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Scaling, localization and bandwidths for equations with competing periods

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  • Thouless, D.J.
  • Tan, Yong

Abstract

The finite size scaling theory of the measure of the spectrum of Harper's equation is reexamined. For sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as (log p)−2 or p−2 as the denominator p is increased. The question of how special are the properties of the Harper equation, is studied. It is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from “localized” to “extended” at a value of the strength of the off-diagonal term that depends on energy, in contrast to the Harper equation where the transition is energy-independent. This has a crucial effect on the measure of the spectrum.

Suggested Citation

  • Thouless, D.J. & Tan, Yong, 1991. "Scaling, localization and bandwidths for equations with competing periods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 177(1), pages 567-577.
    • Handle: RePEc:eee:phsmap:v:177:y:1991:i:1:p:567-577
    DOI: 10.1016/0378-4371(91)90202-N
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    References listed on IDEAS

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    1. den Nijs, Marcel, 1982. "The critical exponent η of the planar model from one-dimensional quantum field theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 111(1), pages 273-287.
    2. Simon, Leo K & Stinchcombe, Maxwell B, 1989. "Extensive Form Games in Continuous Time: Pure Strategies," Econometrica, Econometric Society, vol. 57(5), pages 1171-1214, September.
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