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Experimental realization of the topological Haldane model with ultracold fermions
G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, T. Esslinger
Nature 515, 237–240 (2014);
Observation of topological transitions in interacting quantum circuits
P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, J. M. Martinis
Nature 515, 241–244 (2014)

The discovery of topological phases in condensed-matter systems has changed the modern conception of phases of matter. The Haldane model on a honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter. In fact, the model has provided the conceptual basis for theoretical and experimental research exploring topological insulators and superconductors.

These two works report the experimental simulation of the Haldane model in two different platforms, namely optical lattices and superconducting circuits.

In the case of optical lattices, Jotzu and co-workers 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. In the setup, time-reversal symmetry and inversion symmetry are broken, which opens a gap in the band structure. The competition between the two broken symmetries gives rise to a transition between topologically distinct regimes. The approach, which allows for tuning the topological properties dynamically, is suitable even for interacting fermionic systems. This work represents a crucial step towards cold-atom realizations of exotic phenomena such as fractional quantum Hall phases and fractional Chern insulators.

In the case of superconducting circuits, Roushan and co-workers investigate basic topological concepts of the Haldane model after mapping the momentum space of this condensed-matter model to the parameter space of a single-qubit Hamiltonian. In addition to constructing the topological phase diagram, they visualize the microscopic spin texture of the associated states and their evolution across a topological phase transition. They also study the topology in an interacting quantum system, which requires a new qubit architecture that allows for simultaneous control over every term in a two-qubit Hamiltonian. By exploring the parameter space of this Hamiltonian, they discover the emergence of an interaction-induced topological phase. This work establishes a powerful platform to study topological phenomena in quantum systems.