Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry

Abstract

Abstract. We describe some applications of group-and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e. g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The rôle of additional Z 2 -symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same Z 2 -symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.

Advances in Quantum Mechanics

“…As was already mentioned, in this case the presence of a further symmetry kills the topological obstruction given by the Chern number (2) [23,22]. However, the same symmetry allows to refine the notion of "symmetric Bloch frame" by requiring that it be also time-reversal symmetric (compare Section 2.3).…”

“…The above result shows that if deg([ U]) = 2r ∈ 2Z is even, it is still possible to "unwind" the map U with the help of an auxiliary map X, without breaking the symmetries (τ-equivariance, time-reversal) enjoyed by the frame Φ as in (22). Indeed, it is easily verified that the map X : ∂ B eff → U(m) defined (in the basis where ε is of the form (11) …”

“…Periodicity means that the operator H should commute with the translations associated to a lattice Γ ≃ Z d ⊂ R d , namely the Bravais lattice of the crystal under scrutiny. This symmetry of the Hamiltonian leads to a partial diagonalization of it, by looking at common (generalized) eigenstates for the Hamiltonian and the translations: this procedure, which is reminiscent of the Fourier decomposition, goes by the name of Bloch-Floquet reduction [22]. In this representation, the Hamiltonian becomes a fibered operator, with fibre H(k) acting on a space H f containing the degrees of freedom associated to a unit cell for Γ .…”

“…Indeed, one can associate to any family of projectors satisfying (P 1 ) and (P 2 ) a vector bundle E over the torus T d , called the Bloch bundle, via a procedure reminescent of the SerreSwan construction: the fibre of E over the point k ∈ T d is the m-dimensional vector space Ran P(k) (we refer to [23,22] for details). The geometry of the Bloch bundle for d = 2 is what enters in the theoretical understanding of the quantum Hall effect: the integer n that equals the Hall conductivity (1) in natural units is the (first) Chern number of E , defined as…”

“…We call a vector bundle endowed with such an endomorphism Θ a time-reversal symmetric vector bundle. One can verify that if d = 2 every such vector bundle is trivial, i. e. isomorphic to the product bundle T 2 × C m , since under (P 3 ) the integrand in the definition (2) of the Chern number is an odd function of k, and hence integrates to zero on T 2 [23,22]. However, the Bloch bundle may still be non-trivial as time-reversal symmetric bundle [8,10].…”