Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
Disorder correction
What is the impact of the self-energy Σ B on the spectral function (3.45)?
The imaginary part broadens the Lorentzian
S B (k, ω) =
−2ImΣ B (k, ɛk)
[�ω − ɛk] 2 + [ImΣ B (k, ɛk)] 2.
(3.65)
This merely has an effect on the density of states (3.64), because as function
of ω, the Lorentzian is still normalized. For the evaluation of the integral
(3.63), we can approximate the Lorentzian with a Dirac δ-distribution at
the corrected dispersion relation ɛk = ɛk
ρ(ω) =
� 1 + V 2
0
µ 2 ΛN(k) � :
� d d k
(2π) dδ(�ω − ɛk) = Sd
(2π) d
�
k d−1
� �
�∂ɛk
�
� �
� ∂k �
−1 �
k=k0
. (3.66)
� 1 + V 2
0
µ 2 ΛN(k0) � . To leading order in the
Here, k0 is defined by �ω = ɛk0
disorder strength, equation (3.66) can be expressed as
� 2 V0 ρ(ω) = ρ0(ω) 1 −
µ 2
�
d + k ∂
�
vph(k)
∂k vg(k) Λ(k)
�
k=kω
, (3.67)
with �ω = ɛkω , the phase velocity vph(k) = ɛk/(�k), and the group velocity
vg = ∂kɛk/�. The velocity ratio vg/vph = 1+k2ξ2 /2
1+k2ξ2 is one in the linear soundwave
part of the spectrum and approaches 1/2 in the regime of the quadratic
particle spectrum.
A detailed study of the disorder-averaged density of states in the hydrodynamic
regime will be presented in section 4.1.
80