Realizing the symmetry-protected Haldane phase in Fermi-Hubbard ladders.

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 chain1,2. 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 parameter3,4. 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 entanglement5-7. 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.

Microscopic evolution of doped Mott insulators from polaronic metal to Fermi liquid.

The competition between antiferromagnetism and hole motion in two-dimensional Mott insulators lies at the heart of a doping-dependent transition from an anomalous metal to a conventional Fermi liquid. We observe such a crossover in Fermi-Hubbard systems on a cold-atom quantum simulator and reveal the transformation of multipoint correlations between spins and holes upon increasing doping at temperatures around the superexchange energy. Conventional observables, such as spin susceptibility, are furthermore computed from the microscopic snapshots of the system. Starting from a magnetic polaron regime, we find the system evolves into a Fermi liquid featuring incommensurate magnetic fluctuations and fundamentally altered correlations. The crossover is completed for hole dopings around 30%. Our work benchmarks theoretical approaches and discusses possible connections to lower-temperature phenomena.

Robust Bilayer Charge Pumping for Spin- and Density-Resolved Quantum Gas Microscopy.

Quantum gas microscopy has emerged as a powerful new way to probe quantum many-body systems at the microscopic level. However, layered or efficient spin-resolved readout methods have remained scarce as they impose strong demands on the specific atomic species and constrain the simulated lattice geometry and size. Here we present a novel high-fidelity bilayer readout, which can be used for full spin- and density-resolved quantum gas microscopy of two-dimensional systems with arbitrary geometry. Our technique makes use of an initial Stern-Gerlach splitting into adjacent layers of a highly stable vertical superlattice and subsequent charge pumping to separate the layers by 21 µm. This separation enables independent high-resolution images of each layer. We benchmark our method by spin- and density-resolving two-dimensional Fermi-Hubbard systems. Our technique furthermore enables the access to advanced entropy engineering schemes, spectroscopic methods, or the realization of tunable bilayer systems.

Time-resolved observation of spin-charge deconfinement in fermionic Hubbard chains.

Elementary particles carry several quantum numbers, such as charge and spin. However, in an ensemble of strongly interacting particles, the emerging degrees of freedom can fundamentally differ from those of the individual constituents. For example, one-dimensional systems are described by independent quasiparticles carrying either spin (spinon) or charge (holon). Here, we report on the dynamical deconfinement of spin and charge excitations in real space after the removal of a particle in Fermi-Hubbard chains of ultracold atoms. Using space- and time-resolved quantum gas microscopy, we tracked the evolution of the excitations through their signatures in spin and charge correlations. By evaluating multipoint correlators, we quantified the spatial separation of the excitations in the context of fractionalization into single spinons and holons at finite temperatures.

Imaging magnetic polarons in the doped Fermi-Hubbard model.

Polarons-electronic charge carriers 'dressed' by a local polarization of the background environment-are among the most fundamental quasiparticles in interacting many-body systems, and emerge even at the level of a single dopant1. In the context of the two-dimensional Fermi-Hubbard model, polarons are predicted to form around charged dopants in an antiferromagnetic background in the low-doping regime, close to the Mott insulating state2-7; this prediction is supported by macroscopic transport and spectroscopy measurements in materials related to high-temperature superconductivity8. Nonetheless, a direct experimental observation of the internal structure of magnetic polarons is lacking. Here we report the microscopic real-space characterization of magnetic polarons in a doped Fermi-Hubbard system, enabled by the single-site spin and density resolution of our ultracold-atom quantum simulator. We reveal the dressing of doublons by a local reduction-and even sign reversal-of magnetic correlations, which originates from the competition between kinetic and magnetic energy in the system. The experimentally observed polaron signatures are found to be consistent with an effective string model at finite temperature7. We demonstrate that delocalization of the doublon is a necessary condition for polaron formation, by comparing this setting with a scenario in which a doublon is pinned to a lattice site. Our work could facilitate the study of interactions between polarons, which may lead to collective behaviour, such as stripe formation, as well as the microscopic exploration of the fate of polarons in the pseudogap and 'bad metal' phases.

