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Topological quantum phases underpin many concepts of modern physics. Whilethe existence of disorder-immune topological edge states of electrons usuallyrequires magnetic fields, direct effects of magnetic field on light are veryweak. As a result, demonstrations of topological states of photons employsynthetic fields engineered in special complex structures or externaltime-dependent modulations. Here, we reveal that the quantum Hall phase withtopological edge states, spectral Landau levels and Hofstadter butterfly canemerge in a simple quantum system, where topological order arises solely frominteractions without any fine-tuning. Such systems, arrays of two-level atoms(qubits) coupled to light being described by the classical Dicke model, haverecently been realized in experiments with cold atoms and superconductingqubits. We believe that our finding will open new horizons in severaldisciplines including quantum physics, many-body physics, and nonlineartopological photonics, and it will set an important reference point forexperiments on qubit arrays and quantum simulators.

Quantum Hall phase emerging in an array of atoms interacting with photons