Symmetric Fermi projections and Kitaev’s table: topological phases of matter in low dimensions. (English) Zbl 1507.82074
Summary: We review A. Kitaev’s celebrated “periodic table” for topological phases of condensed matter [AIP Conf. Proc. 1134, 22–30 (2009; Zbl 1180.82221)], which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections that depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for zero- and one-dimensional systems and recover the (weak and strong) topological invariants proposed by Kitaev and others.
©2022 American Institute of Physics
©2022 American Institute of Physics
MSC:
| 82D20 | Statistical mechanics of solids |
| 81V70 | Many-body theory; quantum Hall effect |
| 82D25 | Statistical mechanics of crystals |
| 57Z05 | Relations of manifolds and cell complexes with physics |
Citations:
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