5.2 Lattice Vibrations
103
parity) and a phase factor of −1 (δ = ±π , negative parity). In Fig. 5.8b, for the non-trivial phase, the
parity changes twice within the Brillouin zone, while for the trivial case it does not. In view of later
discussions of electronic band structure we remind the reader that electronic s-states (p-states) have
positive (negative) parity.
The phase change summed up when going through the entire Brillouin zone must a multiple of
2 π since the physical properties are periodic with the Brillouin zone. The property to calculate for
obtaining a proper topological invariant here is the Berry phase [369, 370] γ n for a branch n with the
more general recipe,
γ n = ı
C
v
∗
n ·
∂
∂k
v n
dk ,
(5.29)
where C is a closed loop in parameter space, here it is the integral over the entire Brillouin zone. The
integrand is the one-dimensional equivalent to the ‘Berry connection’.
1 For the diatomic linear chain,
the integral yields
γ ± =
⎧
⎨
⎩
0
C 2 < C 1
−π C 2 = C 1
−2 π C 2 > C 1
.
(5.30)
One should think also about the following: The situation for C 2 > C 1 means that the A–B inter-dimer
spring constant is larger than the intra-dimer one. But our choice of unit cell is free and the situation can
be thought of differently in terms of B–A dimers where then, for exactly the same physical situation,
the inter-dimer spring constant (now C 1 ) is smaller than the intra-dimer one (C 2 ). Thus, the two bulk
situations will not be much different per se and cannot be distinguished, in particular when the A- und
B-sites are equivalent (here M 1 = M 2 ). However, when two media with different topology have an
interface, edge states develop according to the bulk-boundary correspondence; this will be shown for
the DLCM in Sect. 5.2.10.
In the case C 1 = C 2 (Fig. 5.1b), C + = C × = C and the dispersion relation can be simplified to
Fig. 5.8 Phonon dispersion with the phase of the wavefunction (relative phase between B- and A-sites) shown in false
colors. Left: C 1 /C 2 = 2, right: C 1 /C 2 = 1/2. The parity symmetry of the mode with respect to exchange of A- and
B-sites is indicated with ‘+’ (in-phase) and ‘−’ (out-of-phase)
1 For higher dimensions, the integral over a closed path (or surface) can be replaced via Stokes’ theorem with an area (or
volume) integral of the Berry flux. Also it is invariant under a ‘gauge transformation’ where v n is replaced by exp(i θ) v n .
103
parity) and a phase factor of −1 (δ = ±π , negative parity). In Fig. 5.8b, for the non-trivial phase, the
parity changes twice within the Brillouin zone, while for the trivial case it does not. In view of later
discussions of electronic band structure we remind the reader that electronic s-states (p-states) have
positive (negative) parity.
The phase change summed up when going through the entire Brillouin zone must a multiple of
2 π since the physical properties are periodic with the Brillouin zone. The property to calculate for
obtaining a proper topological invariant here is the Berry phase [369, 370] γ n for a branch n with the
more general recipe,
γ n = ı
C
v
∗
n ·
∂
∂k
v n
dk ,
(5.29)
where C is a closed loop in parameter space, here it is the integral over the entire Brillouin zone. The
integrand is the one-dimensional equivalent to the ‘Berry connection’.
1 For the diatomic linear chain,
the integral yields
γ ± =
⎧
⎨
⎩
0
C 2 < C 1
−π C 2 = C 1
−2 π C 2 > C 1
.
(5.30)
One should think also about the following: The situation for C 2 > C 1 means that the A–B inter-dimer
spring constant is larger than the intra-dimer one. But our choice of unit cell is free and the situation can
be thought of differently in terms of B–A dimers where then, for exactly the same physical situation,
the inter-dimer spring constant (now C 1 ) is smaller than the intra-dimer one (C 2 ). Thus, the two bulk
situations will not be much different per se and cannot be distinguished, in particular when the A- und
B-sites are equivalent (here M 1 = M 2 ). However, when two media with different topology have an
interface, edge states develop according to the bulk-boundary correspondence; this will be shown for
the DLCM in Sect. 5.2.10.
In the case C 1 = C 2 (Fig. 5.1b), C + = C × = C and the dispersion relation can be simplified to
Fig. 5.8 Phonon dispersion with the phase of the wavefunction (relative phase between B- and A-sites) shown in false
colors. Left: C 1 /C 2 = 2, right: C 1 /C 2 = 1/2. The parity symmetry of the mode with respect to exchange of A- and
B-sites is indicated with ‘+’ (in-phase) and ‘−’ (out-of-phase)
1 For higher dimensions, the integral over a closed path (or surface) can be replaced via Stokes’ theorem with an area (or
volume) integral of the Berry flux. Also it is invariant under a ‘gauge transformation’ where v n is replaced by exp(i θ) v n .