The basic property of superfluid is that it can flow without dissipation. In a normal fluid, friction and
viscosity arise because particles are randomly scattered. Such scattering events are forbidden in superfluid
because energy and momentum cannot be simultaneously conserved. The key argument of Landau is that in
a superfluid like helium-4, the particles are strongly correlated so that the concept of single particles
becomes meaningless. However, at low enough temperatures, the system is still assumed to be described in
terms of noninteracting “quasiparticles”, which do not correspond to material particles but to many-body
motions (excitations). The energy spectrum of these quasiparticles can be very different from that of
single particles. Using this idea, Landau [247] was able to explain the origin of nondissipative
superflow and the existence of a critical velocity beyond which superfluidity disappears. The
argument is the following. Let us consider a macroscopic body of mass
flowing through the
superfluid. At low temperatures, its velocity
can only be changed in scattering processes
where one or more quasiparticles are created, assuming that the flow is not turbulent. For a
quasiparticle of energy
and momentum
to be created, energy conservation implies that
These two conditions can only be satisfied if
Since the perturbing body contains a macroscopic collection of particles, the mass is very large so
that the second term can be neglected. The resulting inequality cannot be satisfied unless the velocity
exceeds some critical value
At very high momenta, the dispersion relation coincides with that of a normal liquid, Equation (144).
In between, the dispersion relation exhibits a local minimum and is approximately given by
The quasiparticles associated with this minimum were dubbed “rotons” by I.E. Tamm as
reported by Landau [247]. Landau postulated that these rotons are connected with a rotational
velocity flow, hence the name. These rotons arise due to the interactions between the particles.
Feynman [145] argued that a roton can be associated with the motion of a single atom. As the atom
moves through the fluid, it pushes neighboring atoms out of its way forming a ring of particles
rotating backwards as illustrated in Figure 51
. The net result is a vortex ring of an atomic
size.
The roton local minimum has also been interpreted as a characteristic feature of density fluctuations
marking the onset of crystallization [200, 334, 307]. According to Nozières [307], rotons are “ghosts of
Bragg spots”. Landau’s theory has been very successful in explaining the observed properties of superfluid
helium-4 from the postulated energy spectrum of quasiparticles.
In weakly interacting dilute Bose gases, as in ultra-cold Bose atomic gases, the energy of the quasiparticles are given by (see for instance Section 21 of Fetter & Walecka [142])
where Owing to the specific energy spectrum of quasiparticles in both atomic Bose gases and helium-4, the
critical velocity does not vanish, thus explaining their superfluid properties. Landau’s critical velocity,
Equation (143), of superfluid helium-4 due to the emission of rotons, is given by
The previous discussion of the critical velocity of Bose liquids can be easily extended to fermionic superfluids. In the BCS theory, fermions form bound pairs, which undergo Bose condensation when the temperature falls below a critical temperature (Section 8.2). The quasiparticle energies for a uniform Fermi system are given by
where This expression can be derived more rigorously from the microscopic BCS theory [35]. It shows that a
system of fermions is superfluid (i.e. the critical velocity is not zero) whenever the interactions are
attractive, so that the formation of pairs becomes possible. It is also interesting to note that the BCS
spectrum can be interpreted in terms of rotons. Indeed, expanding Equation (149) around the minimum
leads, to lowest order, to an expression similar to Equation (146
). In this case,
is obtained by solving
. The other parameters are given by
and
where
is
the group velocity evaluated at
.
The presence of an “external” potential affects superfluidity. This issue has recently attracted a lot of
theoretical, as well as experimental, interest in the field of optically-trapped ultra-cold atomic Bose
gases [297]. It is also relevant in the context of neutron stars, where the solid crust is immersed in a neutron
superfluid (and possibly a proton superconductor in the liquid crystal mantle, where nuclear “pastas” could
be present; see Section 3.3). In the BCS approximation (128), considering a periodic potential
(induced by the solid crust in neutron stars), the quasiparticle energies still take a form similar to
Equation (149
). However, the dependence on the momentum is no more isotropic. As shown by Carter,
Chamel & Haensel [77
], Equation (150
) for the critical velocity should then be replaced by
The real critical velocity is expected to be smaller than that given by Equation (151) due to finite
temperature and many-body effects beyond the mean field. Likewise, the critical velocity of superfluid
helium-4 obtained in Landau’s quasiparticle model is only an upper bound because in this model, the
quasiparticles are assumed to be noninteracting. The experimentally-measured critical velocities are usually
much smaller, in particular, due to the nucleation of vortices. Indeed, relative motions between superfluid
and the vortices lead to mutual friction forces and, hence, to dissipative effects. In general, any curved
vortex line does not remain at rest in the superfluid reference frame and, therefore, induces dissipation.
