setup consists of a balanced two-component gas of fermionic lithium in the lowest band of a square optical lattice (*21*), with *U*/*t* set to 8.1(2). We selectively image one of the spin states or the total atom distribution (*22*). Entropy redistribution with a digital micromirror device enables a disk-shaped homogeneous system of ~80 sites with temperatures as low as *T*/*J* = 0.50(4) (*11*). We alter the local chemical potential to dope the system, maintaining independent temperature control [section 7.1 of (*14*)]. We determine the doping from the single-particle occupation density and temperature from the nearest-neighbor spin correlator, both by comparing to numerics [section 2 of (*14*)].

## Pattern recognition of geometric strings

We design a pattern recognition algorithm for geometric strings that we apply to real-space snapshots where doublons and one of the two spin states have been removed (Fig. 2A). Because geometric strings describe a relationship between doped and half-filled AFMs, we search for stringlike patterns in the deviation between snapshots of the doped Hubbard model and an approximation to the AFM, an exact checkerboard. For each image, we take the set of sites that deviate and extract string patterns using the following rules: (i) every string pattern is a connected subset of sites forming a path without branching points; (ii) each site can be part of only one string pattern; (iii) longer string patterns are favored; and (iv) every string pattern must have at one end a site that is detected as empty, and therefore consistent with having a hole on that site. We discuss alternate algorithms in section 3.4 of (*14*).

We find that this algorithm is indeed sensitive to hole doping. Figure 2B shows string-pattern length distributions *J* and 0.70(3)*J*. As the sample is doped from half-filling to a doping δ of 10.0(8)%, the number of string patterns increases across the entire range of lengths. The appreciable distribution of string patterns *14*)] and is largely caused by the finite temperature and underlying SU(2) symmetry of the system. We have lessened these contributions by reducing the analysis region to a diameter of seven sites and postselecting on the staggered magnetization. In section 3.3 of (*14*), we show that results are robust to the choice of postselection scheme and that the limited detection of one of the spin states causes only an overall factor decrease in string patterns detected.

Next, we compare our experimental results to the simulation results of three microscopic models. We make predictions by producing artificial images and evaluating them with our string pattern detection algorithm, such that the detection is common to experiment and theoretical simulation. Beginning with the analytic string model, we generate images by randomly placing a number of holes into actual experimental images taken at half-filling, then randomly propagating each hole according to the analytically generated string-length histogram (see Fig. 1C) and appropriately displacing the spins along the hole’s path. Note that this approach preserves the SU(2) symmetry of the system. The resulting string-pattern length distribution agrees with experimental data [see Fig. 2B for 10% doping], even though the theory has no free parameters.

To verify whether our measured signal simply results from the introduction of holes rather than changes to the spin background, we next compare our experimental result with simulations where holes are artificially and randomly placed (“sprinkled”) into experimental data taken at half-filling, equivalent to placing one-site-long strings. The associated string-pattern length distribution

We repeat the measurements for a sample heated before lattice loading to investigate temperature effects. Figure 2C shows experimental data at half-filling and at 10.1(8)% doping, along with the simulated prediction, averaged over samples at temperatures between 1.3(1)*J* and 1.8(1)*J*. In contrast to colder temperatures, there is no statistically significant deviation between the experimental data with and without hole doping;

In Fig. 2D and its inset we plot, respectively, the relative and absolute differences between the pattern-length histograms in the doped and undoped cases; these differences are shown for both the cold and hot datasets used in Fig. 2, B and C. For the undoped case, we use the sprinkled string-pattern length distribution

Focusing on the cold dataset, we now examine the relationship between doping and the number of detected string patterns (Fig. 3A). In this string-pattern count, we omit patterns of one or two sites to avoid contributions from quantum fluctuations such as doublon-hole pairs or spin-exchange processes. The string-pattern count increases with doping and saturates at about 16% doping. This saturation is consistent with a high density of strings and overlapping or adjacent strings scrambling spin order such that pattern detection becomes insensitive to additional strings. The continued agreement between geometric strings and experiment in both the string-pattern count and the absolute difference

The experimental string-pattern count is significantly larger than that of the sprinkled-hole simulation; nonetheless, there is an increase in detected string patterns owing to the additional holes. The string-pattern count from π-flux states shows considerably better agreement with experimental data than with sprinkled holes, exhibiting only a slight excess of string patterns at low doping and a deficit at high doping. The largest deviations occur at low doping, which may be related to the absence of long-range order at zero temperature in π-flux states at half-filling.

