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\DOI{10.5802/crphys.248}
\datereceived{2024-09-11}
\daterevised{2025-02-03}
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\dateposted{2025-04-25}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{history-of-sciences}

\title{The factor two}

\alttitle{Le facteur deux}

\author{\firstname{Yves} \lastname{Pomeau}}
\address{LadHyX, CNRS \&amp; Ecole Polytechnique, 
UMR 7646, IP Paris, 91128, Palaiseau, France}
\email[Y. Pomeau]{yves.pomeau@gmail.com}

\author{\firstname{Sergio}  \lastname{Rica}\CDRorcid{0000-0003-0695-6700}\IsCorresp}
\address{Instituto de F\'isica, Facultad de F\'isica, 
Pontificia Universidad Cat\'olica de Chile, Casilla 306, Santiago, Chile}
\email[S. Rica]{sergio.rica@uc.cl}

\keywords{\kwd{\tralicstex{\textit{SU}(2) Ã— \textit{U}(1)}{$SU(2)$ Ã— $U(1)$} Topological  defects}
\kwd{Quantum of London}
\kwd{Superconductivity}
\kwd{Superfluidity}}

\altkeywords{\kwd{D\'efauts topologiques \tralicstex{\textit{SU}(2) Ã— \textit{U}(1)}{$SU(2)$ Ã— $U(1)$}}
\kwd{Quantum de London}
\kwd{Supraconductivit\'e}
\kwd{Superfluidit\'e}}

\thanks{Fondecyt (ANID) under Grant No. 1220369.}

\begin{abstract}
Examples are given of physical problems where factor two played an
important role. First, we consider the irregularities of the Moon
trajectory that Newton correctly, attributed to the differences in the
attraction by the Sun of Earth and Moon because of the small
differences in their location. Newton's estimate of those (small)
irregularities was off the measures by a factor close to two, which was
restored mater by Clairaut and seen as a major success for Newton's
dynamical theory. Another factor two appears in the Sun's estimate of
the deviation of light. The deviation computed by using special
relativity is half the one measured. As Einstein showed, the factor is
restored by considering general relativity which considers the changes
in the metric due to gravity. Lastly, we consider a more recent
problem, the one of the apparent doubling of electric charges in a
superconductor as measured by the quantum of flux, which is half the
one predicted by London, namely \tralicstex{\textit{â„c}/2\textit{e}}{$\hbar c /2e$} instead of \tralicstex{\textit{â„c}/\textit{e}}{$\hbar c /e$}.
This doubling is classically explained by assuming that, in
superconductors current carrying fermions are in pairs so that the unit
charge becomes 2\textit{e} instead of \textit{e}. We argue instead that if one takes
into account that the fermions carry a spin 1/2 the quantization
element of magnetic flux is divided by two as implied by looking at the
second class of homotopy of the group of transformation \tralicstex{\textit{SU}(2) Ã— \textit{U}(1)}{$SU(2)$ Ã— $U(1)$} of the ground state, as it follows from Toulouse theory of
defects in ordered media.
\end{abstract}

\begin{altabstract}
Nous donnons plusieurs exemples de probl\`emes physiques  o\`u un
facteur 2 joue un  r\^ole important. Nous examinons d'abord la question
des irr\' egularit\' es du mouvement de la lune attribu\' ees
correctement par Newton \`a la diff\'erence de l'attraction du soleil
sur la terre et sur la lune l\' eg\` erement d\' ecal\' ees l'une par
rapport \` a l'autre vues du soleil. L'estimation par Newton de ces
irr\'egularit\'es diff\' erait d'un facteur 2 environ des observations.
Ce facteur 2 a \'et\'e trouv\' e par Clairaut en utilisant la th\'
eorie dynamique de Newton qui fut ainsi \'et\'e pleinement confirm\'ee.
Un autre facteur 2 appara\^it dans le calcul de la d\'eviation d'un rayon
lumineux par la masse du soleil. La relativit\'e restreinte donne un
r\'esultat diff\'erent de l'observation aussi par un facteur 2. Ce
facteur 2 a \'et\'e r\'etabli par Einstein en utilisant la relativit\'e
g\'en\'erale qui tient compte de la d\'eformation de la m\'etrique de
l'espace-temps pr\`es des masses, ce que ne fait pas  la relativit\'e
restreinte. Finalement nous consid\'erons une question de physique plus
r\'ecente, celle du quantum de flux magn\'etique dans un
supraconducteur. Les mesures donnent un quantum moiti\'e de celui
pr\'edit par London, soit  \tralicstex{\textit{â„c}/2\textit{e}}{$\hbar c /2e$}  au lieu de \tralicstex{\textit{â„c}/\textit{e}}{$\hbar c /e$}.
L'explication habituelle de ce facteur 2 est que dans les
supraconducteurs les \'electrons porteurs de charge sont appari\'es, ce
qui doublerait la charge du quantum de London. Nous montrons qu'il
existe une autre possibilit\'e  d'explication de cette division par 2
du quantum de London, bas\'ee sur la th\'eorie de Toulouse des
d\'efauts dans les milieux continus. Dans le cas pr\'esent, le quantum
de flux magn\'etique est li\'e au premier groupe  d'homotopie du
param\`etre d'ordre qui, si l'on inclut la phase de la fonction d'onde
globale et le spin 1/2 a le groupe de sym\'etrie \tralicstex{\textit{SU}(2) Ã— \textit{U}(1)}{$SU(2)$ Ã— $U(1)$} 
qui g\'en\`ere des d\'efauts pour lesquels la circulation vaut la
moiti\'e de celle du groupe \textit{U}(1), qui serait donc limit\'e \`a la
seule phase (le groupe \textit{U}(1) sans les rotations de spin le groupe
\textit{SU}(2)). Cette division par 2 de la circulation de\pagebreak 
la phase remonte
\`a une remarque de Cauchy sur l'\'echange des deux valeurs propres
d'une matrice \text{2 Ã— 2} le long d'un cercle entourant (dans le plan
complexe) un z\'ero du d\'eterminant. 
\end{altabstract}

\maketitle

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\section{G\'erard Toulouse, an inspiring scientist}
The authors of this article have known G\'erard Toulouse and have been
inspired by him over the years both in science and outside of it. This
article is dedicated to his memory and reports work at the end that
could not have been done without one of his deep contributions to
physics. One of us (YP) has known and appreciated G\'erard since their
student time at Ecole Normale in the early sixties. Then G\'erard was
already a quite remarkable student who was not trying to follow the
always changing fashion of the day, but instead tried to put together
new ideas necessary to bring significant progress, a particularly
difficult endeavor in a mature science like physics.   It is also
relevant to point out that at this time the fundamental ideas of
quantum physics had been laid down and shown to be correct for
describing the interaction of matter with radiation, closing so a
chapter of theoretical, physics and leading young physicists to look at
new topics like statistical physics and solid state physics as well. We
had a unique opportunity in the early nineties when G\'erard and a few
others brave scientists launched  an informal group of statistical
physics at Ecole normale physics lab. Each of us brought his knowledge
and we had a very fruitful collaboration. G\'erard  brought many
important ideas at a this time of renewal of concepts in physics. In
statistical physics the concept of frustration brought forward  an
unifying understanding of many diverse observations made on various
models of spin systems and explained at last why they were not all more
or less equivalent to the standard Ising model. In the present article
we use another fundamental idea of G\'erard, the discovery of a deep
mathematical structure in the classification of defects in condensed
matter. This relies on a little known (to physicists)  topological
theory called homotopy. The interested reader may read the original
works of G\'erard and Maurice Kleman on this topic. We shall only
mention that homotopy theory is necessary to show a non trivial result,
namely that smooth fields of unit vectors in 3D are not all
topologically equivalent, namely cannot be deformed continuously into
each other: each field has a topological invariant in the discrete
group $Z$, so that two fields with different homotopy group cannot be
mapped continuously into another field with a different homotopy group.
This mathematical property was discovered by Hopf and can be visualized
by looking at the pre-images in $R^3$ of a given unit vector, that is a
curve and the Hopf index measures the (eventual) entanglement of two
pre-images of different unit vectors. This entanglement was shown to
explain observations  of some defects in cholesteric liquid crystals, a
fairly non trivial point discussed by Pawel Pieranski in this issue of
Comptes rendus Physique. 

This conjunction of deep physical insight and of non trivial
mathematical thinking was typical of G\'erard approach of physics. We
dedicate this article to his memory with the hope that what he sowed
will continue to grow. 

\section{Introduction}
A famous historical instance where a factor two played a role in the
development of our understanding of the world goes back to Newton's 
Principia~\cite{Principia}. There, in Book~1,   Section~3,  
Proposition~11, the author derives the Kepler elliptical, orbits from
his dynamical laws and from the \mbox{assumption} that masses attract each
other with a force inversely proportional to their square distance and
proportional to the product of their mass. Since this has been rightly
seen as a major achievement of mankind scientific inquiry. However, in
some sense, this did not bring anything new, as Kepler laws were
already known at the time and described well trajectories of the
planets. Therefore another result was needed somehow to make Newton's
theory indispensable to explain astronomical phenomena. As well
understood by Newton himself (see in  Book~1,  Section~9, 
Proposition~45,  Corollarium~2,  Section~11,  Proposition~66, and, 
Book~3,  Propositions~25--35) this could be achieved by explaining with
his theory the long observed ``irregularities'' of the motion of the
Moon. Newton understood that those irregularities (with respect to a
regular Keplerian orbit of the Moon around the Earth) was due to the
slightly different solar attraction of Earth and Moon because of their
slightly different location with respect to the Sun. This small
difference introduces a small dimensionless parameter, the ratio of the
distance of the Moon to the Earth to the distance of the pair to the
Sun, a parameter of order of the earth--moon to the earth--sun distances
about $ 1.4\times  10^{-3}$ a small but not infinitesimally small
number. This explains in particular why those irregularities had been
observed and recorded since very ancient times beginning with
Hipparchus  two centuries before CE. The relevance of  understanding
Moon irregularities is for Newton of great importance as highlighted at
the proposition 35 of book 3, by a scholium: ``\textit{I wished to show
by these computations of the lunar motions that the lunar motions can
be computed from their causes by the theory of gravity}''. 

