SUPERFLUIDITY AND
SUPERCONDUCTIVITY IN RELATIVISTIC
FERMION SYSTEMS
D. BAILIN
School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK.
and
A. LOVE
Physics Department, Bedford College, University of London, Regent’s Park, London NWJ, UK.
I
NORTH-HOLLAND PHYSICS PUBLISHING-AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 107, No. 6 (1984) 325—385. North-Holland, Amsterdam
SUPERFLUIDITY AND SUPERCONDUCTIVITY IN RELATIVISTIC
FERMION SYSTEMS
D.BAILIN
School of Mathematical and Physical Sciences, University of Sussex, Brighton. U.K.
and
A. LOVE
Physics Department, Bedford College. University of London, Regent’s Park, London NW!. U. K.
Received December 1983
Contents
I. Relativistic gap equations
1.1. Relativistic gap matrices
1.2. Helicity amplitudes for spin 1/2 scattering
1.3. Gap equations
1.4. jP = ~+ pairing
1.5. The Ginzburg—Landau region
1.6. Gradient terms
1.7. Direct derivation of the Ginzburg—Landau free energy
2. Superfluid neutron star matter
3. Superconducting electron systems
327
327
329
331
334
336
338
341
344
349
3.1. Electromagnetic field fluctuation effects
3.2. Electroweak effects in superconductors
3.3. p-wave superconductivity
4. Pairing in quark matter
4.1. The order parameter for superfluid quark matter
4.2. Superconductivity in quark matter
4.3. Colour superconductivity in quark matter
Appendix A. Angular integrals
Appendix B. J = 2 projections
References
349
355
364
368
368
376
379
383
384
385
Abstract
The derivation of gap equations and Ginzburg—Landau free energies for relativistic fermion systems is reviewed. The cases of superfluid neutron
matter, superconducting electrons and superconducting and colour superconducting quark matter are described in detail.
Single ordersfor this issue
PHYSICS REPORTS (Review Section of Physics Letters) 107, No. 6 (1984) 325—385.
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D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
327
1. Relativistic gap equations
1.1. Relativistic gap matrices
Superfluidity and superconductivity in non-relativistic fermion systems have been much discussed and
many books and review articles have been written on the subject. (See for example, Leggett [1] and
Tinkham [21.)
There are however cases where relativistic effects are significant corrections (as for
neutron star matter), and perhaps even cases where relativistic effects are dominant (as for non-strange
quark matter). There are also situations where one wishes to study effects of relativistic quantum field
theory in a basically non-relativistic system (for instance the effects of electroweak theory in an ordinary
electron superconductor). Here also a relativistic treatment of superfluidity can be helpful. These
systems are discussed in detail in later sections. In this section, we review the treatment of superfluidity
for relativistic fermions keeping the discussion fairly general. To proceed from superfluidity to
superconductivity (or its generalisations when gauge fields other than the photon are involved, such as
the Z°or the colour gluons), is a matter of replacing derivatives by appropriate covariant derivatives in
gradient terms in the Ginzburg—Landau free energy. The method described here is a generalization
[3,4, 5, 6, 7, 8] of one used by Nambu [91for the non-relativistic case.
The origin of superfluidity in a fermion system is a non-zero expectation value for a product of two
fermion fields (describing Cooper pairing). This may be introduced through a term in the action
S4
=
Jd4x
J
d~y[~(x)Li(x,y)~i(y)+h.c.]
(1.1)
where ~/i(x)is a relativistic fermion field operator and 4!!~(x)is the charge conjugate field:
(clic)a(x)= Cap t/Jp(x),
(1.2)
where a and $ are spinor indices and C is a 4 x 4 matrix with the properties (see, for example, Bjorken
and Drell [10])
ty~C=—y~, C~~’—C’~.
(1.3)
C
The object Li (x, y) (which may involve derivatives) is a 4 X 4 matrix in the spinor indices and is referred
to as the gap matrix. In general, we shall allow that ~/i(x)
may have indices other than just spinor indices
(for example, colour indices in the case of quark matter), and correspondingly Li (x, y) will be a matrix in
these additional indices. We shall see later how Li may be calculated in a self-consistent fashion.
The remaining quadratic terms in the Lagrangian for a Dirac spinor field are of the form
‘Zree
t/i(iy~~m)tfr+,.niyot/J,
(1.4)
where Cooper pairing between fermions of equal (or approximately equal) mass m is assumed, and a
chemical potential ~ has been introduced at finite density. We shall use the conventions of Bjorken and
Drell [10] for the metric of Minkowski space, and for Dirac matrices. The interaction terms in the
Lagrangian will depend on the system under discussion.
it is convenient [9] to write the inverse propagator for the fermions as a 2 x 2 matrix acting on the
328
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
column vector (~),and it is also convenient to transform to momentum space. In general, Li(x, y)
depends on two independent spatial variables, and two independent momentum variables are required.
However, if the superfluid is homogeneous then Li (x, y) depends only on x y and not on x + y, i.e.
only on the relative position of the fermions within the Cooper pair, but not on the centre-of-mass
coordinate of the Cooper pair. This is the case we shall consider here deferring the general case to
subsection 1.5. Then only a single momentum variable is required (the relative momentum of the two
fermions in the Cooper pair) and we may rewrite the momentum space inverse propagator acting on (~)
as
—
~
Li(q)
~(q)
15
where
(1.6)
~(q)~y°Lit(q)y°
(1.7)
Li(q)~Jd4ze~Li(x,y)
(1.8)
and
where
z—x—y.
(1.9)
The adjoint Lit is to be taken in both spinor indices and any internal symmetry indices. We shall see in
later sections that provided the interaction producing the Cooper pairing is short-ranged, in a certain
sense, we shall always be interested in
Li(n)~Li(qO=0,~q~=pF)
(1.10)
which is a function only of
(1.11)
n~q.
In writing down the most general form of Li(n) for given angular momentum and other quantum
numbers of the Cooper pair, we must ensure consistency with Fermi statistics. This means that effective
Lagrangian terms as in (1.1) must not vanish if we antisymmetrise in fermion fields. In consequence,
allowed gap matrices have the property
CLi(n) C’
=
LiT(—n)
(1.12)
where the transpose on the right-hand side of (1.12) includes any internal symmetry indices as well as
spinor indices, and the involvement of n here corresponds in coordinate space to derivatives acting on
the fields.
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
329
1.2. Helicity amplitudes for spin 1/2 scattering
In the non-relativistic case, the interaction potential between the fermions enters the gap equation,
though in the end it only affects the value of the critical temperature T~,everything else being model
independent. (See, for example, Leggett [1].) In the relativistic case, the corresponding objects which
enter are the helicity amplitudes for fermion—fermion scattering near the Fermi surface. It will turn out
that these not only affect the value of T~,but also the detailed structure of the relativistic gap matrix
(which can involve more than one Dirac covariant). They do not, however, affect the Ginzburg—Landau
free energy other than through T~.For later reference we present here some relevant facts about the
helicity amplitudes for spin 1/2 scattering, through scalar exchange and through vector exchange.
Consider first the case where the scattering is due to the exchange of a scalar. Then, to leading order,
we have to calculate the scattering amplitude from the one-scalar-exchange diagram and the crossed
diagram (fig. 1). The indices i, j, k, 1 refer to the possibility that the fermions may have internal
symmetry indices.
