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The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which e.g., underlies the theoretical description of graphene, is computed numerically.
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An exception to the single parameter scaling is the two parameter scaling of the integer quantum Hall effect: D. E. Khmelnitskii, JETP Lett. 38
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From the field theory point of view, the topological term in the NL σ \sigma M is responsible for the delocalization in 1 and 2D. The 3D NL σ \sigma M for the symplectic symmetry class does not allow topological terms, but a Chern-Simons type term is possible, which might be responsible for the delocalization. See for example, A. G. Abanov, Phys. Lett. B 492
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A similar switching of Kramers pairs was found in the position of the Wannier states in the ℤ 2 \mathbb{Z}_{2} pumping, L. Fu and C. L. Kane, Phys. Rev. B 74
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