Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

ρµξ d
0.2
0.15
0.1
0.05
1 2 3 4 5
(a) d = 1
¯hω/µ
3.4. Deriving physical quantities from the self-energy
ρµξ d
0.08
0.06
0.04
0.02
1 2 3 4 5
(b) d = 2
¯hω/µ
ρµξ d
0.06
0.05
0.04
0.03
0.02
0.01
1 2 3 4 5
(c) d = 3
¯hω/µ
Figure 3.5.: Bogoliubov density of states (solid line). At low energies �ω ≪ µ, the density
of states is phonon-like ρ ∝ ωd−1 (dashed line), whereas at high energies �ω ≫ µ it is
particle like ρ ∝ (�ω − µ) d
2 −1 (gray line).
3.4.6. Density of states
The spectral function (3.45) can be regarded as the probability of a Bogoliubov
quasiparticle in state k to have energy �ω. By integrating over all
possible states, we obtain the average density of states, i.e. the probability
to find a state at a given energy �ω
� d d k
ρ(ω) =
(2π) d
S(k, ω)
. (3.63)
2π
The spectral function is modified by the corrections to the free dispersion
relation (3.47), i.e. by ImΣ B and ReΣ B = V 2
0 Λ/µ 2 .
Clean density of states
Already in absence of disorder, the density of states shows an interesting
feature, namely the transition from sound-wave like excitations to
particle-like excitations. This implies a transition from ρsw(ω) ∝ ω d−1 to
ρparticle(ω) ∝ ω d
2 −1 . According to equation (3.41), the clean density of states
is found as
ρ0(ω) = Sd
(2π) dkd−1 ω
� �
�
�
∂k �
�
�∂ω
� kω
, �ω = µ kωξ � k 2 ωξ 2 + 2. (3.64)
For low-energy excitations with �ω = c k, this reduces to a phonon-like
dispersion relation ρ ∝ ωd−1 , and for high energies it passes over to the
free-particle density of states ρ ∝ ω d
2 −1 , see figure 3.5.
79