Let the coupling at each interaction vertex be g, and let the propagator associated with the scalar
exchange when the two fermions are on the Fermi surface be D(cos 0), where 0 is the angle of
scattering. Then, in the notation of Goldberger et al. [11] the helicity amplitudes at the Fermi surface
are
=
=
and for J
—
J12’
f~2=
8~
±(-1~]
j(E~+m2) V~ 2J+ 1[(J+ 1) Vj+~+ JV~11}
-
[I
~
±(-1)~]
[(J
+ I)
}
Vj±~
+ J Vj~] p~V1
-
1,
ymp~1+1
1\Jl
8ITEF [1±(-lfl
~
VJ(J+
2J+1 1)~
~
~p~vj
-
VJ_t
2+rn2 [J V~+~
+ (J+ 1) V~
E
1]}
(1.13)
~
Fig. 1. Single-scalar or single-vector.exchange diagrams.
330
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
where
Vj~
f
dzP1(z)D(z)
(1.14)
2)”2
(1.15)
(p~+in
EFEE
and
(1.16)
The upper or lower sign is to be taken according as the wave function of the pair of fermions scattering
is symmetric or antisymmetric in any internal symmetry indices. If there are matrices associated with
internal symmetry at the vertices then they will only affect the value of
With particular reference to the case of quark matter, consider also the case where the scattering is
due to vector exchange with a coupling —igta at each interaction vertex where ta are 3 x 3 matrices
associated with SU(3) of colour. Let the propagator for vector (gluon) exchange on the Fermi surface
have the form
~.
D°°
= (2p~tDE(cos 0)
(1.17a)
D~0= —(2p~)~
ô’~°DM(cos
0)
(l.17b)
where a, $ = 1, 2, 3 are spatial indices, and 0 is the angle of scattering. In a vacuum, DE and DM would
be the same function, but to take account of screening by the medium we allow them to differ. In this
case, the helicity amplitudes at the Fermi surface, calculated from fig. 1, are
~
+~‘
1p~(V~÷1V~~)+2~~
1p~(V~tV~1)}
I6ITpFEF
~‘=
and for J
~
(1)] ~p~(V~+3V~)
2) V~+
2) V~t_p~V~t]}
[(E~+rn
1-p~V~t]+2~ 1 [(E~+m
1,
——~
1±—i~ VJ(J+1) yE
ft2_
—( ~
E
2J+l ~
V3+~)
±(-1Y]{p~V~+
1
V~)
f~2=167rp~EF~
8~.~[
~
~
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
=
331
~ (—1Y] j(E~+m2) V~+p~V M
16irp~E~11
1
+~
1p~(VY1+V~i)+2/~
1~’~ V~+1)}
(1.18)
where
J
V~~vs~
dzPj(z)D”(z)
with p
(1.19)
E, M,
2I4ir for ~ channel
~
~g
1—~g2/4ir
for ~channel
(1.20)
and the upper or lower sign is to be taken according as the wave function of the pair of fermions
scattering is symmetric or antisymmetric in any internal symmetry indices (flavour and colour indices, in
the case of quarks). If the coupling at each interaction vertex had been —ig rather than —ig ta then y
would be replaced by ~ of (1.16).
1.3. Gap equations
Relativistic gap equations may be derived following a generalization [3,4, 5, 6,7, 8] of a method
developed for the non-relativistic case by Nambu [9]. The idea is to use the Dyson equation for the
proper self-energy of a fermion to calculate Li self-consistently. More precisely, it is the off-diagonal
component of the Dyson equation in the space of (~)
that is required (fig. 2). For insertion in the Dyson
equation we need the fermion propagator, S(q). Let us write
S( \_(A(q) B(q)
(121
‘1~\C(q) D(q)
as a 2 x 2 matrix in
(~)
space. Inverting (1.5) gives
C(q)=(g_m_1Li[A(g_~_m)_1Li_(g+u~~m)]1
(1.22)
and we shall not need the other entries. To keep things fairly general let the interaction vertices be
Fig. 2. Single-particle-exchange contribution to the off-diagonal component of the Dyson equation. Cross-hatching denotes the proper self-energy,
and diagonal marking the exact propagator of the fermion.
332
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
_igFA, and the propagator of the exchanged particle be DAB(k — q), where
A and B each denote a
collection of spin and internal symmetry indices (e.g. for colour gluon exchange TA = yt0) This
corresponds to an interaction Lagrangian
(1.23)
where 4°A is the field of the exchanged particle. We shall need the interaction in the space of (~).
Adopting the notation
~=
(~)
(1.24)
then the interaction may be written as
=
g~
(TA
0)
~P4CA
(1.25)
where
TC~
(1.26)
C(f~x)
and C is the charge conjugation matrix of (1.3). The proper self-energy in (1.5) is separated off by
[A
writing
S’(q)= S~t(q)—2(q)
(1.27)
where
S~’(q)=(4+frI~~~rn ~-~_~)
(1.28)
and the proper self-energy
~(~)_(Li~)
~(q))
(1.29)
Then the Dyson equation (fig. 2) gives at finite temperature
~ DAB(k_q)(’~ ç~A)S(q)(’~ J~B)
n
(1.30)
odd
where
i3=(k~TYt.
(1.31)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
The Summation
~5 over
333
Matsubara frequencies, and we are now using q to mean
(1.32)
q=(icon,q)
and similarly for k, where
=
nir/$
(n
odd).
(1.33)
From the off-diagonal component of (1.30), using (1.21) and (1.29) we find
Li(k)= g2$’
J~—~3
~ D~~(k_q)[AC(q)FB.
n
(1.34)
odd
We now notice that for any function f(q
0) we may write
~ f(iw~)=
n
~—
~ dqof(qo) tanh(~$qo)
(1.35)
odd
where it is to be understood that the q0 integration is round an anticlockwise contour including the
poles of f(qo) but not those of tanh ~f3q0.
Thus (1.34) may be rewritten as
Li(k)
2~j4
=
DAB(k
— q) [A C(q) tanh(~f3qo)TB
(1.36)
~ig
where the q
0 integral is to be evaluated round an anticlockwise contour including the poles of C(q) but
not those of tanh(~$qo).
Evaluating the residues at the poles of C(q), with C(q) given by (1.22), yields
J
2
4(k) = ~g
)3 DAB(k
—
q)
[A
{A(q) [~2i + A(q) A(q)]_1/2tanh ~$ [~2f+ A(q) A(q)]1~~2}pB
(1.37)
where
DAB(k
—
q)
DAB(kO— q
0=
0, k — q)
Li(k)=Li(ko__\/k2+m2_,1,k)
and similarly for Li (q) and A(q). Also
2)~t
(,~O+ q
— m)Li(q)(~sy°—
q ‘y + m)
(1.38)
(1.39)
(1.40)
L(q)ov(4p~
and
an
(q2+ rn2)”2— ~.t.
(1.41)
334
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
To zeroth order in g2 we have
=
(1.42)
(p~+m2)t’2= EF,
and in deriving (1.37) we have made the approximations
/~,(Vk2+rn2_~)/~1.
(1.43)
(Only momenta close to the Fermi surface are important.) This means that some of the poles of
have negligible residues.
