Anomalous quantum Hall effects in single-layer and bilayer graphene are
related with nontrivial topological properties of electron states (Berry phases
$\pi$ and 2$\pi$, respectively). It was known that the Atiyah-Singer index
theorem guarantees, for the case of the single-layer, existence of zero-energy
states for the case of inhomogeneous magnetic fields assuming that the total
flux is non-zero. This leads, in particular, to appearance of midgap states in
corrugated graphene and topologically protects zero-energy Landau level in
corrugated single-layer graphene. Here we apply this theorem to the case of
bilayer graphene and prove the existence of zero-energy modes for this case.Comment: minor changes (misprints are fixed, a bit more explanations in the
mathematical part are added
The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems. We address here the simplest nontrivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X. We compute the K 1 (X)valued index in terms of the topological data over the boundary. The second result is universality of the index: we show that the index is a universal additive homotopy invariant for such families, if the vanishing on families of invertible operators is required.
Contents1. ind a (γ 1 ) = ind a (γ 2 ).
Family index for self-adjoint unbounded operatorsIn order to deal with unbounded self-adjoint operators (in particular, with self-adjoint differential operators) directly, one needs an analogue of the Atiyah-Singer theory [2]. Cf.[5], [7], [9]. This section is devoted to such an analogue adapted to our framework. The functor K 1 . Let H be a Hilbert space. Denote by B(H) the space of bounded linear operators H → H with the norm topology.
We give some general criteria of being a homeomorphism for continuous mappings of topological manifolds, as well as criteria of being a diffeomorphism for smooth mappings of smooth manifolds. As an illustration, we apply these criteria to the problems arising in two-and three-dimensional grid generation.
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