Twisted bulk-boundary correspondence of fragile topology

Exploiting topological features in materials is being pursued as a route to build in robustness of particular properties. Stemming from crystalline symmetries, such topological protection renders the properties robust against defects and provides a platform of rich physics to be studied. Recent developments have revealed the existence of so-called fragile topological phases, where the means of classification due to symmetry is unclear. Z.-D. Song et al. and Peri et al. present a combined theoretical and experimental approach to identify, classify, and measure the properties of fragile topological phases. By invoking twisted boundary conditions, they are able to describe the properties of fragile topological states and verify the expected experimental signature in an acoustic crystal. Understanding how fragile topology arises could be used to develop new materials with exotic properties. Science , this issue p. 794 , p. 797 A better understanding of fragile topological phases could lead to the development of materials with exotic properties. A topological insulator reveals its nontrivial bulk through the presence of gapless edge states: This is called the bulk-boundary correspondence. However, the recent discovery of “fragile” topological states with no gapless edges casts doubt on this concept. We propose a generalization of the bulk-boundary correspondence: a transformation under which the gap between the fragile phase and other bands must close. We derive specific twisted boundary conditions (TBCs) that can detect all the two-dimensional eigenvalue fragile phases. We develop the concept of real-space invariants, local good quantum numbers in real space, which fully characterize these phases and determine the number of gap closings under the TBCs. Realizations of the TBCs in metamaterials are proposed, thereby providing a route to their experimental verification.