Fragile Phases as Affine Monoids: Classification and Material Examples

Abstract

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.

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Plain Language Summary

Topological electronic materials have remarkable properties such as perfect conducting surfaces that make them potentially useful in electronic devices, catalysts, and optical technologies. Recently, researchers have proposed strange, new kinds of topological states called “fragile,” but many details of these states remain unclear. Here, we have made progress toward that end by coming up with a way to classify a certain subset of the fragile topological states.

Stable topological insulators are characterized by a set of integer numbers known as topological invariants. These numbers remain unchanged if we add more atoms to the state. However, fragile topological states can be characterized by integer numbers only if some other constraints are satisfied. Unlike the stable topological states, if we add atoms to the fragile states, these other constraints, and hence the topology, can be broken.

We fully analyze the structure of the fragile topological states that can be determined by the symmetry property of the band structure. These fragile states are characterized by inequalities or certain types of equations of integers rather than the integers themselves. With this procedure, we fully classify all such fragile states and identify 100 materials that could host them.

With our classification, physicists could focus further studies on the response of these fragile topological states. Since the topological band theory applies not only to electronic materials but also to any periodic linear system, our procedure also applies to photonic systems and metamaterial systems.