The integral d3q is now separated into a radial and an angular integral,
~~s=
~-~-~(+,u) [(~ + p~~—
rn2]t/2d~
and as in the non-relativistic case (Leggett [1]) the
C(q)
(1.44)
integration is cut off when Jj
= ~,
with
(1.45)
(This is nothing to do with controlling divergences, but is simply a way of approximating the integral.)
Then,
146
(21T)32d
~4
(
.
)
where
dn/d
=
/.LPF!7T2
(1.47)
is the density of states at the Fermi surface. We suppose here that the propagator DAB(k — q) is slowly
varying in the sense that its variation is on a scale large compared to ~ (and we shall choose ~ so that
this scale of variation is large compared to ~).This corresponds to the assumption of a short-range
potential in the non-relativistic case. Then, we may consistently take Li (k) to be a function only of
(1.48)
n’=—k
in what follows, and 4(q) to be a function only of
(1.49)
n~q
since the variation of Li with respect to the magnitude of the momentum is on a scale large compared to
$_t~ Some discussion, at T= 0, of the variation of 4(k) with jkl when long-range forces are present, is
given by Barrois [3].
1.4.
J”=O’
pairing
As an illustration, which will be relevant in later sections, we specialize the results of the last
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
335
subsection to the case of Cooper pairing with .P’ = 0~and no internal symmetry indices. Then the most
general gap matrix consistent with Fermi statistics is
4(n)= Li~~5+
42n YYo
75+4370
(1.50)
y~
with n as in (1.49). (Recall that Li is sandwiched between ~ and i/i, and not between ~ and ~i. Thus,
there is no factor of —1 from the intrinsic parity of the antiparticles.) In this case, by expanding in a
power series in 4(q) 4(q) and performing some Dirac algebra, the matrix structure of (1.37) may be
greatly simplified. We then find
J ~-;~j~
2
Li (n’) =
—
~g
DAB(k
—
q) d(2 + d*d)_l~’2
rA ~ ~
x tanh~$(~2± d*d)l~2.
(~o_~~
~y +2~-)18
(1.51)
where n’ is as in (1.48), and
d=Li
1_P~ELi2_~~~Li3.
~LL
/2
(1.52)
Thus, using (1.46),
t12 tanh ~$ (~2+ d*d)h/2}
d
= _i~g2 ~
(~2 +
d*d)_
(1.53)
~
where
DAB(n, n’) =
DAB(k
— q)IIkl=IqI=,~
(1.54)
and we have used the fact that the propagator is slowly varying.
Individual gap equations for 4~,42 and 43 are very model dependent, depending on the details of
the exchange producing the pairing. However, an almost model independent equation for the order
parameter d may be obtained from (1.53) by taking a suitable trace to project out d. Thus,
d
a~d
=
J
de
(~2+
d*d)~/2tanh ~$ (e~+ d*d)h/2
(1.55)
where
1
a,, =
2
fdQ’ IdQ
,
,
1-6g J-~—---j~—-DAB(n~n)Tr[(,ys+n.vYovs
(1.56)
336
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
and the only model dependence is in the value of the coefficient a~twhich we shall see shortly may be
related to the critical temperature T~.
At zero temperature, $_÷Do, (1.55) is easily solved to give
Id~T=o=2~exp[_(~~a~t)].
(1.57)
At the critical temperature T~,jdl approaches zero, and so T~is given by
kBTc=~oexp[_(~~a~t)]
(1.58)
where ~ arises from
J
d (2)- tanh ~/3
=
In ~$~
(1.59)
—
and
~‘=1.14.
(1.60)
Comparing (1.57) and (1.58), there is the relationship between T~and Id~T=o,
kBTC=~~ldfTo
(1.61)
just as in the non-relativistic case.
1.5. The Ginzburg—Landau region
For Cooper pairing with J~0, the gap equation (1.37) is not usually soluble at a general temperature, nor even at T = 0. The situation is much simpler in the region close to the critical temperature
(the Ginzburg—Landau region) where an expansion in powers of Li may be used. Upon expansion in
powers of 4, (1.37) gives
(1.62)
~
with
R
=
—~ (y0+~n.y_~~)Li(q)(
7o..PE~.7+~)
x [A(q)
(~o+
n
-
~)
4(q)
(yo -
LE~ y +
.
~)
]fl
2)~[(2)- ‘tanh ~$].
22~2n! (d
(1.63)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
Correct to order 43, after carrying out the
4(n’)
=
g2
J
~DAB(n,
n’)
integration this gives
(1.64)
(aS(n)+ b~(n)At(n)A(n))Tl
[A
337
where
a=
ln ~$~
=
J
~
b = 16(ITkBTC)2
=
~
d (2)-’ tanh ~$
(1.65)
J
(1.66)
d
[(2)_t tanh ~$]
with
(1.67)
A(n)=~(y0+~n. 7__,j-)4(n) ~
and DAB(n, n’) as in (1.54).
If we now specialize to the case of 7
subsection 1.4, then
0~Cooper pairs with no internal symmetry indices, as in
=
A(n)= _~d(ys+~n.i
7oys_~-yoys)
(1.68)
with d as in (1.52), and (1.64) yields
J
2d ~DAa(n,
Li(n’)= —~g
fl~)[A
(~
5+~~yyo y~—~-yo
Ys) TB (a +
bd*d).
(1.69)
Projecting d by taking traces as in subsection 1.4, we find
a~’(a+ bd*d)d
1as in (1.56). (This is consistent with the expansion of (1.55) to order d3.)
with
a~
In terms of the critical temperature T~,
d
=
~j~.1.td=bd*dd
(1.70)
(1.71)
where
t=(T—T~)/T~.
(1.72)
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
338
Equation (1.71) is the Ginzburg—Landau equation arising from the Ginzburg—Landau free energy
~
(1.73)
where we have used (1.47). The constant of proportionality in (1.73) is not fixed by the gap equation,
and we shall return to the determination of the normalization of ~ later.
1.6 Gradient terms
So far we have been restricting attention to the case where the superfluid is homogeneous, i.e. where
there is no dependence of the gap matrix on the centre-of-mass coordinates of a Cooper pair.
Equivalently, we have been taking the centre-of-mass momentum of the Cooper pairs to be zero. If we
relax this restriction then gradient terms appear in the Ginzburg—Landau free energy. To calculate these
gradient terms [8], we return to (1.1) and allow Li (x, y) to depend on x + y as well as x — y. Then, we
have to Fourier transform with respect to two variables, and we introduce two momentum variables by
writing
Li(x, y) = 1(2)41 (2)4e
e~Li(p’,p).
(1.74)
As a 2 x 2 matrix in the space of (~)the inverse fermion propagator is
—
Li(p’,p)4~(p’—p)
(~rn +jt)(2ir)
Li(p’,p)
(p-m-~)(2~)~~(p’-p))
(I 7S)
where
(1.76)
Li(p’,p)= yOzlt(p,pl)y0.
The off-diagonal entries describe a pair of particles with momenta p and —p’. When p and p’ were equal
we were able to invert to obtain the fermion propagator (as in (1.21) and (1.22)). Here things are not so
simple. If we write for the propagator
5
~ (A(p’,p)
—
~C(p’,p)
B(p’,p)\
1
D(p’,p))
(177)
and attempt to invert to obtain C(p’, p) we arrive at the difficult integral equation
-1
C(p’, p) 4(j~
— md4+ ~)
“
~d
~
+ (p’ — rn — ~
Li(p’ p)
C(p’, p”)A(p”, p”) (p” - m —
Li(p”, p) =0.
