We ingest tiny crystals of table salt every day. How well do we understand such commonplace substances? More broadly, how do we characterize and classify the insulating states of matter? Recent advances in the topological approach in condensed-matter physics offer a classification based on the winding and the quantum entanglement in the ground-state wavefunction. The nontrivial bulk topology is often manifested as anomalous surface states, but only corners and hinges exhibit gapless modes in the case of "higher-order" topology.
In this talk, we discuss that even absolutely topologically trivial materials may exhibit fractional charges on their corners and hinges. To predict these boundary signatures from the bulk, we develop a general formulation of bulk multipole moments, directly generalizing the "modern theory" formulation of the bulk polarization. As an example, we discuss e/8 fractional corner charges of grains of table salt and propose their direct measurement using atomic force microscopy.
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