Localization and String Theory
Anderson localization has been investigated by the renormalization group method [1â€“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) . 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  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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