To summarize the scientific problem, the perturbation concerns small
changes of the orbit parameter induced by the addition of a small
interaction of the Sun on the Moon, the main interaction responsible of
Moon's trajectory being the interaction with Earth. This makes what is
called in general a three body problem, Sun, Earth and Moon. More
precisely this is a restricted three body problem because the Sun is
too heavy to be moved by the two other masses. When speaking of small
perturbation, one does not refer to perturbation in the ordinary sense
of applied mathematics, because one cannot observe small differences
between a computed trajectory and the real trajectory. The only thing
observed are changes of Moon's trajectory with respect to an ideal
Keplerian elliptic trajectory. There are plenty possibles changes of
this Keplerian trajectory: 

\begin{itemize}
\item 
the focus of the ellipse may drift slowly in time
\item 
the parameters of the ellipse may also change (and they do)
\item  
the orientation of the axis of the ellipse may also change, this is
what we shall consider. 
\end{itemize}

A very interesting property of the Keplerian orbits is that, when
elliptical, they are closed. This property is shared by the harmonic
potential as shown by Newton and does not hold for other interactions.
This is related to the existence of another invariant than the usual
Noether invariants, energy and angular momentum. The new invariant,
known to  Laplace as well as other scientists, is today called the
Runge--Lenz invariant and played a role in science history because it
permitted Pauli to find the energy levels of the hydrogen atom using
the Heisenberg's matricial mechanics, without solving Schr\"odinger
equation. 

To return to the factor 2 in the Moon's orbit, Newton calculated
(geometrically) the precession of the axis of the ellipse run by the
Moon by a very clever method that is equivalent of the adiabatic
variation of parameters. He was seemingly disappointed to find by his
method a rotation of the axis of the ellipse about half as fast as the
observations. Indeed in the Corollary~3, of the  Proposition~45, Newton
concludes: ``\textit{thus in each revolution the upper apsis will move
forward through 1$\text{\textdegree}$31$^{\prime}$28$^{\prime\prime}$.  The
apsis of the moon is about twice as swift}.''\footnote{Cohen and
Whitman~\cite{Principia}, translated from latin by ``\textit{The
advance of the apsis of the moon is about twice as swift}.''}

Being these ten words the well-remembered statement\footnote{For a
more detailed discussion see the works of
Nauenberg~\cite{Nauenberg98,Nauenberg00} who considered in detail the
perturbative methods used by Newton in the original and later editions
of the Principia. As Nauenberg writes, in the  Portsmouth manuscript
Newton developed what is known today as the method of variations of
parameters re-discovered many years later. Although Newton did not
allow the eccentricity to vary, but only the rotation of the major
axis.  That is, Newton allowed variations of the direction of the
Runge--Lenz invariant but not in its modulus.}. Newton's geometrical
method, even though superbly elegant, was replaced during the
eighteenth century by what we call now ``calculus''. Using the modern
methods of calculus, Clairaut~\cite{Clairaut52,Clairaut54} redid Newton
calculation and obtained the correct answer in agreement with
observations. The difficulty in this problem is that the ellipticity of
Moon's orbit is quite small so that the first order perturbation
induced by the gradient of the attraction by the Sun  is of the same
order as this ellipticity, which imposes to consider the perturbation
as affecting the whole ellipse instead of its approximation by a circle
as used by Newton. This was saluted by Euler in a letter to
Clairaut~\cite{EulerLettres} telling that 
\begin{quote}
{j'en suis
maintenant d'autant plus sensiblement penetr\'e; et plus je considere
cette hereuse decouverte, plus elle me parait importante, et \`a mon
avis c'est la plus grande decouverte dans la Theorie de l'Astronomie,
sans laquelle il serait absolument impossible de parvenir jamais \`a la
connaissance des derrangements, que les planetes se causent les unes
les autres dans leur mouvement}\footnote{Which in english, reads:  ``I
am now all the more sensibly penetrated by it; and the more I consider
this happy discovery, the more important it seems to me, in my opinion
it is the greatest discovery  in the theory of astronomy, without which
it would be impossible in knowing the perturbations that the planets
cause to each other motion''.}.
\end{quote}
Clearly Euler~\cite{Euler} saw rightly that the theory of perturbation
is the ultimate proof of validity of Newton's theory.    

A second historical instance\footnote{Another example of a ``factor
two'' is Newton's theory of ocean tides. The gravitational interaction
of the Moon in addition with the daily rotation of the earth creates
two tides per day (a factor 2 that Kepler could not explain).}  when a
factor two took place was in Prague in 1912, when Albert
Einstein~\cite{Einstein1912} published a paper, ``\textit{On the
Influence of Gravitation on the Propagation of Light}''. In this paper,
he stated the equivalence principle and demonstrated that a
gravitational field acts on both masses and energy, thereby influencing
light propagation. Specifically, he showed the so-called redshift
effect and the deflection of a ray of light passing near a massive
object, and he suggested a possible experimental test: 
\begin{quote}
{In particular, it turns out that, according to the theory I am going
to set forth, rays of light passing near the sun experience a
deflection by its gravitational field, so that a fixed star appearing
near the sun displays an apparent increase of its angular distance from
the latter, which amounts to almost one second of arc.}
\end{quote}

The final result predicts that the deflection angle is:
{$$
\alpha = 2\frac{GM}{c^2 R},
$$}\unskip
where $G$ is the gravitational constant, $M$ is the mass of the Sun,
and $R$ is the distance from the ray of light to the center of the 
Sun.
\begin{quote}
{Accordingly, a ray of light traveling past the sun would
undergo a deflection amounting to $4\times10^{-6} = 0.83$ seconds of
arc. This is the amount by which the angular distance of the star from
the center of the sun seems to be increased owing to the bending of the
ray. Since the fixed stars in the portions of the sky that are adjacent
to the sun become visible during total solar eclipses, it is possible
to compare this consequence of the theory with experience. In the case
of the planet Jupiter, the displacement to be expected comes to about
1/100 of the amount indicated. It is greatly to be desired that
astronomers take up the question broached here, even if the
considerations here presented may appear insufficiently substantiated
or even adventurous. Because apart from any theory, we must ask
ourselves whether an influence of gravitational fields on the
propagation on of light can be detected with the currently available
instruments}.
\end{quote}

Interestingly, the value of the deflection angle is exactly the one
predicted by a Newtonian mass with an incident velocity of  $c$,
passing around a massive body. At first glance, it appears that the
mass content of the energy of a light ray is sufficient to perturb
light propagation. However, Einstein's theory is more subtle. According
to special relativity, a light ray possesses a universal speed, making
a Newtonian interpretation inadequate, even if the result coincides
formally.

According to the special relativity there is no absolute time, thus
Einstein kept the concept of time dilation in presence of an
accelerated frame of reference, thus by the equivalence principle  time
must depend on the gravitational field. A primary result is the
well-known redshift and  a second consequence is the deflection of a
ray of light passing near a massive star.

Einstein encouraged the community of astrophysicists to measure such
deflection caused by the Sun's gravitational field, which could be
observed during a solar eclipse. However, due to World War I, the
decisive measurement was postponed. Meanwhile, Einstein developed the
field equations for gravitation, now known as the General Theory of
Relativity. This theory includes geodesics as the propagation paths of
light and involves a gravitational field described by the Einstein
equations. The metric follows an equation where, besides the temporal
component of the metric tensor, denoted usually as $g_{00}$, the
spatial components also depend on the gravitational field source. The
propagation of a ray of light includes both temporal dilation and
spatial contraction. The final result for the deflection angle becomes:
{$$ 
\alpha = 4\frac{GM}{c^2 R},
$$}\unskip
which is bigger than the previous result by a factor 2. 

As is well known, the crucial measurement was carried out by Eddington
in 1919, agreeing with the predicted value  $\alpha = 1.7$ seconds of
arc from the General Theory of Relativity~\cite{Einstein1916}.

Scientists are often confronted with ``factors of two''. In most cases,
these factors are not particularly significant, but sometimes these
``forgotten factors of two'' obscure the correct understanding of a
phenomenon. In the following, we will discuss the case of the quantum
of magnetic flux in a superconductor, where surprisingly, the
experimentally observed value is half (not double) the originally
predicted value. The reader is invited to explore the basis of this
factor of two. 

First, we review the case of the quantum of circulation in spin-0
superfluids in  Section~\ref{Sec:QuantumCirculation}, and subsequently
the quantum of magnetic flux in  Section~\ref{Sec:QuantumLondon}. Next,
we provide an alternative explanation for the one-half factor in the
quantized magnetic flux for a superfluid composed of spin-1/2 particles
in  Section~\ref{Sec:Spinor}. Finally, we conclude.

\section{Quantum of circulation}\label{Sec:QuantumCirculation}
The Landau two-fluid model~\cite{Landau41}  admits irrotational motion
for the superfluid component of liquid helium at temperatures below the
$\lambda$-transition. However, Lars Onsager suggested that the
circulation of velocity in a superfluid, along an arbitrary path
$\mathcal{C}$,  namely:
{$$
\Gamma = \oint_{\mathcal{C}}  \mathbi{v}\cdot 
\mathrm{d}\boldell , 
$$}\unskip
must take values that are multiples of $\Gamma_0 = h/m$, where $h$
represents the Planck constant and $m$ denotes the mass of the atoms.
Particularly for Helium-4, the quantum of circulation is approximately
$\Gamma_0 \approx 10^{-3}~\mathrm{cm}^2/\mathrm{s}$, which is entirely
a macroscopic quantity. It may be useful to recall that the circulation
of vortices at the Kolmogorov scale in fully developed turbulence
reaches a value of the order of the kinematic viscosity,  i.e., $\Gamma
\approx 10^{-2}~\mathrm{cm}^2/\mathrm{s}$ for water.

