One-parameter scaling in the lowest Landau band: Precise determination of the critical behavior of the localization length

Abstract

The localization properties of noninteracting electrons in the lowest Landau band of a disordered twodimensional system are investigated using a new version of the numerical finite-size scaling method which includes a quantitative statistical procedure of data evaluation. In contrast to other suggestions, universal one-parameter scaling behavior is obtained. The localization length diverges at the center of the band with a critical exponent v='2.34±0.04. The critical region is determined to be about half of the bandwidth. Comparison with experimental quantum-Hall-effect data yields the temperature exponent of the phase-breaking time /? »*2.0 ± 0.2. PACS numbers: 71.50.+t, 71.30.+h, 71.55.Jv The one-parameter scaling theory of localization'"^ indicates that in two dimensions all states are localized in the absence of a magnetic field. On the other hand, the quantum Hall effect requires states which can carry a current"^ and which must, therefore, extend throughout the entire system.Ono was the first to suggest the presence of singularities in the localization length ^ near the centers of the Landau subbands £'".^ Extremely large localization lengths in the band centers were also indicated by numerical calculations.^'^ The nature of the singularity was studied by Aoki and Ando^ by adapting the scaling method proposed earlier by MacKinnon and Kramer^ to the disordered Landau model. They found that ^ oc lE-En I ~\ with v<2 and v<4 for AZ=0 and 1, respectively. Hikami^ estimated v= 1.9 ± 0.2 by perturbational calculation of the inverse participation ratio for the lowest Landau band.Ando and Aoki interpreted their data as showing a magnetic-field-induced destruction of the one-parameter scaling behavior of the localization length. A breakdown of one-parameter scaling of the conductance and a singular behavior in the centers of the Landau bands was also proposed by Levine, Libby, and Pruisken'^ using field theory, but no critical exponent was obtained.For slowly varying random potentials, the quantummechanical problem is equivalent to a classical percolation problem.'' While the percolation critical exponent is j,'^ the inclusion of quantum tunneling leads to v= J. '^ A numerical study, taking into account quantum tunneling and interference near the percolation threshold, yielded v = 2.5 ±0.5 and one-parameter scaling,'"* in contrast to the results mentioned previously.One-parameter scaling of transport is also consistent with recent experimental investigations of the temperature behavior of the magnetoresistivity pxx and the Hall resistivity pxy near the centers of different Landau bands.'^ The maxima of dpxy/dB, d^pxx/dB^, and d^pxy/dB^ diverge as T~"\ with K:=p/2v=0.42 and for /t = l, 2, and 3, respectively, which is consistent withwhere L-.^CCT'P^'^ is the phase coherence length.This paper resolves the discrepancies in the scaling behavior discussed in the various approaches mentioned above. Results of a numerical finite-size scaling investigation of the normalized exponential decay length XM/M in the lowest Land...

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