Fragile Phases as Affine Monoids: Classification and Material Examples

Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an elementary band representation (atomic limit band) splits into a particular set of bands. For the first time, we obtain a complete classification of the fragile topological phases, which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure that far surpasses that of stable or strong topological states. We find and enumerate hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues, and we link the mathematical structure of these phases to that of affine monoids in mathematics. Furthermore, for the first time, we predict and calculate (hundreds of realistic) materials where fragile topological bands appear, and we showcase the very best ones.We acknowledge Maia G. Vergniory, Zhijun Wang, and Claudia Felser for collaboration on previous related works. Z. S. and B. B. are supported by the Department of Energy Grant No. desc-0016239; the National Science Foundation EAGER Grant No. DMR 1643312; Simons Investigator Grants No. 404513, No. ONR N00014-14-1-0330, and No. NSF-MRSECDMR DMR 1420541; the Packard Foundation No. 2016-65128; and the Schmidt Fund for Development of Majorama Fermions funded by the Eric and Wendy Schmidt Transformative Technology Fund. L. E. is supported by the Government of the Basque Country (Project No. IT779-13), the Spanish Ministry of Economy and Competitiveness (MINECO), and the European Fund for Economic and Regional Development (FEDER; Project No. MAT2015-66441-P). We are mostly grateful to theMPGcomputing center for allowing us access to the Cobra and Draco Supercomputers. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. The mathematical analysis of this work is supported by the Department of Energy