This remark by Onsager, later becoming famous, is found in a footnote:
``\textit{Vortices in a suprafluid are presumably quantized; the
quantum of circulation is $h/m$ where $m$ is the mass of a single
molecule}.''  of an article on ``\textit{Statistical Hydrodynamics}''
published following a Statistical Physics congress in Florence in May
1949~\cite{Onsager49}. Although this is the most common reference in
the literature, it should be noted that two days earlier, Onsager
proposed for the first time during the general discussions following a
seminar by C.~J.~Gorter~\cite{Onsager49bis}:
\begin{quote}
{The equations $\mathrm{curl} \, \mathbi{p} =0$; $ \mathbi{p}=\hbar\,
\mathrm{grad}\, \phi$, are analogous to the law which governs the
super-currents in a supra-conductor, only simpler, because there is no
hydrodynamic analogous of the magnetic field generated by the
super-current. Hence (unless this one uses a rotating frame of
references), there is no vector potential to be included in
$\mathbi{p}$, the ``local'' momentum per particle. Given the wave
function $\psi_0(\mathbi{r}_1,\ldots,\mathbi{r}_N)$ of the liquid at
rest (ground state), wave-functions for the liquid in ``vortex-free''
motion can be constructed quite readily in the form $ \psi=\psi_0\,
\exp (\rmi\Delta \varphi(\mathbi{r}_j))$~\footnote{It is written $
\psi=\psi_0\, \exp (\rmi\Delta \varphi(\mathbi{r}_j))$ but the authors
think the correct formula is $\psi=\psi_0\, \exp (\rmi\sum_j
\varphi(\mathbi{r}_j))$.  One might say--if it is truh-- that Onsager's
remarks were merely recorded by typist and published likely as they
were, which introduced difficulties in their later interpretation,
particularly when Onsager used Greek letters as symbols that were not
available on mechanical typewriters at the time.}. Here, if the phase
$\varphi(\vec r)$ satisfies the condition $\bnabla\cdot\bnabla \varphi +
(2\uppi k)^2 \varphi=0$ with appropriate boundary conditions, then in
the case $k\neq 0$ the wave-function can be decomposed as usual (by
expansion in Hermite polynomials) into functions which describe
stationary states with $0, 1, 2, \ldots$ ``phonons'' of wave-number $k$
present. The case $k=0$ represents potential motion without
compression, which may take place either in the presence of moving
boundaries or in a multiply connected space. In the latter case
$\varphi(\mathbi{r})$ may be multiple valued; but its increment over
any closed path must be a multiple of $2\uppi$, so that the
wave-function will be singled-valued. Thus the well-known invariant
called hydrodynamic circulation is quantized; the quantum of
circulation is $h/m$, and $\oint \mathbi{v}\cdot \rmd\mathbi{r}$ is always
an integral multiple of this. In case of cylindrical symmetry, the
angular momentum per particles is a multiply of $\hbar$. Now we observe
that a torus can be converted into a simply connected space by
shrinking the hole. If a circulating superfluids is subjected to such a
deformation of its container, it must retain a quantized vortex in its
interior. The position of such a vortex would presumably be more or
less indeterminate; the mathematical description of this indeterminacy
remains to be developed. If we admit the existence of quantized
vortices, then a superfluid is able to rotate; but the distribution of
vorticity is discrete rather than continuous. The critical rate of
creep ($\hbar/2m$) may be identical with the minimum rate flow required
before the liquid film can ``lose energy'' by the development of a
vortex. Finally we can have vortex ring in the liquid, and the thermal
excitation of Helium II, apart from the phonons, is presumably due to
the vortex rings of molecular size. As a possible interpretation of the
$\lambda$-point, we can understand that when the concentration of
vortices reaches the point where they form a connected tangle
throughout the liquid, then the liquid becomes normal}.
\end{quote}

As seen in the previous paragraph~\cite{Onsager49bis}, Onsager assumed
that superfluid helium four is a long-range single state described by a
ground-state wave function of $N$ identical quantum interacting 
particles $\psi_0 (\mathbi{x}_1,\ldots,\mathbi{x}_N)$. Allowing the
possibility of a collective state, then, in the hydrodynamical
approximation concerned with the large-scale motion of this system,
this wave function is ``rigid'' under slow transformations bringing it
to the  form\footnote{London~\cite{london} coined the concept of
``rigidity'', meaning that the ground state wave function does not
change substantially under transformations that induce a non-uniform
flow in a superfluid or a magnetic field in superconductors.}:
{\vspace*{-2pt}}
{\begin{equation}
\psi (\mathbi{x}_1,\ldots,\mathbi{x}_N)=
\psi_0 (\mathbi{x}_1,\ldots,\mathbi{x}_N) 
\rme^{\rmi(\phi(\mathbi{x}_1)+\cdots +\phi(\mathbi{x}_N))},
\label{eq:ExcitedWaveFunctionForBosons}
\end{equation}}\unskip
where $\phi(\mathbi{x},t)$ is a phase field almost constant at
molecular distances. Notice that the sum  $\phi(\mathbi{x}_1)+\cdots +
\phi(\mathbi{x}_N)$  must be symmetric, independent of the quantum
statistics of the particles (bosons or  fermions)\footnote{Nowhere did
Onsager make reference to the statistics of the particles in his
remark. This is a question we will return to in the final discussion.}.
Onsager later showed that because $\phi$ and $\phi+2\uppi$ yield the
same physical state, the circulation in superflows is quantized in
units of $h/m$ ($h$ and $m$ being the Planck constant and the mass of
the atoms respectively). In a sense, this quantization arises from the
fact that a hydrodynamic flow can be considered as a slightly
non-uniform symmetry transformation (phase rotation) on the physical
system.

The momentum density or current reads:
{\vspace*{-2pt}}
{$$
\mathbi{P}= 
\frac{\hbar}{2m \rmi}
\int \rmd\mathbi{x}_2 \cdots \rmd\mathbi{x}_N 
\sum_{i=1}^N 
( \bar \psi \bnabla_i \psi-  \psi \bnabla_i \bar\psi) =
\frac{\hbar}{m} 
\rho (\mathbi{x}_1 ) \bnabla_1 \phi(\mathbi{x}_1  ) .
$$}\unskip
The number density, $ \rho (\mathbi{x}_1 )$, of the liquid is defined
through the one-particle density matrix:
{\vspace*{-2pt}}
{\begin{equation} 
\rho (\mathbi{x}_1 )  = N  
\int \rmd\mathbi{x}_2 \cdots \rmd\mathbi{x}_N 
| \psi_0 (\mathbi{x}_1,\ldots,\mathbi{x}_N)|^2 .
\label{eq:density}
\end{equation}}\unskip
In conclusion, the superfluid velocity is given by:
{\vspace*{-2pt}}
{$$
\mathbi{v}_{\mathrm{s}}(\mathbi{x} ) =  
\frac{\hbar}{m  } \bnabla \phi(\mathbi{x}  ) ,
$$}\unskip
thus, the circulation becomes:
{\vspace*{-2pt}}
{$$
\Gamma =  \frac{\hbar}{m  }  
\oint_{\mathcal{C}}    \bnabla \phi(\mathbi{x}  )\cdot 
\rmd\boldell = 
\frac{\hbar}{m  }  
\oint_{\mathcal{C}}    
\rmd\phi= 2\uppi n 
\frac{\hbar}{m  }, 
$$}\unskip
where in the last step a single-valued wave function is imposed, and
$n$ is an arbitrary integer.

The Onsager quantum of circulation was observed experimentally by Hall
and Vinen~\cite{HallVinen,Vinen} as early as 1956, and it has been
measured with varying degrees of accuracy over the past sixty
years~\cite{Donnelly}.

{\vspace*{-4pt}}

\section{Quantum of magnetic flux---Quantum of
London}\label{Sec:QuantumLondon}

{\vspace*{-3pt}}

The situation for fermions is very similar; the only difference lies in
the spin of the particles. In 1951, F. London, in his well-known book
``\textit{Superfluids, Volume I: Macroscopic theory of
superconductivity}''~\cite{london}, in  Section~26, again in a footnote
on page~152, writes: 
\begin{quote}
{We note that in order for $\psi$ to be
singled-valued function, as required by quantum mechanics, it is
necessary that the moduli of $\chi$ fulfill a kind of quantum
condition}:
{\vspace*{-2pt}}
{$$
\langle \chi\rangle = 
\oint \bar {\mathbi{p}}_{\mathrm{s}} \cdot \rmd\mathbi{s} = K h
$$}\unskip
{where $K$ must be an integer. This means that there exists a
universal unit for the fluxoid:}
{\vspace*{-2pt}}
{$$
\Phi= hc/e \simeq 4 \times 
10^{-7}~\mathrm{gauss}{\cdot}\mathrm{cm}^2.
$$}\unskip
\end{quote}
Here $\chi$ represents the gauge $\mathbi{A}= \sofrac{c}{e} \bnabla \chi
$, thus,  $\langle \chi\rangle = \sofrac{e}{c} \Phi$ is the magnetic
flux or fluxoid.

In conclusion, using his quantum-mechanical theory of
superconductivity, predicted, under the same hypothesis as Onsager,
that the magnetic flux must be quantized in units of $hc/q$, where $q$
is the charge of the particle and $c$ is the speed of light. Finally,
he notes that for superconductors, $q$ may be replaced by $e$, the
charge of an electron. Thus, the flux quantum becomes $hc/e \approx 4
\times 10^{-7}~\mathrm{G}/\mathrm{cm}^2$.

The priority of London was recognized by Onsager, again as a comment in
the Proceedings of the International Conference on Theoretical Physics
Kyoto and Tokyo September, 1953~\cite{OnsagerJapan} (see pages 935 and
936).