(1.78)
However, if we only want to derive the Ginzburg—Landau free energy in the Ginzburg—Landau region,
then we need only keep spatial derivatives acting on the lowest order term in Li. In order to calculate
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
339
these gradient terms it is sufficient to carry out the inversion correct to order Li, and for this purpose
(1.78) may be replaced by
C(p’,p)=
m— ~
—(i’—
4(p’,p) (~—m
+
(1.79)
~)1.
(The order 43 and higher non-gradient tenns are evaluated as before.)
With unequal momenta on the external legs, the Dyson equation for the proper self-energy (fig. 3)
gives
Li(p’, p) = ~ig2
DAB(p — q)[A C(p’
—
p + q, q)~Btanh ~f3q
0
(1.80)
where we have converted the Matsubara frequency sum to a contour integral as in subsection 1.3. The
q0 integration is round an anticlockwise contour which includes the poles of C(p’ — p + q, q) but not
those of tanh ~/3q0.
It is convenient hereafter to consider 4 as a function of the variables
k
(1.81)
~(p’
+ p)
and
(1.82)
K=p’—p.
Working correct to order Li using (1.79) we may write
Li (k, K) =
J
2
—
~ig
~~4
DAB(k
—
x 4(q, K) (g— ~X— m
q) [A(~y
+~
+
-~.K m —
—
TB tanh ~13q
0.
(1.83)
The integration variable has been shifted from q to q + ~K and K0 has been taken zero. (Wehave taken
the liberty of using the same symbol for 4 as a function of the new variables k, K.) Provided we are
only interested in spatial variation of the gap matrix on a distance scale large compared with p~’we
may take
(1.84)
~
Then the dominant poles are at
rn2]”2
q0 [(q —
2
~K)
+
—
an
E
1
(1.85)
Fig. 3. Single-particle-exchange contribution to the off-diagonal component of the Dyson equation for pairing with non-zero centre-of-mass
momentum. Cross-hatching denotes the proper self-energy, and diagonal marking the exact propagator of the fermion.
340
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
and
q0
2
=
—[(q +
~c —E
rn2]t’2 +
+
2.
~K)
(1.86)
Evaluating the residues at these poles gives
~
(1.87)
RIB
with
R
m
=(/2y0± qy—
—~K)Li(q,K)(/2y°—
t
x [4/22 (E~+
q .y+
m
(tanh ~$E~+ tanh ~$E
(1.88)
2)
E2)]
and
Li (k, K) = Li (k0 = 0, k, K0
=
0,
(1.89)
K).
2 yields
Expanding in powers of
R
=
K
up to order K
A(q, K) [~tanh ~$
— -~tanh ~$
K)an(4/22)t
q ~y
sech2 1$~ $2
(1.90)
(q•K)2]
where
A(q,
(/270+
—
rn)4(q, K)(/2y°— q
~y +
m).
(1.91)
If we now include the non-gradient terms at all higher orders in Li, as previously calculated, we get
the final gap equation
Li(k, K) = ~g2I(2ir~~
DAB(k
— q) [A
(1.92)
RIB
with
R
=
A(q,
K)
[~2J
— (16/22)t
+
(q
zl(q, K) S(q, K)]t12 tanh ~1~(2I
~K)2f32
+
e~tanh ~$ sech2~$A(q,K)
(1.93)
and
Li(q,K)=
.yO4t(q K)y°
This generalises (1.37) by including the gradient terms linear in Li.
(1.94)
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
341
In the Ginzburg—Landau region, correct to order 43, (1.92) and (1.93) become
Li(n’, K)= g2
+
J
~
DAB(n,
~i)fA
{aA(n,
K)
bA(n, K) At(n, K) A(n, K) — c ~
(n
.
K)2
A(n, K)}
fB
(1.95)
generalizing (1.64), where
c
=
~
=
~
J
d
tanh ~$ sech2 ~$
(1.96)
and
A(n, K)=~~i~~(/2y0+pFn
~y— m)Li(n, K) (/2y°-pFn y+ m).
(1.97)
If we now specialize to the case of 7 = 0~pairing with no internal symmetry indices, as in subsection
1.4, we may proceed further. Isolating the order parameter d of (1.52) by taking traces as usual, and
projecting out the J = 0 part by replacing KtK’ by ~
we arrive at
d=a~t(a+bd* d_c~_~K2)d
(1.98)
with ~ as in (1.56). The need to project out the original J value on the right-hand side of a gap
equation arises frequently. This projection is justified because admixtures of other J values in the order
parameter are expected to be small, for the same reasons as in the non-relativistic case. (See, for
example, Leggett [1], section yE.)
Since K is a momentum variable associated with the centre-of-mass coordinates of the Cooper pair,
eq. (1.98) is the Ginzburg—Landau equation deriving from the Ginzburg—Landau free energy
~j.cc.i4~!~td*d+ 7(3)/2j1~~d*d2+
2d
32’7T4(kBTC? ~
7~(3)p~ Vd*.Vd
96
/2 (kBTC)2
IT~
199
(
.
)
where
t= (T— T~)/T~
(1.100)
and we have used (1.47). We return to the overall normalization in the next section.
1.7. Direct derivation of the Ginzburg—Landau free energy
In the derivation of the Ginzburg—Landau free energy in previous sections, the overall normalization
was not fixed by the gap equations. In this section, we see how to fix this normalization by a direct
342
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
calculation of the order 42 term in the Ginzburg—Landau free energy. (An alternative method of
calculation is given by Barrois [3].) Although the present calculation provides us with new information
about the normalization of the Ginzburg—Landau free energy, it does not yield the information about
the individual components of the gap matrix (Lii, Li2, 43 in the J” = 0 case) given by the gap equations,
once the pairing force is known.
The free energy density may be calculated from vacuum bubbles. (See, for example, Freedman and
McLerran [12]and references therein.) The contribution of fermion loops to the free energy density of a
fermion system is given to order g2 by
~i~=~$~‘
E
f
n odd
~~
t(q)S(q)}
3{tr
(1.101)
S(q)~(q)+trln S~
where S(q) is the exact fermion propagator, So(q) is the free propagator, and 1(q) is the proper
self-energy at order g2. The sum is over Matsubara frequencies
w~=n1T/$,
(nodd)
(1.102)
and here we are using q to mean
q=(iw~,q).
(1.103)
Using (1.35) to convert the frequency sum to a contour integral, we obtain
=
~i~
tanh ~$q
0{-tr 5(q) 1(q) + tr In S~(q) S(q)}
(1.104)
where the q0 integration is round an anticlockwise contour which includes the poles of the trace factors
but not the poles of tanh ~$qo.
To apply this result to a Cooper paired system, we simply use the matrix propagators and proper
self-energy acting on the space (~) defined in (1.5), (1.27), (1.28) and (1.29). (The factor of ~ relative to
Freedman and McLerran [12]is to avoid double counting in this basis.) To proceed with the evaluation we
need to determine the propagator S(q) of (1.21) by inversion of (1.5). However, we need only carry out this
inversion correct to order 42 if we want to calculate the order 42 term in &~To this order,
(1.105)
(1.106)
1 Li(q) (~+,~t
— m)’
C(q)= —(g—~— m)
A(q) = (g+ ~ — m)t — B(q)Li(q) (~q+~
D(q)= (q’—~—
m)1 —
—
m)’
(1.107)
C(q)A(q) (g—~i—
m)1.
