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.
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We thank X. Wen and B. Zeng for helpful discussions. This work is supported National Key Basic Research Program of China (Grant No. 2013CB921800 and No. 2014CB848700), the National Science Fund for Distinguished Young Scholars (Grant No. 11425523), the National Natural Science Foundation of China (Grants No. 11375167, No. 11227901, No. 11575173, and No. 91021005), the Strategic Priority Research Program (B) of the CAS (Grant No. XDB01030400) and Key Research Program of Frontier Sciences of the CAS (Grant No. QYZDY-SSW-SLH004). Y.W. acknowledges support from the John Templeton foundation No. 39901. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. Y.W. is also supported by the Shanghai Pujiang Program Grant No. KBH 1512328. L.-Y.H. would like to acknowledge support by the Thousand Young Talents Program, and Fudan University.