Relation between the correlation dimensions of multifractal wave functions and spectral measures in integer quantum Hall systems

Abstract

We study the time evolution of wavepackets of non-interacting electrons in a two-dimensional disordered system in strong magnetic field. For wavepackets built from states near the metal-insulator transition in the center of the lowest Landau band we find that the return probability to the origin p(t) decays algebraically, p(t) ∼ t −D 2 /2 , with a non-conventional exponent D 2 /2. D 2 is the generalized dimension describing the scaling of the second moment of the wavefunction. We show that the corresponding spectral measure is multifractal and that the exponent D 2 /2 equals the generalized dimension D 2 of the spectral measure. 71.50.+t,71.30.+h,71.55.Jv Typeset using REVT E X 1

Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

“…It is found that D 2 = 1.66 ± 0.05, D 2 = 0.83 ± 0.03, η = 0.35 ± 0.05, and α 0 = 2.19 ± 0.03. Thus, within the numerical uncertainties, the proposed relations η = 2 − D 2 [13,9], D 2 = 2 D 2 [3], and Λ typ 0 π = (α 0 − 2) −1 [14] hold also for the two-dimensional system with spin-orbit interaction. The model used to calculate the eigenstates has been put forward by Ando [15,16] to simulate two-dimensional systems in n-channel inversion layers on surfaces of III-VI semiconductors.…”

“…The correlation dimension of the spectral measure, D 2 , which is related to the temporal decay [25,3] of the maximum of a wavepacket (probability of return) built from critical eigenstates was calculated for the same energy interval. This exponent is obtained from the scaling relation of the local density of states…”

“…At T = 0 the so called mobility edge separates insulating (localized) from current carrying (extended) states. Recently, the electronic properties directly at the critical point have received increasing attention [1][2][3][4][5][6]. Two-dimensional (2d) systems are very well suited for numerical investigations of the electronic eigenstates at a critical point.…”

“…a two-dimensional disordered system of non-interacting electrons where a strong magnetic field causes the localization length to diverge at the centres of the Landau bands with a universal exponent [11]. For these critical states, characteristic quantities like the generalized correlation dimension of the wavefunction D 2 = 1.62 [8,9], the corresponding generalized dimension of the spectral measure D 2 = 0.81 [3], the exponent η = 0.38 [1,3] governing the energy eigenfunction correlations, and α 0 = 2.3 [9,10] which for −1 < ∼ q < ∼ 1 determines the so called f (α)-distribution are known, partly with sufficient precision. This is, however, not the case for the spin-orbit system mentioned above, mainly because of the small system sizes considered so far (M ≤ 18 a) [7].…”

“…A power law relation is observed for l = λL in the range 2a ≤ l ≤ L/2. The average over all eigenstates from the energy interval gives a correlation dimension D 2 E = 1.66±0.05 which is close to the value of 1.63 reported in [7] for smaller system sizes.The correlation dimension of the spectral measure, D 2 , which is related to the temporal decay [25,3] of the maximum of a wavepacket (probability of return) built from critical eigenstates was calculated for the same energy interval. This exponent is obtained from the scaling relation of the local density of states…”