The average string-pattern length quantifies the size of the region around the hole where the spin pattern is distorted by the string (Fig. 3B). The observed values are comparatively small, influenced by the large contributions from quantum fluctuations at half-filling. The average string-pattern length does not change substantially with doping, consistent with spatially isolated patterns; however, at larger dopings, we observe a slight decrease in average length that coincides with the observed saturation in the string count. This behavior is captured by the geometric-string model for low and intermediate doping. At high doping, the theory exhibits shorter average string lengths than the experiment, which may result from high–string-density effects such as string-string interactions, which are not included in the theory.

We compare these results to a dataset in which geometric strings are not expected to occur. This dataset consists of experimental images taken at various temperatures at half-filling with sprinkled holes to match each desired doping level [for details, see section 3.5 of (*14*)]. Temperatures are chosen to match the measured staggered magnetization and capture the observed loss of AFM order. Notably, the average string-pattern length reveals that this loss through heating occurs in a fundamentally different way than through doping. For all nonzero doping, the temperature-based dataset exhibits shorter average string-pattern lengths than the experimentally measured doping dataset. As doping increases, the average length monotonically decreases. Alternatively, we match the nearest-neighbor spin correlator instead of the staggered magnetization and find an even greater distinction between the doped and temperature-based datasets.

We better understand the role of temperature in string-pattern detection by observing how the string-pattern count varies with temperature at fixed doping. For 10% doping, we plot the difference between the experiment and sprinkled-hole string-pattern counts (Fig. 3C), which are plotted separately in the inset. At our lowest temperatures, the difference is greatest. This high sensitivity is consistent with the greatest spin ordering for the parent AFM at low temperatures, accompanied by a relatively large string-pattern count from the experimental data. The difference decreases steadily with increasing temperature, predominantly owing to the increase in the sprinkled-hole string-pattern count from decreased spin ordering in the parent AFM, vanishing around *T* = *J*.

## Spin correlations and staggered magnetization

An accurate microscopic framework for the Fermi-Hubbard model should also be able to predict more conventional observables such as two-point correlation functions, which have been used with quantum gas microscopes to quantify spin and charge order (*22*–*25*). To that end, we measure the sign-corrected spin-spin correlation function for displacements *22*). Thanks to the sign correction *26*). As the system is doped, all correlators exhibit a reduction in magnitude. *22*, *24*, *27*) and numerics (*24*), and are good benchmarks for the evaluation of theoretical models.

We make predictions for spin correlations from ensembles of non-postselected images with sprinkled holes, geometric strings, or π-flux states. By construction, at half-filling the predictions of sprinkled holes and the string model are the same as those of experimental half-filling data. Away from half-filling, sprinkled holes underestimate the decrease of the correlators because the model fails to account for the disruption of AFM order as the system is doped. By contrast, beginning at intermediate doping values, the string model overestimates the decrease of

Cold-atom experiments provide access to full-counting statistics (FCS) because of their ability to project and measure an entire quantum system at once (*11*). We measure the FCS of the staggered magnetization operator*N* across all experimental realizations as we dope the system (Fig. 4B). As expected, the staggered magnetization distribution narrows, reflecting the finite-size crossover from the AFM-ordered phase (*11*). The sprinkled-hole simulation does not exhibit a major change in the distribution as the system is doped, as it fails to account for holes disrupting the AFM order. By contrast, both π-flux states and geometric strings demonstrate reasonable agreement with the experimentally measured distribution function across all dopings. Across all observables considered, both of these theories perform quite well, especially in comparison to the sprinkled-holes simulation and the naïve phenomenological models detailed in section 5 of (*14*). However, we find the sign change of

## Antimoment correlations

All observables studied in this work thus far have focused on the spin sector of the Hubbard model. Next, we examine correlations in the charge sector. At sufficiently low temperatures, one may expect signatures of pairing (*10*, *28*) or stripe phases (*29*, *30*), which lead to hole bunching. However, anticorrelations of the holes, as observed previously at increased temperatures (*24*), are expected in the strongly correlated metallic regime of the Hubbard model. The transition between these two regimes in the Hubbard model phase diagram is not yet fully understood; however, the currently accessible experimental regime allows us to place more accurate bounds on where this transition can occur. We continue to compare experimental results to predictions of π-flux states, but do not compare to predictions of the geometric string theory because it approximates that charges are uncorrelated. Rather, because each string is associated with a single hole, correlation functions of holes can reveal possible interactions and correlations between geometric strings, should they exist.

In our experiment, doubly occupied sites appear as empty when imaged and the exact hole correlation is not directly accessible; rather, we measure “antimoment” correlations *24*). Doublon-hole pairs beyond nearest neighbors become increasingly unlikely; therefore, to avoid the effects of doublon-hole pairs, we focus on correlations at distances greater than 1. We find the nearest-neighbor antimoment correlator at half-filling to be weaker than predicted according to numerics, which may be caused by imperfect imaging fidelity. However, this effect only weakens the magnitude of the antimoment correlators measured; we therefore focus on qualitative conclusions from the experimental data.

