Introduction

Abstract

In condensed matter physics, a crystal without impurities is often described by the one-electron model (Fig. 1.1), where the transport properties of the material are studied using the energy spectrum of a single electron moving under the influence of a periodic array of atoms [1]. Neglecting interaction between electrons, the one-electron Hamiltonian is given by

$$ {H}_0\equiv K+{V}_0, $$

(1.1)

where K is the kinetic energy of the particle and V₀ is the periodic potential function of the lattice. Throughout this work, we take K ≡  − Δ, where Δ is the discrete Laplacian on the Hilbert space ℋ. A crucial assumption of the one-electron model is that the medium is infinite in space, which allows for the use of statistical mechanics without considerations of finite size and boundary effects. However, a common practice is to approximate the infinite system by a finite one, where one examines the dependence of relevant properties on system size. Such methods are incomplete without a proper scaling approach, which extends the finite volume back to infinity and recovers the properties of the extended system. A primary goal of this book is to point out possible issues related to scaling and propose an alternative approach to transport problems, which does not require finite-size approximations.