(1.108)
From (1.21) and (1.29),
tr S(q) 1(q) = —tr B(q)
4(q) —
tr A(q) C(q)
(1.109)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
343
and using (1.105) and (1.106),
tr S(q)1(q)= 2trA(q) (4— ~ — m)~Li(q)(4+ ~
Also, expanding the logarithm out to order
trin S~(q)S(q)= trA(q) (g— ~
—
—
(1.110)
m)’.
42,
(1.111)
m)~/i(q)(4+ ~ — m)~.
So, returning to (1.104),
3~= 4i~~—~4tanh2$qotr
A(q) (g— t
—
m)’ Li(q)(4+ ~
—
(1.112)
m)~.
Evaluating the residues at the poles leads to
~—j3
tanh2l3 tr[A(q)
S~~’
(16,.t~’J
(/2yo+
q ~y — m)Li(q) (/270—
q ~y
+
m)]
(1.113)
where
2)1”2— p..
(q~+rn
Using (1.46) and carrying out the
(1.114)
=
=
J
—~a ~tr
integration, we arrive at
At(n) A(n)+ O(Li~)
(1.115)
where we have used the notation of (1.65) and (1.67).
If we now consider the case of J” = 0~Cooper pairs with no internal symmetry indices as in
sub~ection1.4, then we obtain
(1.116)
~=_ad*d+O(d4).
In this approach, it remains to introduce a “mass” counterterm, i.e. a counterterm for the coefficient of
d*d. A convenient way of doing this is to subtract this coefficient at T = T~.We then have
=
td* d + 0(d4).
(1.117)
This enables us to establish the normalization of (1.73) or (1.99). Thus the correctly normalized form of
(1.99) is
3~T~vd*Vd
td* d+
3~~
~(d* d)2+964~
(1118)
344
D. Baum and A. Love, Supeifluidity and superconductivity in relativistic fermion systems
2. Superfluid neutron star matter
It is believed [13, 14] that densities in the cores of neutron stars are sufficient to favour a 3P2 paired
neutron superfluid rather than a ~ paired superfluid. The ~ pairing is just J~= 0~,and as no internal
symmetry indices are involved, the appropriate
3P relativistic Ginzburg—Landau free energy functional is
(1.118). We discuss 3P
in this section the case of 2 pairing (J” = 2).
The anisotropic 2 superfluid can have an important influence on the properties of a neutron star,
e.g. by affecting the rate of cooling by neutrino emission [15],
and the characteristic time for the transfer
of angular momentum from the interior to the surface of the neutron star through interactions
of
3P
electrons with vortex cores [161.It is therefore of interest to establish which of the possible 2 paired
superfluid phases is realised. The general form of the Ginzburg—Landau free energy may be written
down by using rotational symmetry. For different values of the parameters in this free energy, unitary
phases and two distinct non-unitary phases are possible (fig. 4). With the non-relativistic B.C.S. values
of the parameters, the unitary phase region of the phase diagram (region III) is selected. Strong
coupling corrections are probably too small to make any difference to this conclusion and go in the
wrong direction anyway [171. Relativistic
corrections
areand
however
much larger.
Fermi energy of
2~O.2
for neutrons,
20% corrections
to At
the aGinzburg—Landau
about
100MeV
we
will
have
(pF/m)
parameters may be expected. If such corrections were to go in the right direction they would be
adequate to move the superfluid into the nearby non-unitary region II of fig. 4. We now study the effect
of relativistic corrections [18] for a general value of PF.
r/ q
Regioni
>P/q
Region U
Unstable
Fig. 4. Phase diagram of the 3P
2 neutron superfluid. Region III corresponds to unitary phases, and regions I and II correspond to distinct
non-unitary phases. The non-relativistic limit is indicated by 0, and the ultra-relativistic limit is indicated by *.
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
The most general form of the gap matrix for J”
Li(n’)=Li~J~T1+4~S11+Li~
~
=
345
2 pairing consistent with Fermi statistics is
~1+Li~X11
(2.1)
where the covariants S, S, T, X, Y, Y, with definite values of L, are defined to be
T~1nn—~
(2.2)
S~=~(ny°y’+ ny°y’)+~n’yy°6~1
(2.3)
=
(2.4)
~(ny~’+ n~y’)—~n’~
y3..,
=
[n~nn~
— ~(n6~k+ n oak + n~
&~)]y°~k
5q)]y
=
[nn~n~—~(nO1k+
njO,k+nkr
=
~i(n’ A
(2.5)
(2.6)
and
Xq
y y~),n,
+
~i(n’A 7 y~n~.
(2.7)
The matrices ~
p = 1,. . , 6 are symmetric and traceless because they describe J = 2 pairing.
Detailed gap equations for the Ginzburg—Landau region may be derived (after much labour) from
(1.64). It is necessary to perform various angular integrals, and to carry out a projection of the J = 2
part of the right-hand side of the gap equation. Relevant formulae for these purposes are summarised in
appendices A and B. The projection of J = 2 is justified because admixtures of other J values in the gap
matrix are expected to be small, for the same reasons [1] as in the non-relativistic case. The gap
equations for the individual matrices 4 ~J~)are highly model dependent, involving the detailed form of
pairing force assumed. However, the gap equations for the possible order parameters may be expressed
in terms of the helicity amplitudes for neutron—neutron scattering in a form which is independent of the
specific pairing force. (This may be checked by employing the scalar and vector exchange forces
described in subsection 1.2.) Gap equations then arise in terms of coupled order parameters, d~8~
(9 =
1,2), with
.
d”~= p~Li~
+
p.(4(2)
+ ~4(3))_
rn (4(4) + ~Li(5))
(2.8)
and
dt2~=
—m(Li~2~
— ~4(3))
where d~°~
and ~
=
are 3
+
X3
p.(4(4) — ~4~5))
(2.9)
matrices. These gap equations are
Fe~~(ad~~0)
+ bD~)
(2.10)
where a, b are as in (1.65) and (1.66), and F°~
is the matrix of helicity amplitudes (in the notation of
Goldberger et al. [11])
F=8-~--( ~21
V
2~2).
/2PF
3f~2
f~
(2.11)
346
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
Also D~°~(0 = 1, 2) are 3 X 3 matrices cubic in d~°~:
= _7d(~*
tr(d~2~)2
+ 14d”~tr(d~)*d~2~)
+ 2d~)*tr(d°~)2
+ 4d~2~
tr(d(2)d~)*)
63p.2D°~
— 4d(2)* tr(d(t)d(2)) — 4d~2~
tr(d(2)*dU)) + 4d”~tr(d(t)* d°~)
— 20(d~2~)2
du)*
+
20d(2)* d~2~
d°~
+ 20d~2~
d(2)* d”~+ 8d~2~
d~)*d~2~
+ 8d”~du)* d~”
+ 16d(t)* (d”~)2
+
16d(2)* d~t~
d~2~
(2.12)
and
189p.2D~2~
= 62d~2~
tr(d(2)* d~2~)
— 23d(2)* tr(d~2~)2
+ 28d~2~
tr(dO)* d(t))
—
8d~°
tr(d~2~
d~)*)— 8d(t)* tr(d”~d~2~)
+ 8d(t) tr(d(t) d(2)*) — 14d(2)* tr(d~°)2
+ 32d(2)* (d~2~)2
—
2d~2~
d(2)* d~2~
+ 40d~)*d”~
d~2~
+ 40d”~d(t)* d~2~
— 32d~)*d~2~
d°~
—
40(d°~)2
d~2~’
+ 16d°~
d(2)* d”~.