The quantum of flux was first observed and measured experimentally by
Deaver and Fairbank in the USA and Doll and N\"abauer
in Germany simultaneously in  1961~\cite{Deaver,Doll}\footnote{The
same value for the  quantum flux, $h/2e$, was observed  in a high-$T_{\mathrm{c}}$
yttrium-based superconductors in 1987~\cite{Gough}.}. Surprisingly, both
agreed that the quantum of magnetic flux in a superconducting loop is
about half of London's prediction, i.e., 
{$$
\Phi_0 = \frac{hc}{2e}.
$$}\unskip
Doll and N\"abauer, conclude  their letter: ``\textit{The
experimentally observed interval, however, reaches only 0.2 oe, that is
about 40\% of the calculated value. So far the reason for this
discrepancy is not  clear}.''   Both experimental results produced an
avalanche of theoretical works, all published in the very same volume 
\textbf{7} of the Physical Review Letters in
1961~\cite{Byers,Onsager61,Blatt,Bardeen,Keller,Brenig} and by
Ginzburg~\cite{Ginzburg} in a Soviet journal.

In Deaver and Fairbank's letter, it becomes clear they knew that the
actual value of the quantum flux could be 1/2 of London's prediction.
Indeed, the paper acknowledges a private communication:  ``\textit{Such
a possibility was mentioned by Lars Onsager to one of us (WMF) at the
conference of superconductivity in Cambridge, England, 1959 
(unpublished)}.'' Prior to the experiment, it seems that Onsager was
the only theoretician who had already suggested  that the actual value
for the quantum of flux in superconducting loops is half the value
first predicted by London in 1951. Onsager argument is found in
Ref.~\cite{Onsager61}:  ``\textit{Schafroth pointed out that electron
pairs held together by attractive interactions would obey Bose
statistics and be capable of superfluid (Einstein)  condensation}.''
Onsager maintains: ``\textit{The discovery of steps just this size
provides a convincing direct proof for the pairing of  electrons}.''
Interestingly, Onsager's letter did not mention any connection with the
Bardeen--Cooper--Schrieffer (BCS) theory of superconductivity~\cite{BCS},
instead referring to Schafroth's
theory~\cite{Schafroth0,Schafroth1,Schafroth2}, a theory that seems to
have been  discarded\footnote{The Swiss physicist, Dr. Max Robert
Schafroth,  passed away in 1959 as a consequence of an airplane
accident in Queensland, Australia. His early theory has regained
interest in the context of BCS-BEC crossover observed in ultra-cold
atoms of fermion type in the early years of the current century.}.

Other references available at the same
time~\cite{Byers,Onsager61,Blatt,Bardeen,Keller,Brenig,Ginzburg} for
the explanation of the one-half factor were all based, at different
levels, on the hypothesis of pairing: ranging from  ``by substituting
the electronic charge, $e$, by $2e\ldots$'' to perhaps the most unique
approaches to the problem in a different and novel manner by Byers and
Yang~\cite{Byers}. Indeed, they proved that the free energy of a
superconductor phase must be periodic in the magnetic flux, and being
locally quadratic: $F\sim \mathcal{P}_n  ( \Phi - n \Phi_0)^2$, where
$\Phi_0= hc/2e$ and the symbol $\mathcal{P}_n$ stands for periodic
repetition from $n\in (-\infty, \infty)$ of the quadratic basic free
energy:  $( \Phi - n \Phi_0)^2$. This periodic dependence on the
magnetic flux has an impact on the critical temperature for the
transition to superconductivity, which also becomes periodic in the
flux, an effect observed by Little and Parks~\cite{Little}.

As history progressed, this one-half factor in the quantum flux caused
great excitement in the scientific community. Briefly, this measurement
became the most common experimental evidence for the existence of
electron pairing, now called the Cooper pairs~\cite{cooper}, which is
the cornerstone of the Bardeen--Cooper--Schrieffer (BCS) theory of
superconductivity~\cite{BCS}. As we observe in this brief review of the
literature:
\begin{enumerate}[(1)]
\item  
Byers and Yang~\cite{Byers}: ``\textit{The quantization of flux is an
indication of the pairing of the electrons in the superconductor}.''
\item 
Parks~\cite{ParksScience64}: 
``\textit{The observation that the magnetic flux in a hollow superconducting
cylinder is quantized in units of $hc/2e$ has confirmed Fritz London's
prediction that superconductivity is a macroscopic quantum phenomenon
and that the superconducting state is a single-valued quantum state. It
has also conclusively demonstrated that the pair interaction of the
Bardeen--Cooper--Schrieffer theory of superconductivity is the important
one, at least for the few superconductors  studied}.''
\item  
In a later review for a Handbook of Physics, Fairbank and Fetter
express~\cite{fairbank}: ``\textit{This factor of 2 is explained as
being due to electron pairing and constitutes a direct proof of pairing
in superconductors}.''
\item  
Such a statement can be found also in books as for
instance the one of Tilley's~\cite{tilley}: 
``\textit{The flux quantum involves $2e$, the charge of a Cooper pair, and
results like those of figure~6.7 are the experimental confirmation that
the superconducting state is built out of Cooper pairs}.''
\item 
Leggett~\cite{Leggett} p.~296: ``\textit{The observation  of this  flux
quantization effect, with a factor $2e$ rather than $e$ in the flux
quantum $\Phi_0$, is one of the most convincing pieces for the BCS
``paring'' hypothesis}.''
\item 
Abrikosov~\cite{Abrikosov}, p.~402: 
``\textit{The phenomenon of flux quantization in a superconductor was first
predicted by F. London (1950). However, being unaware of Cooper
pairing, London assumed the charge carriers to be equal to $e$ instead
of $2e$, and obtained a flux quantum equal  to $2 \Phi_0$}.''
\end{enumerate}
It seems evident that experimental measurements of the quantum magnetic
flux are often taken as evidence for the existence of electron pairs.  
On the other hand, this line of reasoning may not be logically sound:
while pairing certainly implies a one-half factor in the quantum flux,
the presence of a one-half factor in the quantum flux may not be
necessarily a consequence  of electron pairing.

Thus, the question arises: Are electron pairs necessary to produce a
one-half factor in the quantum magnetic flux~?

There is no proof of such a statement, and it is unlikely that one
could exist. Below, we will demonstrate that the one-half factor may
arise as a consequence of a system composed of one-half spin particles,
without any assumptions about pairing.

\section{Spinor order parameter: an alternative explanation for a
factor of 2}\label{Sec:Spinor}
By extending Onsager's original idea to possible slightly non-uniform
rotations of the spins of the particles in the system, in
Ref.~\cite{NuovoCimento}, we have shown that the quantum of circulation
of Onsager is to be divided by two in such spin one-half systems. In
collaboration with Len Pismen from Technion, we explored more carefully
the existence of superfluidity and vortices in this kind of 
interacting Fermi liquid~\cite{Pismen}. The line of reasoning is as
follows.

For bosons, the simplest dynamical model consistent with the basic
principles (Galilean invariance, phase symmetry, etc.) of quantum
systems is the Gross--Pitaevskii
theory~\cite{GinzburgPitaevskii1958,Pitaevskii1961,Gross1957,Gross1958,Gross1961,Gross63}, 
which is formally valid only in the weakly interacting limit of cold
atoms, but remains of a qualitative interest in the context of strong
interactions,  e.g.\ liquid helium. It describes slow variations of a
field transforming the ground state wave-function, as explained above
following Onsager's argument. The basic object of this theory is thus a
complex wave-function, $\Psi(\mathbi{x},t)$, whose square modulus is
the local density, and whose phase is the same local phase
$\phi(\mathbi{x},t)$, and the superfluid velocity $\mathbi{v}_{\mathrm{s}} =
\bnabla \phi$ as already explained. This complex field has topological
defects, called
vortices~\cite{GinzburgPitaevskii1958,Pitaevskii1961,Gross1961}.
Dynamics of vortex lines in scalar fields have been studied in many
different settings, either conservative or dissipative and with either
Galilean or Lorentzian symmetry. Invariably, they are highly unstable
and tend to entangle or collapse in three space dimensions (see Len
Pismen's book for more details~\cite{PismenBook}).

On the contrary, the ground state of spin 1/2 interacting particles has
the $U(1)\otimes SU(2)$ invariance, as a rotation of all the spins of
the system is represented by an element of $SU(2)$. For a pure $SU(2)$,
a spinor field does not have topological defects since the $SU(2)$
group is simply connected  (see Toulouse and Kl\'eman~\cite{Toulouse}).

In Ref.~\cite{Pismen}, we noticed that the known topology of the direct
product $ U(1) \otimes SU(2)$ is more complex. The reason is that a 
$\uppi$ rotation of the scalar phase, which changes the sign of the
``wave function,'' can be compensated by $\pm 2\uppi$ spinor rotation,
and therefore a ``half-circulation'' vortex line is possible. The
topological group of the invariant manifold is therefore $(U(1) \otimes
SU(2) )/Z_2$. 

