Localization and String Theory

  • Shinobu Hikami
Part of the NATO ASI Series book series (NSSB, volume 254)


Anderson localization has been investigated by the renormalization group method [1–3]. The diffusion-diffusion interaction of a particle in a random potential becomes important in two-dimensions (2D), and eventually, the localized state is realized in 2D when the random potential has time reversal invariance and space-inversion symmetry (orthogonal case). For the strong spin-orbit case and for the magnetic impurity or the magnetic field case, the localization behavior belongs to a different universality class (symplectic or unitary case) [4]. These universal localization behaviors are described by an effective Hamiltonian, which predicts the long-distance behavior of a diffusion mode. The diffusion constant is proportional to the conductivity. The effective Hamiltonian becomes equivalent to the non-linear σ-model near 2D from the point of view of the renormalization group analysis for the diffusion constant. The non-linear σ-model [5] has a particular symmetry, and the field variable is restricted to O(N)/O(p) × O(N-p) manifold with the replica limit p = N → 0 due to the random impurity average. We denote this manifold by O(0)/O(0) × O(0), hereafter.


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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Shinobu Hikami
    • 1
  1. 1.Department of Pure and Applied SciencesUniversity of TokyoTokyo153 Japan

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