## ABSTRACT

One of the biggest challenges of physics is the classification of different states of matter. Until recently, the phases of matter could be understood using Landau–Lifshitz theory [1], which characterizes states in terms of their underlying symmetries that are spontaneously broken. For example, a magnet spontaneously breaks rotation symmetry, although the fundamental interactions within the magnet itself are isotropic in nature. Starting from 1980 with the discovery of the quantum Hall effect (QHE) [2, 3], new phases have emerged that are not characterized by broken symmetry, but rather by the presence of a global, or topological, invariant that is contributed to by all of the states in the system. The QHE appears in large magnetic fields when the two-dimensional density of states becomes broken into successive, highly degenerate Landau levels. As is the case in the traditional Hall effect, the presence of an applied in-plane electric field drives a current from one side of the system to the other. In the case of the QHE, the position of the Fermi level relative to the Landau levels determines the type of transport that is observed. When the Fermi level is inside a Landau level, there are many states available to carry current across the system, both within the bulk and at the edge. However, when the Fermi level is between two successive Landau levels, then the kinetic energy of the bulk is quenched and becomes insulating. The resulting conduction becomes quantized, and appears through chiral (unidirectional) edge states observed at the boundary of the sample that are immune against backscattering. The connection between the QHE and topology is made via the TKKN (after Thouless, Kohomoto, Nightingale, and den Nijs) invariant, in which the first Chern number describes the winding of the corresponding Bloch wave functions over the magnetic Brillouin zone [4]. This quantized number corresponds exactly to the number of propagating edge states in the QHE, and is invariant to the details of the underlying system, so long as the two edge states remain spatially segregated. The TKNN invariant thus provides the means of calculating the topological invariant for the QHE.