Spectral weight transfer in the integer quantum Hall effect and its consequences

Abstract

The energy spectrum of a two-dimensional electron gas placed in a transversal magnetic field B consists of quantized Landau levels. In the absence of disorder, the degeneracy of each Landau level is N = B A/φ 0 , where A is the area of the sample and φ 0 = h/e is the magnetic flux quantum. With disorder, localized states appear at the top and bottom of the broadened Landau level, whereas states in the centre of the Landau level (the critical region) remain delocalized. This single-electron theory adequately explains most aspects of the integer quantum Hall effect 1 . One unnoticed issue is the location of the new states that appear in the Landau level with increasing B. Here, we show that they appear predominantly inside the critical region. This situation leads to a 'spectral ordering' of the localized states, which explains the stripes observed in measurements of the local inverse compressibility 2,3 , of two-terminal conductance 4 and of Hall and longitudinal resistances 5 without the need to invoke interactions as done in previous work 6-8 .The spectrum and eigenstates of a disorder-broadened Landau level can be studied with the well-established approach of diagonalizing the single-electron hamiltonianwhere V (r) is the disorder. We choose A(r) = (0, Bx) and apply periodic boundary conditions (PBCs) to a system of area A = L × L. Owing to the PBCs, the degeneracy of each Landau level, N = BL 2 /φ 0 , is an integer. Usually, in theoretical studies the magnetic field is kept fixed and the electron density n e or, equivalently, the filling factor ν = n e A/N, is swept by adjusting the Fermi energy E F .As the experiments mentioned above investigate the behaviour of various quantities in the (n e , B) plane, we need to understand how the spectrum changes when B is also tuned. Given the constraint that an integer number of fluxes must penetrate the sample, B can only change in discrete steps of φ 0 /L 2 . Therefore, we ask the following question: how do single-electron wavefunctions evolve when one more magnetic flux is inserted?Let |i, N , 1 ≤ i ≤ N be the eigenstates of a spin-polarized Landau level corresponding to a given disorder V (r) and a magnetic field B = N φ 0 /L 2 . The states are ordered by their energies E 1 < E 2 < ··· < E N (accidental degeneracies can be lifted with minute changes in V (r)).To see how the wavefunctions evolve when B increases, we calculate their disorder-averaged overlaps:where 1 ≤ i ≤ N and 1 ≤ j ≤ N + 1 label two eigenstates of the hamiltonian (1) with the same disorder potential but different magnetic fields. The overline indicates a disorder average. In the results presented here we typically average over 1,000 disorder realizations, and show results only for the spin-polarized lowest Landau level. Similar results are expected in higher Landau levels. The disorder potential V (r) is modelled as a sum of many shortrange, randomly placed gaussian scatterers. We show four sets of data, for L = 250, 400, 500, 750 nm, N = 50, 128, 200, 550 and therefore B = 1.654 T for the first t...