(2.13)
The gap equations may be diagonalised using the (non-orthogonal) matrix
(xi
2~
\y,
2+ Vi
X
Y2’
1(1
2
+
z
+
z2~t12(1
\
V
+ Vi + Z
V~z
2Z__~
l+Vl+z21
(2.14)
where
z
=
2f~2/(f~2—f~t).
(2.15)
t
(2.16)
Thus,
=
diag(A”~,A~2~)
SFS
with
~
+
Z21
(2.17)
/2PF
and
A(2)~~[f2+f2_(f2_f2)V1+z21
/2PF
(2.18)
We write for the order parameters diagonalising the gap equations
et8~=SOc~d~
(2.19)
and we also write
E~°~
= S°’~D~.
(2.20)
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
347
The diagonalised equations are
t~°~
e~°~
= bE~°~
(2.21)
tt0’ = (T— T~°~)/T~:~
(2.22)
ln ~f3~°~~
= [~A~°)
dn/d]t.
(2.23)
where
and
The order parameter e~°~
with the higher critical temperature is the one which orders at the phase
transition. Then, in (2.21), in the gap equation for the relevant order parameter, the other order
parameter should be set to zero. Both possibilities, e°~
ordering or e~2~
ordering may be subsumed in a
single formula. We return later to the question of which of e°~
and e~2~
orders in the realistic case. Now
the gap equation for order parameter e~°~
is
te
= ..±!~
+
[6(x4+ x2y2 + 2y4) e * tr e2
6 (2x4 + 14x2y2 + 5y4) e tr e*e — 9(8x2y2 — y4) (e2 e*
+
e* e2)]
(2.24)
where
kt9~= 1, ~
for 9 = 1,2
(2.25)
and we have suppressed the index 0 on e, t, T~,k, z and y. To arrive at (2.24) we have used the identity
[19]for traceless 3 x 3 matrices e,
ee*e+e2e*+e*e2 =~e*tre2+etre*e
(2.26)
to eliminate ee*e. The Ginzburg—Landau free energy corresponding to the gap equation (2.24) is
= ~
t tr e*e
—
315p.~~
Itr e2~2+ q (tr e*e)2 + r tr(e*2e2)],
(2.27)
where the constant of proportionality, which is not determined by the gap equation, has been obtained
using (1.115). The index 0 has been suppressed, and
p=x4+x2y2—2y4
(2.28)
q
(2.29)
=
2x4 + 14x2y2 +
and
r = —3(8x2y2 — y4).
(2.30)
348
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
The quantities x9 and y~are as in (2.14).
To identify the order parameter_for the realistic case of neutron star matter we consider the
non-relativistic limit in which z—~2V6.Then, using (2.19), (2.14), (2.8) and (2.9) we find that
m
(j(3)_
4(5))
(2.31)
4(2))
(2.32)
and
m (j(4)_
2~
are respectively pure L = 3 and
Now
from (2.l)—(2.7)
in the
pure we
L =see
1 order
parameters.that
Thus,
thenon-relativistic
realistic 3P limit e(t) and e~
2 pairing is described by the order parameter ~
Accordingly we shall now restrict attention to the Ginzburg—Landau free energy for the order
parameter ~ For this case, the non-relativistic limit of (2.28), (2.29) and (2.30) gives
p=O,
(2.33)
r=—q
in agreement with Sauls and Serene [17], and the system is in region III of fig. 4, corresponding to a
unitary phase. In general, the criterion for region III is
4p+2~p~+r<0.
(2.34)
It is easy to check that this criterion is always satisfied by (2.28) and (2.30) for the physically allowed
values of
z
0~<z<2V6.
(2.35)
Thus, even after taking account of relativistic effects, the system is always in a unitary phase. This is
despite a striking variation in the Ginzburg—Landau parameters in going from the non-relativistic limit
of (2.33), to the ultra-relativistic limit (z -~0) where
r:q:p=3:5:—2
(2.36)
(see fig. 4).
The gradient terms in the Ginzburg—Landau free energy may be evaluated [20] using (1.95). They
add to (2.27) the terms
2+ 5y2)ake~3keu
~grad
=
~
[(2x
+
(8x2 + 6y2)~
1e~
a~eJk]
(2.37)
where we have suppressed the index 0 on e~°~
etc. 21
In ordering
arriving at
have projected
out the order
J=2
we(2.37),
have we
deleted
the “irrelevant”
part,
diagonalised
the
gap
equations,
and
for
e~
parameter e”~,and vice versa. The coefficients x, y are given by (2.14), and the constant c by (1.96).
D. Baum and A. Love, Superfluidity and superconductivity in
relativistic fermion systems
349
3. Superconducting electron systems
3.!. Electromagnetic field fluctuation effects
The effects which are peculiar to a superconductor, as opposed to a superfluid, arise because the
Cooper pair is coupled to the electromagnetic field. The precise form of the coupling is dictated by the
local gauge invariance of the Lagrangian (1.1). Under a gauge transformation the electron field e(x) is
transformed according to
e(x)—*exp[ieA(x)] e(x)
(3i)
where —e is the charge of the electron. Thus gauge invariance requires that the (bilocal) Li (x, y)
transforms according to
4(x, y)—* exp[—ie A (x) — iefl (y)] Li(x, y).
(3.2)
The local theory which has been developed in section 1 presupposes negligible variation of Li over
distances of the order of the pair size. In any case the Ginzburg—Landau theory developed in 1.6
required slow variation (1.84). Thus in terms of the centre-of-mass coordinate R and relative coordinate
r
R an ~(x +
y)
(3.3a)
ranx—y
(3.3b)
we have
4 (R, r)-4 exp[—2ieA
(R)] Li (R,
(3.4)
r),
and the same transformation law applies to each component of Li. This means that in the case of
ordinary 7 = 0~pairing the combination d, defined in (1.52), also obeys the transformation law
d(R,
r)—*exp[—2ieA(R)] d(R, r).
Under the same gauge transformation the vector potential
(3.5)
A(R)
transforms as
A(R)—*A(R)+VRA(R).
(3.6)
Thus gauge invariance requires that in the expression (1.118) for the free energy density ~ we must
replace the ordinary derivatives Vd by the covariant derivative
Dd=Vd+2ieAd.
(3.7)
In addition there are contributions to ~ from the (local) magnetic field
BVAA
(3.8)
350
D. Baum and A. Love, Superfluidity
and superconductivity in relativistic fermion systems
and from the normal phase. Thus
~+~-td~’d
~=
+32~~T)2(d*d)2
2 (Vd* —
~96 ~
2ieAd*) (Vd + 2ieAd) + ~-~-- B2 — p.~H2
(3.9)
.
(kBTC)
measures the free energy density relative to the energy density of the applied external magnetic field H.
~ is the contribution from the normal (i.e. non-superfluid) phase.
