longitudinal
phonon
crystal
diffusion
Re[ω]
Im[ω]
Figure 1. A sketch of the two hydrodynamic modes appearing in the longitudinal sector of the
QNMs of our system. In orange the longitudinal damped sound ω = ±c
L
k−i D
p
k
2
and in green the
diffusive mode ω = −i D
Φ
k
2
. The arrows indicate the tendency when increasing the momentum k.
3.1 Hydrodynamic regime of quasi-normal modes
First we analyse the quasi-normal modes in the low-frequency and long-wavelength regime.
In particular the spectrum exhibits:
1. A pair of longitudinal damped sound modes with dispersion relation
ω = ±c
L
k − i D
p
k
2
+ . . . , (3.1)
with the speed c
L
and the diffusion constant D
p
.
5
The ellipsis stands for higher
momenta corrections. The mode (3.1) is exactly a longitudinal damped phonon
mode, which is expected both in fluids and solids.
2. A diffusive mode with dispersion relation
ω = −i D
Φ
k
2
+ . . . , (3.2)
where D
Φ
is the diffusion constant. The ellipsis stands for higher momenta cor-
rections. The presence of such a diffusive mode is typical of systems which break
translational invariance spontaneously. This hydrodynamic mode can be viewed as
the diffusive mode of the Goldstone parallel to the momentum.
The two modes presented above are shown in figure 2, for the specific potential N = 3
and for different values of the dimensionless parameter m/T . We obtain similar results
for other values of N with N > 5/2 (see figure 1 for a schematic representation of the
hydrodynamic modes of our system).
5
Sometimes people refer to the sound attenuation constant Γ
S
which corresponds to Γ
S
≡ 2 D
p
.
– 6 –