To gain insight on their response and surface states, topological insulators (TIs) and semimetals are frequently re-expressed as the periodic tuning cycles of lower-dimensional insulators with nontrivial topology or multipole moments. The recent discovery of 2D higher-order polarization phases with corner charges, and the recognition that these phases can be pumped to realize 3D higher-order TIs (HOTIs) and semimetals poses a clear question: “what are the minimal conditions required to realize a 2D insulator with higher-order polarization topology, and correspondingly, what kinds of 3D insulators and semimetals are realized by pumping this minimal 2D phase?” To answer this question, I work backwards in complexity: I first pose a new model of a quadrupole insulator (QI) based on atomic orbitals and band representations (Topological Quantum Chemistry), which I then tune to realize conventional Dirac semimetals with quadrupolar hinge states. I then show how this new QI model is closely related to “fragile”
topological phases, which are revealed to also exhibit corner charges. Paring away symmetries, I then focus on the simplest HOTIs and higher-order (nodal-line) semimetals, which are revealed to be derived from the well-studied magnetic “axion insulator” (AXI). Using nested
Wilson loops, I then show that an AXI, and thus the simplest HOTI, can be re-expressed as the pumping cycle of a fragile TI with corner modes. For each of the strong and semimetallic phases discussed, I provide representative real material candidates. Finally, time permitting, I explain how the fragile pumping formulation directly implies the crystal defect and threaded flux responses of 3D AXIs, HOTIs, and weak fragile TIs.
Results presented in this seminar are drawn from the following works:
Z. Wang*, B. J. Wieder*, J. Li, B. Yan, and B. A. Bernevig, arXiv:1806.11116 (2018)
B. J. Wieder and B. A. Bernevig, arXiv:1810.02373 (2018)
B. J. Wieder*, Z. Wang*, J. Cano*, X. Dai, L. M. Schoop, B. Bradlyn, and B. A. Bernevig, In Review
F. Schindler, S. S. Tsirkin, T. Neupert, B. A. Bernevig, and B. J. Wieder, In Preparation