The Haldane model on the honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter [1]. It describes a mechanism through which a quantum Hall effect can appear as an intrinsic property of a band-structure, rather than being caused by an external magnetic field [2]. Although an implementation in a material was considered unlikely, it has provided the conceptual basis for theoretical and experimental research exploring topological insulators and superconductors [2][3][4][5][6]. Here we report on the experimental realisation of the Haldane model and the characterisation of its topological band-structure, using ultracold fermionic atoms in a periodically modulated optical honeycomb lattice. The model is based on breaking time-reversal symmetry as well as inversion symmetry. The former is achieved through the introduction of complex next-nearest-neighbour tunnelling terms, which we induce through circular modulation of the lattice position [7]. For the latter, we create an energy offset between neighbouring sites [8]. Breaking either of these symmetries opens a gap in the band-structure, which is probed using momentum-resolved interband transitions. We explore the resulting Berrycurvatures of the lowest band by applying a constant force to the atoms and find orthogonal drifts analogous to a Hall current. The competition between both broken symmetries gives rise to a transition between topologically distinct regimes. By identifying the vanishing gap at a single Dirac point, we map out this transition line experimentally and quantitatively compare it to calculations using Floquet theory without free parameters. We verify that our approach, which allows for dynamically tuning topological properties, is suitable even for interacting fermionic systems. Furthermore, we propose a direct extension to realise spin-dependent topological Hamiltonians.In a honeycomb lattice symmetric under time-reversal and inversion, the two lowest bands are connected at two Dirac points. Each broken symmetry leads to a gapped energy-spectrum. F. D. M. Haldane realised that the resulting phases are topologically distinct: A broken inversion symmetry (IS), caused by an energy offset between the two sublattices, leads to a trivial band-insulator at half-filling. Time-reversal symmetry (TRS) can be broken by complex next-nearest-neighbour tunnel couplings (Fig. 1a). The corresponding staggered magnetic fluxes sum up to zero in one unit-cell, thereby preserving the translation symmetry of the lattice. This gives rise to a topological Chern-insulator, where a non-zero Hall conductance appears despite the absence of a net magnetic field [1,2]. When both symmetries are broken, a topological phase transition connects two regimes with a distinct topological invariant, the Chern number, which changes from 0 to ±1, see Fig. 1b. There, the gap closes at a single Dirac point. These transitions have attracted great interest, as they cannot be described by Landau's theory of phase transitions, owing to the ...

Dirac points are central to many phenomena in condensed-matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the electrons behave as relativistic Dirac fermions. In solids, the rigid structure of the material determines the mass and velocity of the electrons, as well as their interactions. A different, highly flexible means of studying condensed-matter phenomena is to create model systems using ultracold atoms trapped in the periodic potential of interfering laser beams. Here we report the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice. Using momentum-resolved interband transitions, we observe a minimum bandgap inside the Brillouin zone at the positions of the two Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking inversion symmetry. Moreover, changing the lattice anisotropy allows us to change the positions of the Dirac points inside the Brillouin zone. When the anisotropy exceeds a critical limit, the two Dirac points merge and annihilate each other-a situation that has recently attracted considerable theoretical interest but that is extremely challenging to observe in solids. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials in which the topology of the band structure is crucial, but also provide an avenue to exploring many-body phases resulting from the interplay of complex lattice geometries with interactions.

