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 striking non-equilibrium phenomena have been discovered or predicted in opticallydriven quantum solids 1 , ranging from light-induced superconductivity 2,3 to Floquetengineered topological phases 4-8 . These effects are expected to lead to dramatic changes in electrical transport, but can only be comprehensively characterized or functionalized with a direct interface to electrical devices that operate at ultrafast speeds 1-8 . Here, we make use of laser-triggered photoconductive switches 9 to measure the ultrafast transport properties of monolayer graphene, driven by a mid-infrared femtosecond pulse of circularly polarized light. The goal of this experiment is to probe the transport signatures of a predicted light-induced topological band structure in graphene 4,5 , similar to the one originally proposed by Haldane 10 . We report the observation of an anomalous Hall effect in the absence of an applied magnetic field. We also extract quantitative properties of the non-equilibrium state. The dependence of the effect on a gate potential used to tune the Fermi level reveals multiple features that reflect the effective band structure expected from Floquet theory. This includes a ∼60 meV wide conductance plateau centered at the Dirac point, where a gap of approximately equal magnitude is expected to open. We also find that when the Fermi level lies within this plateau, the estimated anomalous Hall conductance saturates around ∼1.8±0.4 e 2 /h.Optical driving has been proposed as a means to engineer topological properties in topologically trivial systems 4-8 . One proposal for such a 'Floquet topological insulator' is based on breaking time-reversal symmetry in graphene through a coherent interaction with circularly polarized light 4 . In this theory, the light field drives electrons in circular trajectories through the band structure (Fig. 1a). Close to the Dirac point, these states are predicted to acquire a non-adiabatic Berry phase every optical cycle, which is equal and opposite for the upper and lower band. This time-averaged extra phase accumulation amounts to an energy * These authors contributed equally to this work

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.

We create an artificial graphene system with tunable interactions and study the crossover from metallic to Mott insulating regimes, both in isolated and coupled two-dimensional honeycomb layers. The artificial graphene consists of a two-component spin mixture of an ultracold atomic Fermi gas loaded into a hexagonal optical lattice. For strong repulsive interactions, we observe a suppression of double occupancy and measure a gapped excitation spectrum. We present a quantitative comparison between our measurements and theory, making use of a novel numerical method to obtain Wannier functions for complex lattice structures. Extending our studies to time-resolved measurements, we investigate the equilibration of the double occupancy as a function of lattice loading time.