The extension of Onsager's argument passes by a non local
transformation of the ground state wave function $\psi_0$. Because of
the spin variables, the ground state wave function reads:
{$$
\psi_0 (\mathbi{x}_1,  \mathbi{s}_1,\mathbi{x}_2,  \mathbi{s}_2, 
\ldots,\mathbi{x}_N,   \mathbi{s}_N),
$$}\unskip
where the $ \mathbi{s}_i$ is  the eigenvector of the component of the
spin operator of the $i$th particle in the direction of the 3D vector 
$\mathbi{s}_i $ of norm 1/2~\cite{Dirac}.  
The introduction of a slow
rotation of the spins plays the same role as the slowly varying phase
in space in Onsager's theory. Therefore the wave function becomes,
{$$
\psi (\mathbi{x}_1,  \mathbi{s}_1,\mathbi{x}_2,  \mathbi{s}_2,
\ldots,\mathbi{x}_N,   \mathbi{s}_N)=  
\psi_0 (\mathbi{x}_1,  \mathbi{s}_1(\mathbi{x}_1),
\ldots,\mathbi{x}_N,   \mathbi{s}_N(\mathbi{x}_N) )\cdot 
\rme^{\rmi(\phi(\mathbi{x}_1,t)+\cdots +\phi(\mathbi{x}_N))} .  
$$}\unskip

Therefore the slowly varying wave function reads 
{\begin{equation}
\psi (\mathbi{x}_1,\ldots,\mathbi{x}_N)= \psi_0 
(\mathbi{x}_1, \rme^{\frac{\rmi}{2}  
\theta(\mathbi{x}_1) \hat{\mathbi{n}} (\mathbi{x}_1) 
\cdot \bsigma}   \mathbi{s}_1, \cdots,\mathbi{x}_N,  
\rme^{\frac{\rmi}{2}  \theta(\mathbi{x}_N,t) 
\hat{\mathbi{n}} (\mathbi{x}_N,t)  \cdot \bsigma}   \mathbi{s}_N )\cdot 
\rme^{\rmi(\phi(\mathbi{x}_1)+\cdots +\phi(\mathbi{x}_N))},
\label{eq:ExcitedWaveFunctionForFermions}
\end{equation}}\unskip
where the spins transform according to the $SU(2)$ representation group
of rotations in three dimensional space:
{$$ 
\mathbi{s}_1(\mathbi{x}_1) = 
\rme^{\frac{\rmi}{2}  
\theta(\mathbi{x}_1) \bsigma \cdot  
\hat{\mathbi{n}} (\mathbi{x}_1)} \mathbi{s}_1 =
[ \cos ( \tfrac{1}{2}  \theta(\mathbi{x}_1) )+ 
\rmi (  \hat{\mathbi{n}} (\mathbi{x}_1) \cdot \bsigma    ) 
\sin( \tfrac{1}{2}  \theta(\mathbi{x}_1) ) ] \mathbi{s}_1  .
$$}\unskip

Here, $\phi(\mathbi{x},t)$ represents the local variations of the 
$U(1)$ phase, $\bsigma$ represents the three Pauli matrices,
$\theta(\mathbi{x},t)$ is a local rotation angle, and
$\hat{\mathbi{n}}(\mathbi{x},t)$ is the rotation axis. 

Using the rules of transformation of the spins, the energy of this wave
function becomes:
{\begin{equation}
\Delta E = \frac{\hbar^2}{2 m} \int 
\rho(\mathbi{x}) 
\left(\frac{1}{4} 
(\bnabla \theta(\mathbi{x}) )^2 + 
(\bnabla \phi(\mathbi{x}) )^2 + 
\sin^2(\theta/2) (\partial_i n_j)^2 
\right)  \, \rmd \mathbi{x}, 
\label{eq:EnergyIncrement}
\end{equation} }\unskip
where $ \rho(\mathbi{x})$ is the particle density defined through the
one-particle density matrix (\ref{eq:density}). In the context of
superfluidity,  $\sofrac{\hbar^2}{2m}  \rho(\mathbi{x})  (\bnabla \phi
)^2$ is the kinetic energy density of the superfluid component, thus 
$\rho(\mathbi{x})$ is the superfluid number density field with
dimensions of the inverse of a volume. Moreover, in the context of
magnetism, the magnetic energy density,  $\sofrac{\hbar^2}{2 m}  \int  
\rho(\mathbi{x})  (\sfrac{1}{4}  (\bnabla \theta(\mathbi{x}) )^2 +
\sin^2(\theta/2) (\partial_i n_j)^2 ) $ implies that $\rho(\mathbi{x})$
also plays the role of a ``spin stiffness''.

Accordingly, the basic object, with a $U(1) \otimes SU(2)$ symmetry, is
a $2\times 2$ matrix field presented as:
{\begin{equation}
\Psi (\mathbi{x},t) = \sqrt{\rho(\mathbi{x},t)}
\rme^{\rmi\phi(\mathbi{x},t)} 
\rme^{\frac{\rmi}{2} \theta(\mathbi{x},t) 
\hat{\mathbi{n}} (\mathbi{x},t) \cdot \bsigma}
\equiv
\sqrt{\rho(\mathbi{x},t)}
\rme^{\rmi\phi(\mathbi{x},t)} 
\left[\cos\left(\frac{\theta(\mathbi{x},t)}{2}\right) 
+ \rmi\hat{\mathbi{n}} \cdot \bsigma
\sin\left(\frac{\theta(\mathbi{x},t)}{2}\right) \right],
\label{eq:Psi}
\end{equation}}\unskip
Here, $\rho(\mathbi{x},t)$ denotes the density field,
$\phi(\mathbi{x},t)$ represents the velocity potential,
$\theta(\mathbi{x},t)$ is the rotation angle, and
$\hat{\mathbi{n}}(\mathbi{x},t)$ is unitary the direction of the
rotation axis. All these fields are slowly varying functions of space
and time. Finally, both $\theta$ and $\hat{\mathbi{n}}$ can be made
complex to include Lorentz boosts, but this is not needed in the
context of a non-relativistic limit~\cite{Pismen}.

The dynamics of this order parameter follow from a unique action
invariant under $U(1) \otimes SU(2)$ and the Galilean 
group\footnote{In Ref.~\cite{Pismen} we derive the Galilean invariant
action:
{$$
S= \int 
\rmd\mathbi{x}\rmd t 
\left[ \mathrm{Re}\,
( \rmi \hbar \, \mathrm{Tr}
(\Psi^\dagger\partial_t\Psi )) -
\frac{\hbar^2}{2m}  \mathrm{Tr} 
( \partial_i \Psi^\dagger\partial_i\Psi ) -
\mu( \mathrm{Tr} (\Psi^\dagger\Psi) ) \right]
$$}\unskip
where $\Psi^\dagger$ is the Hermitian conjugated matrix, $\mathrm{Re}$
means the real part, $\mathrm{Tr}$ the trace, $\mu(\cdot)$ is the
density energy of the liquid. The general case would be a matrix
defined by 8 real functions, but here we have only 5 independent
functions. The restriction here is that we have fixed the mean value of
the magnetization $m_i =  \mathrm{Tr} (\Psi^\dagger\sigma_i\Psi)$ to be
zero.  Although this model has strong similarities with the
Gross--Pitaevskii theory, the general Onsager's argument, reproduced
earlier, does not need weak interactions but the existence of a
suitable ground state wave function. Nevertheless, we use a
Gross--Pitaevskii framework because it allows us to carry out explicit
calculations to the end.}:  
{\begin{equation}
S= -\int \rmd\mathbi{x}\rmd t
\left(\hbar \rho  \partial_t\phi  +
\frac{\hbar^2}{2m}
\left[ \frac{(\bnabla\rho)^2}{4\rho } +
\rho\left( (\bnabla \phi )^2 + 
\left(\frac{\partial_i\theta}{2}\right)^2+ 
\sin^2(\theta/2) (\partial_i n_j)^2\right)\right] +
\mu(\rho ) \right).
\label{eq:Action}
\end{equation}}\unskip

Assuming an axi-symmetric and multi-connected domain as shown in 
Figure~\ref{Fig:fig1}. The Euler--Lagrange equations of previous action
(\ref{eq:Action}) admits a solution of the form:
{\begin{equation}
\rho = \varrho_{\mathrm{s}} , \quad 
\phi(\varphi) = {\ell }\varphi,\quad  
\theta(\varphi)= s \varphi ,\quad  
\hat{\mathbi{n}}  =   \hat{\mathbi{k}} ,
\label{eq:vortex2}
\end{equation}}\unskip
where $ \varrho_{\mathrm{s}} $ is constant and represents the superfluid density,
$\ell$ and $s$ are constant. The indices $\ell$ and $s$ cannot be
arbitrary, because of the uniformity of  the wave-function as $\varphi$
varies from $0 $ to $2\uppi$.

\begin{figure}
\includegraphics{fig01}
\caption{\label{Fig:fig1}Scheme for a half-quantum of circulation in a
multi-connected domain. The angular coordinate $0\leq \varphi\leq
2\uppi$. The $U(1)$ phase variation, $\phi$, and the spin phase,
$\theta$ for a one half quantum of flux are drawn explicitly.}
\end{figure}

By varying the angular coordinate $\varphi$  from $0$ to $2 \uppi$, the
wave function (\ref{eq:Psi}),
{$$
\Psi  = \sqrt{\varrho_{\mathrm{s}}}
 \rme^{\rmi 2\uppi \ell } 
 \rme^{\rmi s \uppi  { \sigma}_3}
\equiv
 \sqrt{\varrho_{\mathrm{s}}},
$$}\unskip
must be single valued. Therefore, the matrix  $\rme^{\rmi 2\uppi \ell }
\rme^{\rmi s \uppi  { \sigma}_3}$ must be the identity matrix.
Expanding the spinorial part by the usual formula:  $\rme^{\rmi 2\uppi
\ell }  \rme^{\rmi s \uppi  { \sigma}_3} =  \rme^{\rmi 2\uppi \ell }
( \cos(s \uppi)  + \rmi  { \sigma}_3 \sin( s\uppi) )$,
therefore the conditions are: $s$ and $ 2\ell$ must be integers to
suppress the $ { \sigma}_3$ and the imaginary parts of the matrix.