Figure 5A shows the antimoment correlation for 3% (top) and 19% doping (bottom) at a temperature *31*). Here we do not consider geometric-string theory or sprinkled holes because both introduce uncorrelated holes by construction. Additionally, in the comparison to π-flux states, we do not include doublon-hole pairs to avoid unintended artifacts in the antimoment correlator. For reference, we plot the predicted hole-hole correlation function for a phenomenological model of spinless fermionic chargons with nearest-neighbor hopping of strength *t* and temperatures between 0.5*J* and 0.7*J* (*32*). Here, strong anticorrelations result from Pauli repulsion between the fermionic chargons, but qualitatively similar behavior is expected for bosonic chargons with hard-core interactions. We find that both theories qualitatively describe the experimental result.

The emergence of this repelling behavior can be characterized by plotting the antimoment correlation as a function of doping for *d* = 1 evidences against holes tunneling preferentially between sites of one sublattice, as predicted by theories of pointlike magnetic polarons with a dispersion minimum at (π/2, π/2) in the Brillouin zone (*33*–*36*).

Finally, we plot a normalized *37*) is used to define a tight-binding hopping model of the polaron. Figure 5C shows that our data are incompatible with this model, which predicts significantly weaker hole-hole anticorrelations. Similar behavior is predicted by the π-flux theory, which models the doped holes as pointlike objects moving in a quantum spin liquid of singlets.

Next, we examine a picture of free chargons, motivated by considering magnetic polarons with a finite extent that results from the spinon-chargon bound state predicted by geometric-string theory. At sufficiently large chargon density, or doping, the chargons are expected to interact, and their hard-core character will introduce anticorrelations. In this regime, geometric strings are also expected to overlap substantially and modify the dispersion relation of the chargons to be independent of spinons, yielding spinless chargons. We find that the experimental results demonstrate consistency with these free chargons, in agreement with earlier theoretical work in the strange-metal regime (*32*, *38*). This extension to the geometric string theory may be able to explain the deviations from experiment seen in other observables at high doping, but additional analyses are required.

## Conclusions and outlook

The string-pattern–based observables introduced here complement established observables such as correlation functions or full counting statistics. Across the observables considered, we find better agreement with experimental data between both the geometric-string theory and π-flux states, as compared to sprinkled holes.

At intermediate doping values, we find evidence for hole-hole repulsion. Although signatures of other phases such as stripe phases, incommensurate spin order, or nematic fluctuations have not yet been observed in this system, they are predicted to emerge at lower temperatures.

The ideas presented can be extended to other real-space patterns—for example, patterns that reflect the underlying physics of other candidate microscopic theories for the doped Hubbard model. Moreover, machine learning techniques could be used to directly compare sets of raw experimental atom distributions to theoretical models without the need for intermediate observables (*39*). This class of techniques is highly promising as quantum simulations of the Hubbard model continue to probe lower temperatures within the pseudogap and strange-metal phases, but can also be applied to spatially resolved studies of quenches across phase transitions (*40*), dynamical phase transitions (*41*), and higher-order scattering processes (*42*). Possible extensions of our work include systems with anisotropic spin interactions (*16*) or doped SU(*N*) spin models (*43*).

## Supplementary Materials

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

## References and Notes

**Acknowledgments:**We thank M.Kanász-Nagy for Heisenberg QMC code. We thank A. Hébert, S. Sachdev, Z.-Y. Weng, and J. Zaanen for insightful discussions.

**Funding:**We acknowledge support from AFOSR grant nos. FA9550-14-10035 and FA9550-16-10323; DoD NDSEG; the Gordon and Betty Moore Foundation EPIQS program and grant no. 6791; NSF GRFP and grant nos. PHY-1506203, PHY-1734011, and DMR-1308435; ONR grant no. N00014-18-1-2863; SNSF; Studienstiftung des deutschen Volkes; and the Technical University of Munich–Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy–EXC-2111–390814868, through DFG grant no. KN1254/1-1, and DFG TRR80 (Project F8).

**Author contributions:**C.S.C., G.J., M.X., and D.G. performed the experiment. C.S.C. and A.B. curated the data and, along with F.G., developed the pattern finding algorithm. C.S.C., G.J., A.B., M.X., and D.G. analyzed the data. F.G. and E.D. developed the geometric-string theory. A.B. and F.G., together with M.K. and E.D., developed the theory simulations. M.G. and D.G. supervised the work. All authors contributed to the interpretation of the results and writing of the manuscript.

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

**Data and materials availability:**All experimentally measured site-resolved atom distributions and analysis code are available (

*44*).