ABSTRACT - DRUM - University of Maryland

product of det U(Γ) where Γ is a particle-hole symmetric momentum (Γ = −Γ + G
where G is reciprocal lattice vector). We can relate the det U to the Hamiltonian.
Define a matrix W (Γ) = H(Γ)Ξ. First we can show that it is anti-symmetric:
W T (Γ) = Ξ T H T (Γ) = −Ξ T Ξ −1 H(−Γ)Ξ = −H(Γ)Ξ = W (Γ).
On the other hand, we have
W (Γ) = H(Γ)Ξ = U(Γ)D(Γ)U † (Γ)Ξ.
Notice that ΞU(k) = U ∗ (k)Λ, so
U(Γ)D(Γ)U † (Γ)Ξ = U(Γ)D(Γ)ΛU T (Γ).
One can show that D(Γ)Λ is anti-symmetric. Therefore
Pf W (Γ) = Pf [ U(Γ)D(Γ)ΛU T (Γ) ] = det U(Γ)Pf [D(Γ)Λ] .
Combining, we finally obtain
e iπC = ∏ Γ
Pf W (Γ).
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