ABSTRACT - DRUM - University of Maryland

which allows further simplification of C:
C = 1 ∑
(∫ π
dk x a m (k x , π) −
π
m
0
∫ π
0
)
dk x a m (k x , 0) .
We can relate the gauge fields a m to the eigenstates of the Hamiltonian. Define
the unitary matrix U(k) as
U † (k)H(k)U(k) = D(k),
where D(k) is the diagonal matrix of eigenvalues ordered in descending order of
value. It is easy to show that
∑
A m (k) = −i∇ k ln det U(k).
m
Therefore
As a result,
∑
a m (k, θ) = −i∂ k ln det U(k, θ).
m
C = 1 [∫ π
dk ∂ k ln det U(k, π) −
πi
0
∫ π
It can be written in a more natural form:
0
]
dk ∂ k ln det U(k, 0) = 1 det U(π, π) det U(0, 0)
ln
πi det U(0, π) det U(π, 0) .
e iπC =
det U(π, π) det U(0, 0)
det U(0, π) det U(π, 0) .
(A.3)
which is the desired result.
Although in the derivation we choose a certain gauge, the result is certainly
gauge-invariant.
We now further simplify this results. (A.3) basically means that the parity
of Chern number of a 2D particle-hole symmetric insulator is determined by the
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