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We discuss, for a two-dimensional Dirac Hamiltonian with random scalarpotential, the presence of a $Z_2$ topological term in the non-linear sigmamodel encoding the physics of Anderson localization in the symplectic symmetryclass. The $Z_2$ topological term realizes the sign of the Pfaffian of a familyof Dirac operators. We compute the corresponding global anomaly, i.e., thechange in the sign of the Pfaffian by studying a spectral flow numerically.This $Z_2$ topological effect can be relevant to graphene when the impuritypotential is long-ranged and, also, to the two-dimensional boundaries of athree-dimensional lattice model of $Z_2$ topological insulators in thesymplectic symmetry class.Comment: 4 pages, 4 figure

physical review letters

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>Topological Term, the Global Anomaly, and the Two-Dimensional Symplectic Symmetry Class of Anderson Localization

Z2Topological Term, the Global Anomaly, and the Two-Dimensional Symplectic Symmetry Class of Anderson Localization