Imaging topology of Hofstadter ribbons

Abstract

Physical systems with non-trivial topological order find direct applications in metrology[1] and promise future applications in quantum computing[2,3]. The quantum Hall effect derives from transverse conductance, quantized to unprecedented precision in accordance with the system's topology[4]. At magnetic fields beyond the reach of current condensed matter experiment, around 10^4 Tesla, this conductance remains precisely quantized but takes on different values[5]. Hitherto, quantized conductance has only been measured in extended 2-D systems. Here, we engineered and experimentally studied narrow 2-D ribbons, just 3 or 5 sites wide along one direction, using ultracold neutral atoms where such large magnetic fields can be engineered[6-11]. We microscopically imaged the transverse spatial motion underlying the quantized Hall effect. Our measurements identify the topological Chern numbers with typical uncertainty of 5%, and show that although band topology is only properly defined in infinite systems, its signatures are striking even in nearly vanishingly thin systems.