The thermodynamic properties of a superconductor in an external field are derived from the Gibbs
free energy density [21]
(3.10)
~an~-H.M,
where M is the average magnetization, defined by
B,a
0H+M.
(3.11)
Thus
2
=
3~+ ad*d + ~$(d*d)2 + y(Vd* — 2ieAd*). (Vd + 2ieAd)+ 2p.~(B — p.0H)
(3. l2a)
where
(3.l2b)
7~(3)
de 16(1rkBT~)2
-_dn
(3.12c)
~(3) Pf - PL Yd%(~.k~T~)2p.26/22$
=
3 12d
~
(~.
In the normal phase (n) d is zero and B = p.
0H. Thus
(3.13)
The superconducting phase (s) is characterized by the Meissner effect, in which the magnetic flux B is
excluded from the interior of the superconducting medium. In this phase A is zero, and in the interior of
the medium we may take d to be constant, independent of position. This constant value of d is found,
as in (1.71), by minimising (1.73). This gives
2=—a113
(3.14)
d1
when T
T~(and therefore a is negative). It follows that
~~(H)’
~n
a2I2I3+~p.oH2
(3.15)
D. Baum and A. Love, Superfluidity and
superconductivity in relativistic fermion systems
351
and therefore that
— ~~(H)
(3.16)
_a2/2f~+ ~L0H2.
=
At a fixed temperature, whether or not the system is in the normal or superconducting phase is
determined by whether ~ is greater or less than
The field H~bat which ‘the Gibbs energies are equal
is given by
~.
H~b
=—a(p.
(3.17)
2
0f3)”
so that
~~(H)
~n(H) = ~/.Lo(W —
H~b).
(3.18)
H is less than the critical field HCb the system is in the superconducting phase, and, in type I
superconductors, when H is larger than H~bthe system is in the normal phase. The transition between
these phases is a first order transition and the order parameter changes discontinuously from —a/f3 to
zero. It follows from (3.17) and (3.12) that the critical field in a relativistic system is given [221by
When
B~=
(3.19)
4p.pF(kBTCt)217 ~(3)
(in the units where Ii = c = = 1, used in section 1).
As the name implies, not all superconductors are type I. In type II superconductors (alloys) the
normal state is not restored as soon as the critical field B~is exceeded. Rather, there is an intermediate
(mixed) state in which the magnetic flux gradually penetrates the superconductor in vortices. This
intermediate state persists until the field reaches an upper (second) critical value B~
2beyond which the
normal phase is fully restored. As the applied field approaches B~2the supercurrent density approaches
zero continuously so that the gap d also approaches zero continuously. Thus in this case the transition
to the normal phase is second order.
The second critical field B~2 may be calculated by dropping the quartic terms in ‘~&, since d
approaches zero as B approaches B~2. Then the minimum of ~ with respect to variation of d is
achieved by balancing spatial variation of d (bending energy) against the magnetic energy of the texture
caused by its minimal coupling. If we take B in the z-direction we may take
A=Bx9
(3.20)
and then d needs only x-dependence. The (second-order) phase transition will then occur when the
minimum of the d-dependent part
of ~
2B2x2d*dl
(3.21)
~(d)Ead*d+y[dl*dl+4e
is zero. Since the term in square brackets is proportional to the Hamiltonian of a simple harmonic
oscillator, it follows that the required field satisfies
a + 2y(~hw) a + 2yeB~
2= 0.
(3.22)
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
352
Then using (3.12) and (3.19) we find [22]
B~- 4y2e2
/3
2 K2
—
7 ~(3)
(e2/4ir)
3
//~7~\2(p.\5
~
p. ) ~)
361T
( 3 .2~)
3
so that
K
=
95.3
(—~—-~)(p-)
kT
5/2
(3.24)
.
If B~
2is larger than B~(K
ultra-relativistic limit p.
>
PF
1IV~)the superconductor is type II, whereas if K
and we have a type I superconductor provided
kBTC/p. <7.4 X i0~.
< iiV~
it is type I. In the
(3.25)
The interpretation of B~2in a type I superconductor (in which B~2< B~)is that it gives the critical
supercooling field. As the applied field is reduced below B~the normal phase ceases to be a global
minimum of ~, although it remains a local minimum. The above calculation shows that B~2is the lowest
field in which ~ has a minimum for small non-zero d. Thus it is the lowest field in which the normal
phase is stable, and at this field there is a second order transition to the superconducting phase.
If the applied magnetic field is zero, it is clear from (3.17), or directly from (3.9), that the phase
transition to the superconducting state occurs when t is zero. Thus when T is reduced below T. there is
a second order phase transition as the minimum of ~ at d = 0 ceases to be a global (or local) minimum.
The above conclusions follow from the mean field theory in which we take A to be zero since there is no
applied field. However although the mean value of A is zero there are still local fluctuations in the
electromagnetic field. Since these fluctuations are coupled to d, they can modify the form of the free
energy, and we shall see that they can have the effect of making the phase transition first order.
Before proceeding to calculate the effect of these fluctuations we must determine the circumstances
in which it is permissible to retain fluctuations in A while neglecting those in d. Any fluctuation is
characterized by a coherence length ~ and the fluctuation is significant only if
kBT.
(~— ~
(3.26)
For temperatures close to T~this gives
2)t”3
(3.27)
(2/3kBTC/a
using (3.15). Fluctuations in d are only significant when the bending terms of S~are comparable with the
bulk terms and this occurs in a temperature range given by
adt—7/~.
(3.28)
So
ad
4f32(kT)2/ 3
7
(3.29)
D. Baum and A. Love,
Superfluidity and superconductivity in relativistic fermion systems
353
Fluctuations in A are clearly characterised by a scale ~A given by
4e2ld~2y= 4e2yIaAj/13
=
(3.30)
using (3.14). Thus the temperature range in which these fluctuations are significant is given by
(3.31)
2~
8T~)
The relative magnitude of the two ranges is
aA
(4e~y)3(2k
=
ad/a,.
(3.32)
(V2K)6
using (3.23).
For a good type I superconductor
(3.33)
V~K~l,
and it follows that we may ignore the effects of fluctuations in d and consider only those in A [23].Thus
we may take d to be constant and the free energy then has the form
F= F
1(d) + F2[d, A]
(3.34a)
where
J
3x [~+ a d~2+~jd~4]
(3.34b)
F~(d)=’ d
F
2[d,
Al =
J d3x d3y A.(x) M,1(x,
(3.34c)
y) A1(y)
with
2IdI2
4, + ~
— O~V~]
O(x —y).
M,~(x,y)=~[8’ye
Since A is not an external field we may integrate the partition function
(3.34d)
Z
over the configurations of
A(x), and thereby define an effective free energy Fen(d):
Z
=
exp{—F~~(d)/k~T}
=
J ~2’Aexp{—Fl(d)IkBT} exp{—F2[d, A1/kBT}.
The functional integral is easily performed, as F2 is quadratic in
Fe~ F1 +
J d3x J
~-~s
kBTC ln(k2 +
8y2e2 d~2).
(3.35)
A,
and we obtain (in Landau gauge)
(3.36)
354
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
The required integration is effected by differentiation with respect to Id~ and we find that the
d-dependent part of the effective free energy density is given by
(3.37)
~effaIdI~jdI+~/3IdI
where
-
a =
ldnT-T,,
-
(3.38)
———
2d
T~
with T~different from T~because of cut-off dependent contributions from the fluctuations.
