6.2 Electrons in a Periodic Potential
143
Fig. 6.6 Brillouin zone of a two-dimensional rectangular lattice and mapping to a torus
Fig. 6.7 Brillouin zone of
a rectangular lattice with a
a constant function and b a
function that changes sign.
On the right the topology of
the situation is visualized
1
2 π
BZ
f (k) d
2 k = n .
(6.28)
If for example another function f that is independent of k x and changes sign in k y -direction with
f dk y = 0, similar to the curvature of the torus, is integrated over the Brillouin zone, the results will
be zero. This is schematically shown in Fig. 6.7 if the integrand is interpreted as curvature.
The generalization of the Berry phase [369] to Bloch states has been made in [465, 466]. For a twodimensional system with Bloch bands and with Bloch functions u m (k) as in (6.3), the integrand leading
to a topological invariant is given by the Berry connection (cmp. (5.29)) A m = =u m (k)|ı ∇ k |u m (k)
and its Berry curvature or Berry flux in three-dimensional notation F m = ∇ k × A m . The Chern number
C m for a band (separated by gaps from other bands), defined as integral over the Brillouin zone,
C m =
1
2 π
BZ
F m d
2 k ,
(6.29)
takes only integer values and is a topological invariant. That means that small variations of the Hamilton
operator behind the band structure do not change its value. In the case of degeneracies, still the sum of
Chern numbers over all occupied bands, n =
m n m , is a topological invariant as long as the empty