Feynman [143
] derived the critical velocity associated with the formation of a vortex ring in a channel of
radius
,
Superfluidity is closely related to the phenomenon of Bose–Einstein condensation as first envisioned by Fritz
London [272]. In the superfluid phase, a macroscopic collection of particles condense into the lowest
quantum single particle state, which (for a uniform system) is a plane wave state with zero momentum
(therefore a constant). Soon after the discovery of the superfluidity of liquid helium, Fritz London
introduced the idea of a macroscopic wave function , whose squared modulus is proportional to the
density
of particles in the condensate. This density
, which should not be confused with the
superfluid density
introduced in the two-fluid model of superfluids (Section 8.3.6), can be rigorously
defined from the one-particle density matrix [322]. The wave function is defined up to a global phase factor.
The key distinguishing feature of a superfluid is the symmetry breaking of this gauge invariance by
imposing that the phase
be local. The macroscopic wave function thus takes the form
Applying the momentum operator to this wave function shows that superfluid carries a net
momentum (per superfluid particle)
This implies that the superflow is characterized by the condition
In the absence of any entrainment effects (as discussed in Section 8.3.6),
the momentum is given by , where
is the mass of the superfluid
“particles”8
and
is the velocity of superfluid. Equation (155
) thus implies that the flow is irrotational. This means,
in particular, that a superfluid in a rotating bucket remains at rest with respect to the laboratory reference
frame. However, this Landau state is destroyed whenever the rotation rate exceeds the critical threshold for
the formation of vortices given approximately by Equation (152
). Experiments show that the whole
superfluid then rotates like an ordinary fluid. The condition (155
) can therefore be locally violated as first
suggested by Onsager [310] and discussed by Feynman [143]. Indeed, since the phase of the macroscopic
wave function is defined modulo
, the momentum circulation over any closed path is quantized
In a rotating superfluid the flow quantization implies the formation of vortex lines, each
carrying a quantum of angular momentum, the quantum number
being the number of
vortices (the formation of a single vortex carrying all the angular momentum is not energetically
favored). The size of the core of a vortex line is roughly on the order of the superfluid coherence
length (see Section 8.2.3 for estimates of the coherence length). In some cases, however, it
may be much smaller [110, 134
], so that the coherence length is only an upper bound on the
vortex core size. In the presence of vortices, Equation (155
) must therefore be replaced by
As shown by Tkachenko [405, 406], quantized vortices tend to arrange themselves on a
regular triangular array. Such patterns of vortices have been observed in superfluid helium and
more recently in atomic Bose–Einstein condensates. The intervortex spacing is given by
Let us remark that the condition (155) for superfluids also applies to superconductors, like the proton
superconductor in the liquid core and possibly in the “pasta” mantle of neutron stars (Section 3.3). The
momentum of a superconductor is given by
(in this section, we use SI units), where
,
, and
are the mass, electric charge and velocity of “superconducting” particles
respectively9,
and
the electromagnetic potential vector. Introducing the density
of superconducting particles and
their electric current density (referred to simply as “supercurrent”)
, Equation (155
) leads to the
London equation
These flux tubes tend to arrange themselves into a triangular lattice, the Abrikosov lattice, with a spacing given by
Averaging at length scales much larger than
For neutron superfluid in neutron stars, superfluid particles are neutron pairs, so that . As early
as in 1964, Ginzburg & Kirzhnits [161, 162] suggested the existence of quantized vortex lines inside neutron
stars. The critical velocity for the nucleation of vortices can be roughly estimated from
where
is the radius of a neutron star. For
,
, which is, by several
orders of magnitude, smaller than characteristic velocities of matter flows within the star. The
interior of neutron stars is thus threaded by a huge number of vortices. The intervortex spacing is
In this section, we will discuss the nonrelativistic dynamics of superfluid vortices. The generalization to
relativistic dynamics has been discussed in detail by Carter [71]. According to the Helmholtz theorem, the
vortex lines are frozen in superfluid and move with the same velocity unless some force acts on them. The
dynamics of a vortex line through the crust is governed by different types of forces, which depend on the
velocities ,
and
of the bulk neutron superfluid, the vortex and the solid crust,
respectively.
