We consider pure Fermi systems (either normal ones or mean-field superconductors), described by general secondary quantized quadratic Hamiltonians. The goal is to classify all such Hamiltonians having a gap at Fermi energy. We restrict ourselves to the situation where all translational symmetries factorize, in which case the problem reduces to the homotopy classification of the space of gapped Hamiltonians in zero dimension (essentially quantum dot). Following Zirnbauer, we eliminate all unitary symmetry constraints and consider only the remaining meta-symmetries. The remaining constraints stem from particle-hole conjugation, anti-unitary or chiral symmetries. An algebraic structure for these meta-symmetries was proposed by Kitaev. We extended this construction for all ten known symmetry classes. This proves the completeness of Zirnbauer classification and allows us to address the question: Are topological insulators always labeled by $Z$ or $Z_2$ ?

Lab. de Physique des Solides, Orsay