Therefore, the final condition is that $2 \ell + s$ must be an even
number, i.e.
{\begin{equation}
\ell = k-\frac{s}{2},
\quad\mbox{with}\ 
k \ \mbox{and}\  s \in \mathbb{Z} .  
\label{eq:ConditionForL}
\end{equation}}\unskip

Therefore the single value condition is satisfied for $\ell $ being an
integer or a half-integer (see Table~\ref{tab1}, for other possible 
combinations). For instance for $s=1$ and $\ell=1/2$, the  
spin phase $\theta$ varies from $0$ to $2 \uppi$ as does the variable
$\varphi$. This results in a change in the sign of the order parameter
$\Psi$, thus necessitating a corresponding change in the angular
momentum phase $\phi$ from $0$ to $\uppi$ to ensure the full order
parameter remains a single-valued wave-function.

\begin{table}
\caption{\label{tab1}Admisible values of $\ell$ for various values 
of $k$ (columns) and $s$ (rows)}
\begin{tabular}{cccccc}
\tbody
\inlinefig{fx01}
\end{tabular}
\end{table} 

The super-current in  this multi-connected geometry becomes 
{$$
\mathbi{j}_{\mathrm{s}} =  \varrho_{\mathrm{s}} 
\frac{\hbar}{m} \bnabla \phi .
$$}\unskip
Therefore, only the phase $\phi$ contributes to the circulation:
{$$
\Gamma =    
\oint_{\mathcal{C}}   \mathbi{v} _{\mathrm{s}} \cdot \rmd\boldell = 
\oint_{\mathcal{C}}  
\frac{\mathbi{j}_{\mathrm{s}}}{ \varrho_{\mathrm{s}}}  \cdot \rmd\boldell  =  
\frac{\hbar}{m }  \oint_{\mathcal{C}}   
\bnabla \phi \cdot \rmd\boldell   \cdot \rmd\boldell  =  
\frac{\hbar}{m  } \oint_{\mathcal{C}}    \rmd\phi=
\frac{ \hbar}{m  } 2\uppi  \ell . 
$$}\unskip
Therefore, the basic quantum of circulation for $\ell=\pm1/2$ is:
{$$
\Gamma =    \oint_{\mathcal{C}}   \mathbi{v} _{\mathrm{s}} \cdot 
\rmd\boldell =\pm \frac{ h}{2m  } . 
$$}\unskip

\subsection{Superfluid half-circulation vortices}  
A simple vortex solution arises in cylindrical coordinates $(r,
\varphi, z)$ by taking
{\begin{equation}
\rho(r) ,\quad 
\phi(\varphi) = \frac{\varphi}{2},\quad  
\theta(\varphi)= \varphi ,\quad  
\hat{\mathbi{n}}  =   \hat{\mathbi{k}} ,
\label{eq:vortex}
\end{equation}}\unskip
where $\hat{\mathbi{k}}$ is the unit vector parallel to the $z$-axis.
Lastly, the field $\rho(r)$ follows an ordinary differential equation
that describes the core of the vortex solution, and it was derived in
Ref.~\cite{Pismen}.

The density obeys a Bernoulli like-equation (with the quantum pressure
included):
{\begin{equation}
-\frac{\hbar^2}{2m} 
\left( \frac{1}{ \sqrt\rho} \bnabla^2 \sqrt{\rho}  +
\frac{ \ell^2 + \frac{s^2}{4}}{r^2}  \right) + 
\mu'(\rho )- \mu_0 =0.
\end{equation}}\unskip
Setting $\rho(r)=\varrho_{\mathrm{s}} [D(u)]^2$ (where $u= r/\xi$, and $\xi $  is
the healing length that depends on the parameter of the model and it
makes the equation free of any scale), $\ell= 1/2$, $s=1$, and by
expanding the free energy $ \mu(\rho ) $ in a Taylor series,  yields
the following ordinary dimensionless differential equation for the
function $D(u) $:
{\begin{equation}
D''(u) +\frac{1}{u}D' -
\frac{1}{2u^2} D + D-D^3 = 0.
\label{eq:VortexPismen}
\end{equation}}\unskip
This equation should be solved with the boundary conditions $D(0)=0$,
and $D(u)\rightarrow 1$ as $u \rightarrow\infty$.

The outer asymptotics $(u\gg 1)$ approaches to the unity as $D(u)
\approx 1-1/4u^2$, while the inner asymptotics $(u\ll 1)$ is
$D(u)\approx \lambda u^{{1}/{\sqrt{2}}}$, denoting a non-analytical
behavior of the vortex near the axis. The constant $\lambda$ is found
by solving Equation~(\ref{eq:VortexPismen}) numerically. In
Ref.~\cite{Pismen} this is done obtaining $\lambda\approx 0.766$.

The one-half quantum of circulation was measured by Packard and
collaborators in 1991~\cite{Packard} in superfluid He$^{3}{B}$. As
expected due to the spin-1/2 nature of He$^{3}$ atoms, the quantum of
circulation is  $ \sbfrac{h}{2 m_3}$, with $m_3$ representing the
atomic mass of He$^{3}$ atoms.

\subsection{The case of a charged superfluid} 
Finally, the addition of a magnetic field, by breaking the global
$U(1)$ invariance, transforms the action~(\ref{eq:Action})  into:
{\begin{eqnarray}
S &amp;=&amp; -\int \rmd\mathbi{x}\rmd t\times\,
\left(\hbar \rho  \partial_t\phi  +
\frac{\hbar^2}{2m}  
\left[\frac{(\bnabla\rho)^2}{4\rho } +
\rho\left( \left(\bnabla \phi - 
\frac{e}{\hbar c} \mathbi{A} \right)^2 + 
\left(\frac{\bnabla\theta}{2}\right)^2
+ \sin^2\left(\frac{\theta}{2}\right) 
(\partial_i n_j)^2\right)\right]
\right.\nonumber\\&amp;&amp; \left.
+ \,
\mu(\rho ) + \frac{1}{8 \uppi }  |\mathbi{B} |^2 \right),
\label{eq:ActionWithB}
\end{eqnarray}}\unskip
where $\mathbi{B} = \bnabla\times \mathbi{A}$.

Assuming a superconducting ring  immersed in a uniform magnetic field
$B\hat{\mathbi{k}} $, see  Figure~\ref{Fig:fig2}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{Fig:fig2}Scheme for the measurement  of the quantum
magnetic flux in a  superconducting ring. Because of Meissner effect
the magnetic field is expelled from the superconductor. The magnetic
field  is $B\hat{\mathbi{k}}$ as $z\to \pm\infty$.}
\end{figure}

In what follows we show that the postulated order parameter admits a
half quantum of flux $\Phi_0$. We show this in two different path, a
first following, London, Onsager, Bardeen, Keller-Zumino and 
Ginzburg~\cite{london,Onsager61,Bardeen,Keller,Ginzburg} and a second
endeavor which follows Byers and Yang~\cite{Byers} argue the value of
the quanta of flux.

\subsubsection{Quantum of flux derived from the supercurrent}
The current becomes
{$$
\mathbi{j}_{\mathrm{s}} = e \rho  
\frac{\hbar}{m}  
\left(\bnabla \phi -  \frac{e}{\hbar c} \mathbi{A} \right),
$$}\unskip
therefore, the line integral of the previous expression becomes
{$$ 
\oint_{\mathcal{C}} \frac{m }{\hbar e  }
\frac{ \mathbi{j}_{\mathrm{s}}}{\rho} \cdot 
 \rmd \boldell   =   
 \oint_{\mathcal{C}} \bnabla \phi  \cdot 
 \rmd \boldell   -  \frac{e}{\hbar c} 
 \oint_{\mathcal{C}} \mathbi{A} 
 \cdot \rmd \boldell .
$$}\unskip
Because of the Meissner effect, the left-hand side vanishes in the bulk
of the superconductor, and as a consequence of the Ansatz
\eqref{eq:vortex},  $ \oint_{\mathcal{C}} \bnabla \phi \cdot \rmd
\boldell =\pm \uppi $, hence, by applying the Stokes theorem to the last
term:
{$$
\oint_{\mathcal{C}} \mathbi{A} \cdot 
\rmd \boldell  = \iint_{S({\mathcal C}) } \mathbi{B} \cdot 
\rmd \mathbi{n}\,  \rmd S = 
\pm  \frac{\uppi \hbar c}{e} = 
\pm \frac{h c}{2e},
$$}\unskip
which means the magnetic flux crossed by an arbitrary  contour
${S(\mathcal{C}) }$ is half of the  quantum of London without assuming
any pairing.

\subsubsection{Quantum of flux derived from the energy}
Assuming an axi-symmetric solution in a multi-connected domain admits a
solution of the form (\ref{eq:vortex2}), together with the magnetic
vector potential 
{$$
A_\varphi(r) = \frac{\Phi}{2\uppi r} 
$$}\unskip
inside the superconductor.

Therefore, the superconductor energy per unit length 
{\begin{eqnarray}
\frac{E}{L} 
&amp; = &amp;
\varrho_{\mathrm{s}} \frac{\hbar^2}{2m} \int 
\left( \left(\ell  - \frac{e \Phi }{2\uppi \hbar c}  \right)^2  + 
\frac{s^2}{4} \right)   
\frac{1}{r^2} 2\uppi  r\, \rmd r  +
\int \mu(\varrho_{\mathrm{s}})  2\uppi  r\, \rmd r ,
\nonumber \\
&amp; = &amp;
2\uppi \varrho_{\mathrm{s}} \frac{\hbar^2}{2m} 
\left( \left(\ell  - \frac{ \Phi }{h c/e}  \right)^2 
+ \frac{s^2}{4} 
\right) \log(R/a_0)  + \mathcal{E}_0. 
\label{Energy}
\end{eqnarray}}\unskip
Here $L$, $R$ and $a_0$ are, respectively, the length, the outer
radius, and the inner radius of the cylinder,  and  $
\mathcal{E}_0=\uppi (R^2-a_0^2) \mu(\varrho_{\mathrm{s}} )$. 