Many exotic phenomena in strongly correlated electron systems emerge from the interplay between spin and motional degrees of freedom [1, 2]. For example, doping an antiferromagnet gives rise to interesting phases including pseudogap states and high-temperature superconductors [3]. A promising route towards achieving a complete understanding of these materials begins with analytic and computational analysis of simplified models. Quantum simulation has recently emerged as a complementary approach towards understanding these models [4][5][6][7][8]. Ultracold fermions in optical lattices offer the potential to answer open questions on the lowtemperature regime of the doped Hubbard model [9][10][11], which is thought to capture essential aspects of the cuprate superconductor phase diagram but is numerically intractable in that parameter regime. Already, Mott-insulating phases and short-range antiferromagnetic correlations have been observed, but temperatures were too high to create an antiferromagnet [12][13][14][15]. A new perspective is afforded by quantum gas microscopy [16][17][18][19][20][21][22][23][24][25][26][27][28], which allows readout of magnetic correlations at the site-resolved level [25][26][27][28]. Here we report the realization of an antiferromagnet in a repulsively interacting Fermi gas on a 2D square lattice of approximately 80 sites. Using site-resolved imaging, we detect (finite-size) antiferromagnetic long-range order (LRO) through the development of a peak in the spin structure factor and the divergence of the correlation length that reaches the size of the system. At our lowest temperature of T/t = 0.25(2) we find strong order across the entire sample, where the staggered magnetization approaches the ground-state value. Our experimental platform enables doping away from half filling, where pseudogap states and stripe ordering are expected, but theoretical methods become numerically intractable. In this regime we find that the antiferromagnetic LRO persists to hole dopings of about 15%, providing a guideline for computational methods. Our results demonstrate that quantum gas microscopy of ultracold fermions in optical lattices can now address open questions on the low-temperature Hubbard model.The Hubbard Hamiltonian is a fundamental model for spinful lattice electrons describing a competition between kinetic energy t and interaction energy U [29]. In the limiting case of half-filling (average one particle per site) and dominant interactions (U/t 1) the Hubbard model maps to the Heisenberg model [1]. There, the exchange energy J = 4t 2 /U can give rise to antiferromagnetically ordered states at low temperatures [30]. This order persists for all finite U/t, where charge fluctuations reduce the ordering strength [31]. Away from half-filling, the coupling between motional and spin degrees of freedom is expected to give rise to a rich many-body phase diagram (see Fig. 1a), which is challenging to understand theoretically due to the fermion sign problem [32]. Even so, in the thermodynamic limit com...

Quantum magnetism originates from the exchange coupling between quantum mechanical spins. Here, we report on the observation of nearest-neighbor magnetic correlations emerging in the many-body state of a thermalized Fermi gas in an optical lattice. The key to obtaining short-range magnetic order is a local redistribution of entropy, which allows temperatures below the exchange energy for a subset of lattice bonds. When loading a repulsively interacting gas into either dimerized or anisotropic simple cubic configurations of a tunable-geometry lattice, we observe an excess of singlets as compared with triplets consisting of two opposite spins. For the anisotropic lattice, the transverse spin correlator reveals antiferromagnetic correlations along one spatial axis. Our work facilitates addressing open problems in quantum magnetism through the use of quantum simulation.

Exotic phases of matter can emerge from strong correlations in quantum many-body systems. Quantum gas microscopy affords the opportunity to study these correlations with unprecedented detail. Here we report site-resolved observations of antiferromagnetic correlations in a two-dimensional, Hubbard-regime optical lattice and demonstrate the ability to measure the spin-correlation function over any distance. We measure the in-situ distributions of the particle density and magnetic correlations, extract thermodynamic quantities from comparisons to theory, and observe statistically significant correlations over three lattice sites. The temperatures that we reach approach the limits of available numerical simulations. The direct access to many-body physics at the single-particle level demonstrated by our results will further our understanding of how the interplay of motion and magnetism gives rise to new states of matter.PACS numbers: 37.10. Jk, 67.85.Lm, 71.10.Fd, 75.10.Jm, Quantum many-body systems exhibiting magnetic correlations underlie a wide variety of phenomena. Hightemperature superconductivity, for example, can arise from the correlated motion of holes on an antiferromagnetic (AFM) Mott insulator [1,2]. It is possible to probe physical analogs of such systems using ultracold atoms in lattices, which introduce a degree of control that is not available in conventional solid-state systems [3,4]. Recent experiments exploring the Hubbard model with cold atoms are accessing temperatures where AFM correlations form, but have only observed these correlations via measurements that were averages over inhomogeneous systems [5,6]. The recent development of site-resolved imaging for fermionic quantum gases [7][8][9][10][11][12][13] provides the ability to directly detect the correlations and fluctuations present in a fermionic quantum many-body state. As demonstrated in experiments with both bosons [14,15] and fermions [12,13,16], microscopy gives access to the spatial variation in observables that occurs in an inhomogeneous system, yielding precise comparisons with theory. The low energy scales in cold atom systems also allow for time-resolved observations of many-body dynamics, which typically occur on millisecond timescales. In bosonic systems temporal and spatial resolution have been combined to observe strongly correlated quantum walks [17], to measure entanglement entropy [18], and to study the dynamics of magnetic correlations [19,20].Here we report the first trap-resolved observations of magnetic correlations in a Fermi-lattice system. Fermionic atoms in a two-dimensional optical lattice can be well described by the Hubbard Hamiltonian, a simple model in which there is a competition between the nearest-neighbor tunneling energy t and the on-site interaction energy U . Despite the apparent simplicity of the Hubbard model it has a rich phase diagram, containing for example the transition from a metal to a Mott insulator. AFM spin correlations begin to appear near half-filling when the temperature scale becomes comp...