Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4.4. Particle regime
which coincides with [23, eq. (34)]. All quantities related to the interaction
have disappeared. The energy scale µ has been replaced with the kinetic energy
ɛ0 k of the particle and the only remaining length scales are combined in
the reduced momentum kσ. In contrast to the sound-wave regime [subsection
4.1.2], the inverse mean free path of particles diverges at low energies.
Slow particles are scattered strongly, and the perturbation theory breaks
down [22, 23].
4.4.2. Renormalization of the dispersion relation in the
Bogoliubov regime
Let us calculate the real part of the self-energy (3.39), i.e. the disorder correction
of the dispersion relation, in the non-interacting limit µ → 0. Apart
from W (1) , we also need the anomalous scattering element Y (1) (2.45d),
which does not vanish as one might expect for non-interacting particles.
Instead, we find
Y (1)
k+q k ≈
� Vq
� k 2 + (k + q) 2 � /q 2
Vq(kξ) 2
qξ ≫ 1 , (4.37a)
qξ ≪ 1 . (4.37b)
The real part of the self-energy (3.39), together with the correction fixing
the particle number (3.59), is an integral over the momentum transfer q.
The correlator Cd(qσ) provides a sharp cutoff at q = 2/σ, and also the
smoothing factors contained in the envelope functions suppress the integrand
for qξ ≫ 1. Thus, we can approximate the integrand for small q. Using
(4.35b), (4.37b) and (2.50), we find
M (0)
N
:= ɛk
µ ΛN =
� d d q
(2π) dσd q
Cd(qσ)
2ξ2 (2 + q2ξ2 ) 2.
(4.38)
Only the anomalous coupling y (1) and the second order w (2) have contributed
to this leading order, but not the normal scattering w (1) . Equation (4.38)
holds for kξ ≫ 1 and kσ not too small. It is a strictly positive function of
σ/ξ. In the leading order, there is no dependence on the momentum kξ, thus
it does not affect quantities like the group velocity and the effective mass
(inverse of the second derivative of the dispersion relation). It is interpreted
as a correction of the chemical potential in
�
ɛk
(2.36)
= ɛ 0 k + µ ↦→ ɛ 0 k + µ
1 + M (0)
N
V 2
0 /µ 2
�
. (4.39)
The integral (4.38) has the same form as the correction (3.59), due to fixing
the particle number. However, it has the opposite sign. The correction
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