All forces considered above are given per unit length of the vortex line. Let us remark that even
in the fastest millisecond pulsars, the intervortex spacing (assuming a regular array) of order
cm is much larger than the size of the vortex core
fermis. Consequently
the vortex-vortex interactions can be neglected.
The dynamic evolution of a vortex line is governed by
where On a scale much larger than the intervortex spacing, the drag force acting on every vortex line
leads to a mutual friction force between the neutron superfluid and the normal constituents. Assuming that
the vortex lines are rigid and form a regular array, the mutual friction force, given by
, can be
obtained from Equation (173
) after multiplying by the surface density
. Since
, the
inertial term on the left-hand side of Equation (173
) is proportional to
and can be
neglected. Solving the force balance equation yields the mutual friction force (per unit volume) [18
]
Different dissipative mechanisms giving rise to a mutual friction force have been invoked: scattering of
electrons/lattice vibrations (phonons)/impurities/lattice defects by thermally excited neutrons in vortex
cores [141, 189, 222], electron scattering off the electric field around a vortex line [48], and coupling
between phonons and vortex line oscillations (Kelvin modes) [139, 223]. In the weak coupling limit
, the vortices co-rotate with the bulk superfluid (Helmholtz theorem), while in the
opposite limit
, they are “pinned” to the crust. In between these two limits, in a frame
co-rotating with the crust, the vortices move radially outward at angle
with respect to
the azimuthal direction. The radial component of the vortex velocity reaches a maximum at
.
Vortex pinning plays a central role in theories of pulsar glitches. The strength of the
interaction between a small segment of the vortex line and a nucleus remains a controversial
issue [5, 138, 333, 134, 120, 121, 122, 27]. The actual “pinning” of the vortex line (i.e., )
depends not only on the vortex-nucleus interaction, but also on the structure of the crust, on the rigidity of
lines and on the vortex dynamics. For instance, assuming that the crust is a polycrystal, a rigid vortex line
would not pin to the crust simply because the line cannot bend in order to pass through the nuclei,
independent of the strength of the vortex-nucleus interaction! Recent observations of long-period precession
in PSR 1828–11 [387
], PSR B1642–03 [371
] and RX J0720.4–3125 [180
] suggest that, at least in those
neutron stars, the neutron vortices cannot be pinned to the crust and must be very weakly
dragged [372, 266].
Let us stress that the different forces acting on a vortex vary along the vortex line. As a consequence,
the vortex lines may not be straight [198]. The extent to which the lines are distorted depends on the
vortex dynamics. In particular, Greenstein [175] suggested a long time ago that vortex lines may twist and
wrap about the rotation axis giving rise to a turbulent flow. This issue has been more recently addressed by
several groups [323, 324, 288, 19]. In such a turbulent regime the mutual friction force takes the
form [173]
One of the striking consequences of superfluidity is the allowance for several distinct dynamic components.
In 1938, Tisza [404] introduced a two-fluid model in order to explain the properties of the newly discovered
superfluid phase of liquid helium-4, which behaves either like a fluid with no viscosity in some
experiments or like a classical fluid in other experiments. Guided by the Fritz London’s idea that
superfluidity is intimately related to Bose–Einstein condensation (which is now widely accepted),
Tisza proposed that liquid helium is a mixture of two components, a superfluid component,
which has no viscosity, and a normal component, which is viscous and conducts heat, thus,
carrying all the entropy of the liquid. These two fluids are allowed to flow with different velocities.
