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We study topological insulators characterized by the integer topological invariant Z, in even and odd spacial dimensions.
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A Wigner transform of a function of two variables f ( x , x ′ ) f(x,x^{\prime}) is defined as f ( k , R ) = ∫ d r e i k r f ( R + r / 2 , R − r / 2 ) f(k,R)=\int dr\,e^{ikr}f(R+r/2,R-r/2)
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