How to use this book xv 1 Introduction: mesoscopic physics 1 1.1 Interference and disorder 1 1.2 The Aharonov-Bohm effect 4 1.3 Phase coherence and the effect of disorder 7 1.4 Average coherence and multiple scattering 9 1.5 Phase coherence and seif-averaging: universal fluctuations 12 1.6 Spectral correlations 14 1.7 Classical probability and quantum crossings 15 1.7.1 Quantum crossings 17 1.8 Objectives 18 2 Wave equations in random media 31 2.1 Wave equations 2.1.1 Electrons in a disordered metal 31 2.1.2 Electromagnetic wave equation-Helmholtz equation 32 2.1.3 Other examples of wave equations 2.2 Models of disorder 36 2.2.1 The Gaussian model 2.2.2 Localized impurities: the Edwards model 39 2.2.3 The Anderson model Appendix A2.1: Theory of elastic collisions and single scattering A2.1.1 Asymptotic form of the solutions 44 A2.

We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase Z(k ), which is a Berry phase across an appropriately chosen one dimensional Brillouin zone, and the existence of a localized state of momentum k at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one dimensional toy model as well as for graphene ribbons with zigzag edges. The range of k for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, it confirms and generalizes the results of several previous works.