Author Correction: Direct observation of incommensurate magnetism in Hubbard chains.

In this Letter, the affiliation for Christian Gross should have been 'Max-Planck-Institut für Quantenoptik, Garching, Germany' instead of 'Fakultät für Physik, Ludwig-Maximilians-Universität, Munich, Germany'; this has been corrected online.

Direct observation of incommensurate magnetism in Hubbard chains.

The interplay between magnetism and doping is at the origin of exotic strongly correlated electronic phases and can lead to novel forms of magnetic ordering. One example is the emergence of incommensurate spin-density waves, which have wavevectors that do not belong to the reciprocal lattice. In one dimension this effect is a hallmark of Luttinger liquid theory, which also describes the low-energy physics of the Hubbard model1. Here we use a quantum simulator that uses ultracold fermions in an optical lattice2-8 to directly observe such incommensurate spin correlations in doped and spin-imbalanced Hubbard chains using fully spin- and density-resolved quantum gas microscopy. Doping is found to induce a linear change in the spin-density wavevector, in excellent agreement with predictions from Luttinger theory. For non-zero polarization we observe a reduction in the wavevector with magnetization, as expected from the antiferromagnetic Heisenberg model in a magnetic field. We trace the microscopic-scale origin of these incommensurate correlations to holes, doublons (double occupancies) and excess spins, which act as delocalized domain walls for the antiferromagnetic order. In addition, by inducing interchain coupling we observe fundamentally different spin correlations around doublons and suppression of incommensurate magnetism at finite (low) temperature in the two-dimensional regime9. Our results demonstrate how access to the full counting statistics of all local degrees of freedom can be used to study fundamental phenomena in strongly correlated many-body physics.

Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators.

Topological phases, like the Haldane phase in spin-1 chains, defy characterization through local order parameters. Instead, nonlocal string order parameters can be employed to reveal their hidden order. Similar diluted magnetic correlations appear in doped one-dimensional lattice systems owing to the phenomenon of spin-charge separation. Here we report on the direct observation of such hidden magnetic correlations via quantum gas microscopy of hole-doped ultracold Fermi-Hubbard chains. The measurement of nonlocal spin-density correlation functions reveals a hidden finite-range antiferromagnetic order, a direct consequence of spin-charge separation. Our technique, which measures nonlocal order directly, can be readily extended to higher dimensions to study the complex interplay between magnetic order and density fluctuations.

Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains.

The repulsive Hubbard Hamiltonian is one of the foundational models describing strongly correlated electrons and is believed to capture essential aspects of high-temperature superconductivity. Ultracold fermions in optical lattices allow for the simulation of the Hubbard Hamiltonian with control over kinetic energy, interactions, and doping. A great challenge is to reach the required low entropy and to observe antiferromagnetic spin correlations beyond nearest neighbors, for which quantum gas microscopes are ideal. Here, we report on the direct, single-site resolved detection of antiferromagnetic correlations extending up to three sites in spin-1/2 Hubbard chains, which requires entropies per particle well below s* = ln(2). The simultaneous detection of spin and density opens the route toward the study of the interplay between magnetic ordering and doping in various dimensions.

Microscopic Observation of Pauli Blocking in Degenerate Fermionic Lattice Gases.

The Pauli exclusion principle is one of the most fundamental manifestations of quantum statistics. Here, we report on its local observation in a spin-polarized degenerate gas of fermions in an optical lattice. We probe the gas with single-site resolution using a new generation quantum gas microscope avoiding the common problem of light induced losses. In the band insulating regime, we measure a strong local suppression of particle number fluctuations and a low local entropy per atom. Our work opens a new avenue for studying quantum correlations in fermionic quantum matter both in and out of equilibrium.