Experimentally probing topological order and its breakdown through modular matrices

The modern concept of phases of matter has undergone tremendous developments since the first observation of topologically ordered states in fractional quantum Hall systems in the 1980s. In this paper, we explore the following question: in principle, how much detail of the physics of topological orders can be observed using state of the art technologies? We find that using surprisingly little data, namely the toric code Hamiltonian in the presence of generic disorders and detuning from its exactly solvable point, the modular matrices—characterizing anyonic statistics that are some of the most fundamental fingerprints of topological orders—can be reconstructed with very good accuracy solely by experimental means. This is an experimental realization of these fundamental signatures of a topological order, a test of their robustness against perturbations, and a proof of principle—that current technologies have attained the precision to identify phases of matter and, as such, probe an extended region of phase space around the soluble point before its breakdown. Given the special role of anyonic statistics in quantum computation, our work promises myriad applications both in probing and realistically harnessing these exotic phases of matter. Fundamental fingerprints of topological orders may be characterized uniquely and purely by experimental means. Here the authors provide a proof of principle demonstration using interferometric measurement in a two-dimensional lattice system.