The Haldane model on a honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter1. 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 field2. Although physical implementation has been considered unlikely, the Haldane model has provided the conceptual basis for theoretical and experimental research exploring topological insulators and superconductors2,3,4,5,6. Here we report the experimental realization of the Haldane model and the characterization of its topological band structure, using ultracold fermionic atoms in a periodically modulated optical honeycomb lattice. The Haldane model is based on breaking both time-reversal symmetry and inversion symmetry. To break time-reversal symmetry, we introduce complex next-nearest-neighbour tunnelling terms, which we induce through circular modulation of the lattice position7. To break inversion symmetry, we create an energy offset between neighbouring sites8. Breaking either of these symmetries opens a gap in the band structure, which we probe using momentum-resolved interband transitions. We explore the resulting Berry curvatures, which characterize the topology of the lowest band, by applying a constant force to the atoms and find orthogonal drifts analogous to a Hall current. The competition between the two 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 us to tune the topological properties dynamically, is suitable even for interacting fermionic systems. Furthermore, we propose a direct extension to realize spin-dependent topological Hamiltonians.
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We thank H. Aoki for drawing our attention to the relevance of their proposal for optical lattices and N. Cooper, S. Huber, L. Tarruell, L. Wang and A. Zenesini for discussions. We acknowledge the SNF, the NCCR-QSIT and the SQMS (ERC advanced grant) for funding.
The authors declare no competing financial interests.
Extended data figures and tables
The experiments shown in the main text in Fig. 2c and d are simulated using the semi-classical equations of motion (equations (6)–(9)). The band structure and the Berry curvature are those of the effective Haldane Hamiltonian (equation (1)). The atomic ensemble is modelled by a zero-temperature Fermi distribution. Data are mean ± s.d. of three simulations containing 4 × 104 trajectories. The differential drift is computed when breaking either IS (a) or TRS (b). The former shows no differential drift, in agreement with the experimental data of Fig. 2c. For the latter, the different curves take into account the transfer to the higher band by excluding trajectories passing through regions where the bandgap lies below a certain threshold. If this transfer is not taken into account (purple line), the differential drift varies sharply around ϕ = 0° where the Chern number changes. However, as the threshold is raised to 0.5ΔE (red line) and ΔE (green line) where ΔE/h = 114.6 Hz is the energy offset per site driving the Bloch oscillation, this sharp feature progressively smoothens and qualitatively reproduces the experimental measurements. c, When TRS is maximally broken (ϕ = 90°) and ΔAB varies, the transfer is also responsible for the differential drift extending beyond the topological phase. Without any transfer (purple line), the differential drift changes sharply around the topological phase transition (vertical dashed line), while it extends significantly in the topologically trivial phase when the threshold is set at ΔE (green line) or 3ΔE (blue line). d, Influence of the transverse trapping frequency ωx/2π. The frequency used in the experiment is indicated by a purple arrow. For much larger frequencies the differential drift can vanish, as the transverse oscillation time becomes comparable to the Bloch period.
The measured drift used to obtain the differential drift in Figs 2c, d and 4 of the main text is individually shown for positive and negative forces in the qy direction. Data for positive (negative) force is shown in blue (red). The central plot, showing the Haldane phase diagram, indicates the region which is scanned when breaking either IS (purple arrow) or TRS (brown arrow) in our system. a, We break IS by introducing a sublattice offset and show measurements with modulation frequency of 4.0 kHz and 3.75 kHz. Although the opposite Berry curvatures at the two Dirac points sum up to zero within the first Brillouin zone, we clearly see a drift depending on the size of ΔAB. Data show mean ± s.d. of at least 6 (4.0 kHz) or 2 (3.75 kHz) measurements. b, By changing the modulation phase difference ϕ we break TRS and the system enters the topologically non-trivial regime. Drift data for positive (negative) force is shown in blue (red) for a modulation frequency of 4.0 kHz and 3.75 kHz. Data show mean ± s.d. of at least 21 (4.0 kHz) or 6 (3.75 kHz) measurements. Schematics below show the expected orthogonal drifts caused by driving the atoms through the Berry curvature distribution. Red (blue) indicates positive (negative) Berry curvature. If only IS is broken (a) the Berry curvature distribution is point-antisymmetric and changes sign when changing the sign of the sublattice offset. For opposite forces this leads to the same direction of the drift, as indicated by the white arrows. If only TRS is broken (b) the Berry curvature distribution at each Dirac point has the same sign, which is changed when reversing the rotation direction. In this case the opposite forces lead to opposite directions of the drift.
a, Entropy increase associated with loading into the modulated lattice and reversing the loading procedure. b, Entropy increase rate in the modulated lattice for long holding times. The modulation frequency ω = 2π × 1,080 Hz opens a gap of h × 44 Hz in the non-interacting band-structure. This value, in units of the tunnelling, is similar to the measurements of the main text. The dashed lines show the measured heating in a lattice without modulation with identical interaction strengths.
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Jotzu, G., Messer, M., Desbuquois, R. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014). https://doi.org/10.1038/nature13915
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