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:fig3}Scheme for the Byers--Yang energy dependence on
the dimensionless magnetic flux: $\Phi/(hc/e)$. Some parabolae are
labeled by the value of $\ell$ below.}
\end{figure}

In Figure~\ref{fig:fig3} is sketched the relevant dependence of the
energy in terms of the normalized  magnetic flux $\Phi/\Phi_0$. As the
magnetic flux increases the system passes from the one local minima of
the energy to the next, making this energy periodic with energy minima
realized for values: 
{$$
\Phi=
\{\cdots -3/2,-1,-1/2,0, 1/2,1, 3/2,\ldots\} 
\times\frac{hc}{e} .
$$}\unskip
From one minima to the next, the system overcomes an energy barrier of
the order of: 
{$$
\frac{\Delta E}{L} = 
\frac{ \uppi \hbar^2}{16m}  \varrho_{\mathrm{s}} 
\log(R/a_0).
$$}\unskip
The interpretation of this periodic energy has been already given by 
Byers and Yang~\cite{Byers}, and observed experimentally by Little and
Parks~\cite{Little}. 

\vspace*{3pt}

\subsubsection{The Josephson effect}

\vspace*{3pt}

Consider two pieces of superconductor 1 and 2 at an electric potential 
$V_1$ and $V_2$ linked by a weak junction characterized by a tunneling
constant $K$. The action of the superconductor junction may be written
directly from \eqref{eq:ActionWithB}. The electric potential appears as
a gauge field: $\partial _t\phi \to \partial _t\phi - 
\sofrac{e}{\hbar} V$, and the tunneling is represented by a term of the
form 
\vspace*{4pt}
{$$ 
K \,\, \mathrm{Tr} 
( \Psi_2^\dagger \cdot \Psi_1 + \Psi_1^\dagger \cdot \Psi_2),
$$ }\unskip
thus the action of the two superconductors reads
\vspace*{4pt}
{\advance\jot by 5pt\begin{eqnarray}
\mathcal{L} 
&amp;=&amp; 
\hbar \rho_1 
\left(\partial_t \phi_1-
\frac{e {V_1}}{\hbar}\right)+
\hbar \rho_2
\left(\partial_t \phi_2-
\frac{e {V_2}}{\hbar}\right)
\nonumber\\
&amp;&amp; +\,
2 K \sqrt{\rho_1\rho_2} \cos (\phi_2-\phi_1) 
\left(\cos\left(\frac{\theta_1}{2}\right) \cos 
\left(\frac{\theta_2}{2}\right)+
\sin \left(\frac{\theta_1}{2}\right) \sin
\left(\frac{\theta_2}{2}\right) 
\mathbi{n}_1\cdot \mathbi{n}_2 \right)+ \cdots 
\label{eq:josephsonLagrangian}
\end{eqnarray}}\unskip
where the dots represent spatial variations of the order parameter near
the boundaries of the junction.

The Euler--Lagrange conditions lead to:
\vspace*{4pt}
{\advance\jot by 5pt\begin{eqnarray*}
\partial_t\phi_1 -\frac{ e {V_1}}{\hbar}+
\frac{K}{\hbar} \sqrt{\frac{\rho_2}{\rho_1} } 
\cos(\phi_2-\phi_1)
\left( \cos \left( \frac{\theta_1-\theta_2}{2} \right)   + 
(\mathbi{n}_1\cdot \mathbi{n}_2 -1) 
\sin\left(\frac{\theta_1}{2}\right)
\sin\left(\frac{\theta_2}{2}\right)  \right) 
&amp; = &amp; 0 ,\\
\partial_t\rho_1 -
\frac{2 K }{\hbar} \sqrt{\rho_1 \rho _2} \sin(\phi_2-\phi_1)  
\left( \cos \left( \frac{\theta_1-\theta_2}{2} \right)   + 
(\mathbi{n}_1\cdot \mathbi{n}_2 -1) 
\sin\left(\frac{\theta_1}{2}\right)\sin
\left(\frac{\theta_2}{2}\right)  \right) &amp; = &amp;0.
\end{eqnarray*} }\unskip
Similar equations hold for the superconductor 2, by simply exchanging 1
by 2. 

The second equation defines the electric current from the
superconductor 1 to  2 would be  $I =  e \partial_t\rho_1= - e
\partial_t\rho_2$, which after assuming  $\mathbi{n}_1\cdot
\mathbi{n}_2\approx 1$, reduces to the first Josephson's equation:
\vspace*{4pt}
{\begin{equation*}
I = 
\frac{2 e K }{\hbar} \sqrt{\rho_1 \rho _2} \sin(\phi_2-\phi_1) 
\cos \left( \frac{\theta_2-\theta_1}{2} \right) .
\end{equation*}}\unskip
Re-arranging the trigonometric functions lead to the Josephson current:
{\begin{eqnarray}
I =
\frac{ e K }{\hbar} \sqrt{\rho_1 \rho _2} 
\left[ \sin \left( \phi_2-\phi_1+ 
\frac{\theta_2-\theta_1}{2} \right) + 
\sin \left( \phi_2-\phi_1- 
\frac{\theta_2-\theta_1}{2} \right)   \right].
\label{eq:JosephsonEqn1}
\end{eqnarray}}\unskip
The second Josephson equation follows by computing
$\partial_t(\phi_2-\phi_1)$ which after neglecting the factor he factor
$( \sqrt{\sfrac{\rho_2}{\rho_1} } -\sqrt{\sfrac{\rho_1}{\rho_2} } )
\approx 0$, leads to 
\vspace*{4pt}
{\begin{eqnarray}
\partial_t(\phi_2-\phi_1)=
-\frac{ e( V_2-V_1) }{\hbar},
\label{eq:JosephsonEqn2}
\end{eqnarray}}\unskip
hence the phase grows at a rate: $ \phi_2-\phi_1 =  -\sofrac{ e(
V_2-V_1) }{\hbar}t$. 

Although the action does not provide any dynamical equation for the
spin variables, a Josephson junction requires a battery that imposes an
electric field, topologically connecting both ends of superconductors 1
and 2. Therefore, the requirement that the spinor wave function, $\Psi$
of \eqref{eq:Psi}, must be single-valued implies that, if the phase
$\phi_1 - \phi_2=   - \sofrac{ e( V_2-V_1) }{\hbar}t$, then, the phase
difference $\theta_2-\theta_1$ follows it at twice its rate, thus,
$\theta_2-\theta_1= - \sofrac{2 e( V_2-V_1) }{\hbar}t$, so that  $I  =
\sofrac{ e K }{\hbar} \sqrt{\rho_1 \rho _2}  \sin  ( \sofrac{ 2e(
V_2-V_1) }{\hbar}t )$. 

Finally, as we know, the Josephson effect is sensitive to magnetic
fields. Equation~\eqref{eq:JosephsonEqn1} 
\mbox{becomes:}
{\begin{equation*}
I =
\frac{ e K }{\hbar} \sqrt{\rho_1 \rho _2} 
\left[ \sin \left( \phi_2-\phi_1 + \frac{e}{\hbar c} 
\int_1^2 \mathbi{A} \cdot \rmd\boldell  + 
\frac{\theta_2-\theta_1}{2} \right) +
\sin \left( \phi_2-\phi_1+ 
\frac{e}{\hbar c} 
\int_1^2 \mathbi{A} \cdot \rmd\boldell   - 
\frac{\theta_2-\theta_1}{2} \right)   \right].
\end{equation*}}\unskip
By repeating the topological arguments, it is obtained that the current
is periodic in the quantum of magnetic flux,  $({h c})/({2e})$.
Furthermore, this same argument is still valid for neutral Fermions,
the mass flow must be periodic with respect the circulation $h/(2 m) $.

\section{Discussion}
In this article we provide three examples of physical problems where a
factor two (or one-half) played an important role. Although, usually a
factor discrepancy comes form elementary mistakes, in some occasions
these  discrepancy may have deep consequences as we tried to illustrate
in these three different topics. We end this paper by mention the,
still not well understood, relation between  superfluidity and the
statistics of the particles. 

The first theoretical interpretation, which is not merely of historical
value, is due to Fritz London in 1938~\cite{London1}. London's work, 
entitled   \textit{The $\lambda$-Phenomenon of Liquid Helium and the
Bose--Einstein Degeneration}, draws an analogy between the macroscopic
quantities of the ideal Bose--Einstein gas and the properties of Helium
II. The phenomenon of Bose--Einstein condensation describes the fact
that, for a given temperature below a critical temperature $T_0$, a
fraction of the particles is in the ground state.

In particular, London noticed that in Bose--Einstein condensation, the
specific heat presents a discontinuity in the first derivative at a
temperature $T_0 = 3.12\text{\textdegree}$~K. The numerical values of
the critical temperature and entropy ($S = 1.283 Nk_B$) at the
transition point are of the same order of magnitude as those of liquid
helium. 

In contrast, for liquid helium, the specific heat diverges, indicating
a second-order phase transition, which is not the case for an ideal
Bose--Einstein gas. However, London concludes:
\begin{quote}
{Though actually the $\lambda$-point of helium resembles rather a phase
transition of second order, it seems difficult not to imagine a
connexion with the condensation phenomenon of the Bose--Einstein
statistics. The experimental values of the temperature of the
$\lambda$-point (2.19\textdegree) and of its entropy (${\sim}0.8 R$)
seem to be in favour of this conception}.
\end{quote}
And at the end, he poses the conjecture: 
\begin{quote}
{The conception here proposed might also throw a light on the peculiar
transport phenomena observed with He II (enormous conductivity of heat,
extremely small viscosity and also a strange fountain phenomenon
recently discovered by Allen and Jones)}.
\end{quote}

Subsequently, Tisza~\cite{Tisza}, then at the Coll\`ege de France,
suggested that Helium II could be considered a degenerate Bose--Einstein
gas, in which a fraction of the atoms would remain in the ground state,
with the value depending on the temperature, and these atoms could flow
without any friction.

However, it is clear that helium cannot be considered a perfect gas of
bosons, as it is well known that the liquid state is only possible
thanks to the interactions between atoms or molecules. 

