The Integer Quantum Hall Effect

Abstract

Learning goals • We know the pumping argument of Laughlin and the concept of spectral flow. • We know that there is always a delocalized state in each LL. • We know that xy is given by the Chern number. • We understand why the Chern number is an integer. • K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980) 3.1 Laughlin's argument for the quantization of xy In the following we try to understand the pumping argument presented by R. Laughlin [1]. 3.1.1 Spectral flow The idea of spectral flow is central to the pumping argument of Laughlin. We try to understand this idea on the example of a particle on a ring threaded by a flux H = 1 2 (@ eA) 2) n () = 1 p 2 e in with ✏ n = 1 2 ✓ n 0 ◆ 2 . (3.1) After the insertion of a full flux quantum 0 , the Hamiltonian returns to itself. However, if we follow each state adiabatically, we see that the first excited and the ground state exchanged their positions. This situation is called spectral flow: While the spectrum has to be the same for = 0 and = 0 , the adiabatic evolution does not need to return the ground state to itself! This is illustrated in Fig. 3.1. While the example of a particle on a ring is particularly simple, the same situation can occur for a general setup where after the insertion of a flux 0 the original ground state is adiabatically transferred to an excited state. Let us now see how this spectral flow e↵ect applies to the quantum Hall problem. 3.1.2 The ribbon geometry Laughlin proposed that if xy is quantized, it should not depend on the details of the geometry. One is therefore allow to smoothly deform a rectangular sample in the following way: where in the last step we replaced the applied voltage V ! @ t with the electromotive force of a time-dependent flux through the opening of the " Corbino " disk. Let us see what happens when we inset this flux. We make use of the eigenfunctions in the radial gauge 1 ⇠ z m exp(z ⇤ z/4), where z = (x + iy)/l which we can also write as 1 See exercise 9.2.