Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

Abstract

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 which 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 0-and 1-dimensional systems, and recover the (weak and strong) topological invariants proposed by Kitaev and others.(T -symmetry).If ε T = 1, this T -symmetry is said to even, and if ε T = −1 it is odd.Definition 1.2 (Charge-conjugation symmetry). Let C : H → H be an antiunitary operator such thatWe say that a continuous map P :If ε C = 1, this C-symmetry is said to even, and if ε C = −1 it is odd.Definition 1.3 (Chiral symmetry). Let S : H → H be a unitary operator such that S 2 = I H . We say that P : T d → G n (H) satisfies chiral or sublattice symmetry, or in short S-symmetry, ifThe simultaneous presence of two symmetries implies the presence of the third. In fact, the following assumption is often postulated [RSFL10]: Assumption 1.4. If the Hilbert space H is endowed with anti-unitary operators T and C as in Definitions 1.1 and 1.2 respectively, then we assume that their product S := T C is an operator as in Definition 1.3, that is, S is unitary and S 2 = I H .Remark 1.5. This assumption is tantamount to require that the operators T and C commute or anti-commute among each other, depending on their even/odd nature. Indeed, the product of two anti-unitary operators is unitary, and the requirement that T C behaves as in Definition 1.3 readsThe same sign determines whether S commutes or anti-commutes with T and C.