Duality between the deconfined quantum-critical point and the bosonic topological transition

The authors thank Chong Wang and Meng Cheng for helpful discussions. Y. Q. Q., Y,-Y.H., Z.-Y.L., and Z. Y. M. are supported by the Ministry of Science and Technology of China under Grant No. 2016YFA0300502, the National Science Foundation of China under Grants No. 91421304, No. 11421092, No. 11474356, No. 11574359, and No. 11674370, and the National Thousand-Young-Talents Program of China. Y. Q. Q. would like to thank Boston University for support under its Condensed Matter Theory Visitors Program. Y.-Y. H. acknowledges the Cultivation of Outstanding Innovative Talents Program 2016 of Renmin University of China. The work of A. S. is partly supported through the Partner Group program between the Indian Association for the Cultivation of Science (Kolkata) and the Max Planck Institute for the Physics of Complex Systems (Dresden). A. W. S. is supported by the NSF under Grant No. DMR-1410126 and would also like to thank the Institute of Physics, Chinese Academy of Sciences for visitor support. C. X. is supported by the David and Lucile Packard Foundation and NSF Grant No. DMR-1151208. We thank the following institutions for allocation of CPU time: the Center for Quantum Simulation Sciences in the Institute of Physics, Chinese Academy of Sciences, the Physical Laboratory of High Performance Computing in the Renmin University of China, the Tianhe-1A platform at the National Supercomputer Center in Tianjin, and the Tianhe-2 platform at the National Supercomputer Center in Guangzhou. (2016YFA0300502 - Ministry of Science and Technology of China; 91421304 - National Science Foundation of China; 11421092 - National Science Foundation of China; 11474356 - National Science Foundation of China; 11574359 - National Science Foundation of China; 11674370 - National Science Foundation of China; National Thousand-Young-Talents Program of China; Boston University; Indian Association for the Cultivation of Science (Kolkata); Max Planck Institute for the Physics of Complex Systems (Dresden); DMR-1410126 - NSF; DMR-1151208 - NSF; David and Lucile Packard Foundation)