This model was subsequently developed by Landau [247, 246] and justified on a microscopic
basis by several authors, especially Feynman [144]. Quite surprisingly, Landau never mentioned
Bose–Einstein condensation in his work on superfluidity. According to Pitaevskii (as recently
cited by Balibar [33]), Landau might have reasoned that superfluidity and superconductivity
were similar phenomena (which is indeed true), incorrectly concluding that they could not
depend on the Bose or Fermi statistics (see also the discussion by Feynman in Section 11.2 of his
book [144]).
In Landau’s two-fluid model, the normal part with particle density and velocity
is identified
with the collective motions of the system or “quasiparticles” (see Section 8.3.1). The viscosity of the
normal fluid is accounted for in terms of the interactions between those quasiparticles (see, for
instance, the book by Khalatnikov [236] published in 1989 as a reprint of an original 1965 edition).
Following the traditional notations, the superfluid component, with a particle density
and a
“velocity”
, is locally irrotational except at singular points (see the discussion in Section 8.3.2)
The confusion between velocity and momentum is very misleading and makes generalizations of the
two-fluid model to multi-fluid systems (like the interior of neutron stars) unnecessarily difficult. Following
the approach of Carter (see Section 10), the two-fluid model can be reformulated in terms of the real
velocity of the helium atoms instead of the superfluid “velocity”
. The normal fluid
with velocity
is then associated with the flow of entropy and the corresponding number
density is given by the entropy density. At low temperatures, heat dissipation occurs via the
emission of phonons and rotons. As discussed in Section 8.3.1, these quasiparticle excitations
represent collective motions of atoms with no net mass transport (see, in particular, Figures 50
and 51
). Therefore, the normal fluid does not carry any mass, i.e., its associated mass is equal to
zero.
Following the general principles reviewed in Section 10, the momentum of the superfluid helium
atoms can be written as
Comparing Equations (180) and (178
) shows that the “superfluid velocity” in the original two-fluid
model of Landau is not equal to the velocity of the helium atoms but is a linear combination of both
velocities
and
Entrainment effects, whereby momentum and velocity are not aligned, exist in any fluid
mixtures owing to the microscopic interactions between the particles. But they are usually
not observed in ordinary fluids due to the viscosity, which tends to equalize velocities.
Even in superfluids like liquid Helium II, entrainment effects may be hindered at finite
temperature11
by dissipative processes. For instance, when a superfluid is put into a rotating container, the presence of
quantized vortices induces a mutual friction force between the normal and superfluid components (as
discussed in Section 8.3.5). As a consequence, in the stationary limit the velocities of the two fluids become
equal. Substituting in Equation (180
) implies that
, as in the absence of
entrainment.
A few years after the seminal work of Andreev & Bashkin [20] on superfluid 3He – 4He mixtures, it was
realized that entrainment effects could play an important role in the dynamic evolution of neutron
stars (see, for instance, [363] and references therein). For instance, these effects are very important for
studying the oscillations of neutron star cores, composed of superfluid neutrons and superconducting
protons [14]. Mutual entrainment not only affects the frequencies of the modes but, more surprisingly,
(remembering that entrainment is a nondissipative effect) also affects their damping. Indeed, entrainment
effects induce a flow of protons around each neutron superfluid vortex line. The outcome is
that each vortex line carries a huge magnetic field
1014 G [8]. The electron scattering off
these magnetic fields leads to a mutual friction force between the neutron superfluid and the
charged particles (see [18] and references therein). This mechanism, which is believed to be the
main source of dissipation in the core of a neutron star, could also be at work in the bottom
layers of the crust, where some protons might be unbound and superconducting (as discussed in
Section 3.3).
It has recently been pointed out that entrainment effects are also important in the inner
crust of neutron stars, where free neutrons coexist with a lattice of nuclear clusters [79, 78
].
It is well known in solid state physics that free electrons in ordinary metals move as if their
mass were replaced by a dynamic effective mass
(usually referred to as an optical mass in
the literature) due to Bragg scattering by the crystal lattice (see, for instance, the book by
Kittel [241]). The end result is that, in the rest frame of the solid, the electron momentum is given by
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