Triviality of Bloch and Bloch–Dirac Bundles

Abstract

In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the problem has a positive answer in any dimension d ≤ 3, thus generalizing a previous result by G. Nenciu. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity.

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).…”

“…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…”

“…When d = 2, the above integer characterizes the isomorphism class of E as a vector bundle over T 2 [23]. Since both quantum Hall and quantum spin Hall systems are 2-dimensional, in the following we will mostly restrict ourselves to d = 2, where in particular the previous characterization holds.…”

“…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].…”

“…There are by now several techniques that are able to construct analytic frames out of continuous ones preserving moreover all the symmetries, for example by convolution with suitable kernels [6,7]. These are all incarnations of the more general Oka's principle, which states that in fair generality the obstruction to the triviality of a vector bundle in the continuous category can be lifted to the analytic one [23].…”