A few years later, in 1941, Lev Landau~\cite{Landau41,Landau41bis}
published a theory of superfluidity based on the excited states of a
quantum system described by elementary excitations, which characterize
the collective behavior of the liquid. Landau, in two different stages:
in 1941~\cite{Landau41}  and 1947~\cite{Landau47}, deduced the
excitation spectrum for superfluid helium, which essentially contains
phonons in the long-wave limit and the so-called ``roton spectrum'' at
the atomic scale. By fitting the data, Landau found the essential
parameters of this spectrum, known as the Landau spectrum for
superfluid helium. An essential aspect of Landau's theory was the
existence of a critical velocity that depends on the excitation
spectrum:
{$$
v_{\mathrm{c}} =  \min_p \frac{\epsilon (p) }{p} .
$$}\unskip

Thus, if the flow velocity is such that $v&lt; v_{\mathrm{c}}$, no excitations are
created, and a superflow state may be observed. The necessary and
sufficient conditions for the existence of superfluidity are that the
spectrum does not vanish for a finite momentum, i.e.,  $\epsilon(p)&gt;0 $
for $p&gt;0$, and  $\lim_{p\to0}  \sofrac{\epsilon (p) }{p} &gt;0.$ The
thermal properties of heat capacity imply that a phonon spectrum exists
in the long-wave limit: $\epsilon \approx c p$, for $p\to 0$.
Therefore, superfluidity appears to be a generic feature of quantum
fluids at low temperature.

As Landau already noted~\cite{Landau41}, a non-interacting Bose gas at
$T=0$  is not a superfluid because the energy spectrum is 
$\epsilon(p)= \sfrac{p^2}{2m}$, which possesses zero critical velocity.
On the other hand, Bogolyubov~\cite{Bogolyubov1947} had shown, by the
method of second quantization, how to compute the elementary excitation
spectrum in a weakly interacting Bose gas. The development of cold
atoms allowed Ketterle and his group to provide evidence of a critical
superfluid velocity in a Bose--Einstein condensate~\cite{Ketterle1999}.

Incidentally, as  mentioned previously, the statistics of the particle
does not seem to play any role in Onsager argument, neither in the
approach on Leggett's inequalities~\cite{Leggett70,Leggett98} who
provided formal bounds for the superfluid fraction in an arbitrary many
body system. From Leggett's bounds one identifies simply the field
$\rho(\mathbi{x})$ by the superfluid density $\varrho_{\mathrm{s}}$ in the case of
an homogeneous liquid or gas.  

Finally, we end with a brief account of the last 20 years of the
research of ultra-cold atoms of fermion type.  By tuning the
interactions, via the Feschbach resonance, of ultracold gases it is
possible to pass from a regime in which the scattering length, $a_{\mathrm{s}}$,
crosses from an attractive to a repulsive case. This is known as the
BCS-BEC crossover~\cite{Parish,Strinati}. This physical situation,
predicted in the sixties by Keldysh and co-workers in the context of
excitons~\cite{Keldysh1964,Keldysh1968} and Leggett in
1980~\cite{Leggett1980}, is currently understood in weakly interacting
regime, $| a_{\mathrm{s}}| \rho^{1/3}\ll 1$~\footnote{Here $\rho$ is the number
density, as in \eqref{eq:EnergyIncrement}.}, via the current BCS theory
characterized by paring, for the attractive case ($a_{\mathrm{s}}&lt;0$). However, if
the interactions are repulsive and if the density of the gas is high
enough, two fermions with opposite spin may form a ``molecule'' of
spin-zero. Subsequently, as Schafroth
predicted~\cite{Schafroth0,Schafroth1,Schafroth2}, this spin-zero
particle rules a Bose--Einstein condensation becoming a superfluid. A
remarkable experiment was undertaken by Ketterle and collaborators in
2005~\cite{Ketterle2005}, in which reported the observations of a
vortex lattice in a rotating Fermi. Quantitative measurements of the
circulation are in agreement with $h/(2 m)$ as expected.

This remarkable experiment offer the possibility to be described
alternatively by the spinor order parameter proposed in the current
paper.  For the attractive case,  the basic wave function should be
\eqref{eq:ExcitedWaveFunctionForFermions}, with the conclusions that
arise in the current paper, in particular, a quantum of circulation
$h/(2 m)$ as observed by Ketterle and co-workers~\cite{Ketterle2005}. 

However, in the attractive case because two close fermions may form a
bound state the wave function \eqref{eq:ExcitedWaveFunctionForFermions}
is strongly modified. The kinetic term of the $N$ particle
Schr\"odinger equation split in such a way that $N/2$ pair of molecules
become:
{$$
-\frac{\hbar^2}{2m} 
\sum_{k=1}^{N/2}
\left(  \frac{1}{2} \bnabla^2_{R_k} + 2 \bnabla^2_{u_k} \right) . 
$$}\unskip
Where $\mathbi{R}_1 =  \stfrac{\mathbi{x}_1+\mathbi{x}_2}{2} $,
$\mathbi{R}_2 =  \stfrac{\mathbi{x}_3+\mathbi{x}_4}{2} $,  {etc.} are
the center of mass of the first, second,  {etc.} molecule, and
$\mathbi{u}_1 = \mathbi{x}_1-\mathbi{x}_2 $, $\mathbi{u}_2 =
\mathbi{x}_3-\mathbi{x}_4 $,  {etc.} correspond to the relative
coordinate of the molecule. The center of mass part of the wave
function space coordinates of the wave function is completely
symmetric, and  the individual pair of fermions is basically the
anti-symmetrized product of the pair-wise wave function:
{$$
\psi_{2}
( \mathbi{x}_1,\sigma_1, \mathbi{x}_2 , \sigma_2  ) =
\varphi(|\mathbi{x}_1-\mathbi{x}_2 |) 
\mathop{\lvert}\sigma_1,\sigma_2{\mathop{\rangle}},
$$}\unskip
because the spatial relative coordinate, $|\mathbi{x}_1-\mathbi{x}_2
|$, is also a symmetric wave function, the spin part must be
antisymmetric: $ \mathop{\lvert}\sigma_1,\sigma_2{\mathop{\rangle}}= 
\sofrac{1}{\sqrt 2}  (\mathop{\lvert} +-{\mathop{\rangle}} -
\mathop{\lvert}-+{\mathop{\rangle}} )$, hence the molecule constitutes
a spin zero particle  of mass $2m$~\cite{Leggett1980}.

In conclusion, the ground state wave function is of the form:
{$$ 
\psi_0(\mathbi{x}_1,  \mathbi{s}_1,\mathbi{x}_2,  \mathbi{s}_2, 
\ldots,\mathbi{x}_N,   \mathbi{s}_N) = 
\psi_{CM}
\left( \frac{\mathbi{x}_1+\mathbi{x}_2}{2} , 
\frac{\mathbi{x}_3+\mathbi{x}_4}{2} ,\ldots  \right)  
\varphi(|\mathbi{x}_1-\mathbi{x}_2 |) 
\frac{1}{\sqrt 2} 
\left(\mathop{\lvert}+-{\mathop{\rangle}} -
\mathop{\lvert}-+{\mathop{\rangle}} \right) \cdots
$$}\unskip
where the dots, $\cdots$, correspond to the product of the second, the
third, $\ldots,$ the $N/2$th  ``molecule''.

A fundamental difference arises with the wave function
\eqref{eq:ExcitedWaveFunctionForFermions}, the internal degrees of
freedom of relative coordinate and spin state of each molecule become
``freezed'' and cannot vary in space as it happened in 
\eqref{eq:ExcitedWaveFunctionForFermions}. The final quantum collective
state is characterized by a single Bose wave function, $\Psi_{CM}$, for
$N/2$ spin-zero particles of mass $2m$, as in the ground state
\eqref{eq:ExcitedWaveFunctionForBosons}. At the end superfluid
properties emerge as in the Gross--Pitaevskii theory for a weakly
interacting gas of bosons.

However, an important issue remains. For the molecular Fermi gas the
critical temperature is of the order of Bose--Einstein critical
temperature~\cite{nozieres1985}, namely $k_BT_{\mathrm{c}} \sim 
\sofrac{\hbar^2}{m} \rho^{2/3} \sim \epsilon_F$ the Fermi energy, while
for the BCS theory, is a non-analytic expression on interactions:
$k_BT_{\mathrm{c}} \sim  \epsilon_F \exp (-{K}/(| a_{\mathrm{s}}| \rho^{1/3})) $, $K$ a
numerical constant of order unit making $k_BT_{\mathrm{c}}\ll \epsilon_F$. We do
not know if a mean field application of our theory for the spinor wave
function \eqref{eq:Psi}  with fluctuations can produce such a
non-analytic dependence on the critical temperature.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Funding}
SR was supported by Fondecyt (ANID) under Grant No. 1220369.

\section*{Acknowledgements}
We acknowledge both anonymous referees for raising the issue of BCS-BEC
crossover. 

\CDRGrant[ANID]{1220369}

\section*{Note added in proof} 
During the final revision of the current paper, Microsoft announced the
Majorana 1 chip~\cite{majorana1}. It has been suggested that
topological defects could be used to enhance the stability of q-bits.
An alternative approach to creating such defects may be to use 
topological defects in the wave function of electrons on a circuit-like
structure, such as benzene  ($\mathrm{C}_6  \mathrm{H}_6$). As is well
known, these molecules are stabilized by the delocalization of valence
electrons circulating around the hexagonal structure, which lowers
their quantum kinetic energy.

A potential method for constructing molecular q-bits could involve
manipulating the spins of electrons moving along the periphery of the
hexagon. By twisting the spin orientation by $\uppi$ over one full
turn, a topological defect could be introduced, potentially leading to
a stable 
\mbox{configuration.} Since this mechanism operates on molecular
scales, it may be feasible to induce such a transition by illuminating
the molecule with an electromagnetic wave at a frequency corresponding
to the transition between the defect-free state and the state
containing the topological defect. The transition rate would likely
depend on the polarization of the wave.

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