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Symmetric Fermi projections and Kitaev’s table: Topological phases of matter in low dimensions

Gontier, David; Monaco, Domenico; Perrin-Roussel, Solal (2022), Symmetric Fermi projections and Kitaev’s table: Topological phases of matter in low dimensions, Journal of Mathematical Physics, 63, 4. 10.1063/5.0084326

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Journal of Mathematical Physics
Volume
63
Number
4
Publisher
American Institute of Physics
Publication identifier
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Author(s)
Gontier, David cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Monaco, Domenico
Dipartimento di Matematica [Roma TRE]
Perrin-Roussel, Solal cc
Abstract (EN)
We review Kitaev’s celebrated “periodic table” for topological phases of condensed matter, 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.