(3.12c) and
(8ye2)312.
=
/3
is given in
(3.39)
The cubic terms in ~eff, which are induced by the fluctuations, have the effect of giving ~eff a
maximum and minimum away from the origin (IdI 0), as well as a minimum at the origin (Idt = 0). A
first order phase transition occurs [23] when the minimum with non-zero Idl is degenerate with_ the
normal phase (IdI = 0). This occurs at a temperature T~which is higher than the temperature T. at
which a second order transition would otherwise have occurred. T~is given by the solution of
(3.40)
= ~2/~3
and at that transition we find
dIT~=~/f3.
(3.41)
The relationship between T~and T~is easily found from (3.37) to be [22]
=I
+
(3.42)
114
where
2~
(3.43)
T=O
Using the previously given expressions for the quantities involved this gives
—
-
243
ir5
(k
2/p.~7
8T~\
784 ~(3)2 (e2/4~ ~ p.
)
~PF)
344
(.
)
In the non-relativistic limit is very large and the temperature I’,, differs very little from T~.However in
the ultra-relativistic_limit (i~=PF) we have ~ I when (k~T~/p.)~
10~,and T. may well differ
substantially from T~.Thus we expect the fluctuation effects to be more important in the ultra-
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
355
relativistic regime than in the non-relativistic one. This expectation is borne out by comparisons of
various quantities at the first order transition with the natural scales of the system at T = 0.
For example we may compare the order parameter at the transition, given by (3.41), with the order
parameter obtained by minimising ~eff at T = 0. This too is fixed by and we find
A
I A
— 411
UT/14T03t1TksV9E)
(1
~Z \t/21—t
J
When is less than/of order unity, as it is in the ultra-relativistic case, the ratio is also of order unity,
indicating that in this case the phase transition is strongly first order. In_the non-relativistic case when
is large, the ratio is small. Similarly, we may compare the latent heat (Tc AS) released in the first order
phase transition with the T = 0 condensation energy, with the result
32
T AS
~eø(T0)~
16
128
~
32
3/2 -t
]
.
(3.46)
Again we find in the ultra-relativistic limit that the phase transition is strongly first order.
3.2. Electroweak effects in superconductors [24]
In addition to their electromagnetic interactions, the electrons in a Cooper pair also participate in
weak interactions. As their name implies, weak effects are typically small compared with other
interactions, but because parity is not conserved there are some novel features which are intrinsically
interesting and which might assist in their detection. In what follows we shall assume that the
electroweak interactions are correctly given by the standard model [25].
One such feature is that the gap matrix Li is strictly speaking not an eigenstate of parity. In the case of
an ordinary superconductor we should therefore expect that 4 is predominantly JP = 0~with a small
admixture of 7 = 0 which arises from the parity-violating weak interaction between the electrons of
the pair. Thus instead of (1.50) we have that
Li(n, K)= Li175+ 42pj ~y7O75+Li37oy5+Li4J+44~ •77o~46n y
(3.47)
is the most general J = 0 gap matrix consistent with Fermi statistics. In (3.47) n an ~ is the direction of
the relative momentum of the pair, while K is their total centre-of-mass momentum. Lii (1 = 1,. , 6)
are in general functions of K (but not of n). In the non-relativistic limit (3.47) reduces to
. .
(3.48)
Li=Li1—43—(Li5+Li6)noshowing that 4~,3characterise s-wave pairing (P = +) while Lis.6 describe p-wave pairing (P
=
—). For an
ordinary superconductor, therefore, we have
4t,Li3~’Li5,4o,
(3.49)
since 45,6 are non-zero only because of weak interactions. The general gap equation has been derived in
(1.95). The physical situation of a dominant phonon exchange, with a small parity-violating interaction
(arising from Z-boson exchange between the electrons) may be modelled by having two terms on the
356
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
right-hand side of (1.95). The (scalar) phonon contribution is modelled by making the electron—phonon
interaction of the form
gEA
=
gQI
(3.50)
and the propagator of the form
DAB(n,n’)= V(nn’),
(3.51)
while the weak contribution is modelled by an electron Z boson vertex of the form [251
g~A= y~(g~
+ g.~y
5)
(3.52a)
with
2 0~
— ~).
=
(3.52b)
2 c~~ (2 sin
2 c~0~
(3.52c)
where O~is the weak mixing angle and g = e/sin 0~is the semi-weak coupling in the standard model.
The propagator for Z-exchange has the form
(3.53)
DAB(n, n’) = gasX(n ‘n’).
As in section 2, the individual gap equations depend sensitively on the details of the model, although
the form of the order parameters involved is model-independent. The actual order parameters involved
in the present case are
d~Li
1—~Li2—~-Li3
/2
/2
(3.54a)
d(2Li5_~Li4+~~46,
(3.54b)
p.
where m is the electron mass, PF is the Fermi momentum and p. is the (relativistic) chemical potential.
We find that the gap equations for these quantities are diagonalised by combinations, analogous to
(2.19),
4~V~AP~
e~°=
~(1)
2~
= d~2~
+~
e~
gç
/2
X V
2~
V~
0 1~d~
~
g~ p. V
d”~
1-Vo
(3.55a)
(3.55b)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
357
where
J
V1an~ dzP1(z)’~’(z)
Xoan~J
(i=0,1)
(3.56a)
dz~(z)
(3.56b)
analogously to (1.14). Clearly e(t~is predominantly J’°= 0~,since2~
is justGFm~,
the order
isd(t)
of order
whereparameter
GF is the
obtained
in
(1.52)
and
the
admixture
of
the
J”
=
0
combination
d~
(Fermi) weak coupling constant and m~is the phonon mass. Similarly e~2~
is predominantly J” = 0. In
an ordinary superconductor, therefore, we know that the combination e°~
orders before ~ Thus in the
gap equations for the components zi, (i = 1~. , 6) we may set the “irrelevant” order parameter to zero
as in section 2. The result is that the gap matrix Li is given by [26]
. .
4 =[(p.2+m2)V_p2V]_ttp.V7+pVn.yyymVyy
2~
(V~— Vo)+p~V
X
— 4gvgA
0+/2PFV1 n ~7yo+ lflpFVt ~ y]}~e”
0 V0)~
.
(3.57)
1’~. In the nonwhere
e°~
is obtained
by from
minimizing
the free energy (1.118) with d replaced by e
relativistic
limit,
it follows
(3.48) that
A
11
L
4gVgA
XOVS %J\VFfl
2
~J(%J
g~
~oi~~1
ti1 (5)
1e
J
YQ)
m. Thus the parity-violating component of the order
since in the
non-relativistic
limit PF
parameter
is small
not only because
it is a weak effect, but also because it is of order vF/c and vanishes
in the static limit. For this reason it would be extremely difficult to detect, even if we had been able to
devise tests which might be sensitive to its presence.
However there are other manifestations of the electroweak interactions which look more promising.
These derive from the observation that superconductivity spontaneously breaks a local gauge invariance. In the context of QED this has the effect of giving the gauge field (the electromagnetic field) a
non-zero mass, and this leads to the Meissner effect. It is now generally believed that electromagnetism
is only one aspect of a unified electroweak theory [25]based on the gauge group SU(2) x U(1). In vacuo
this symmetry is spontaneously broken down to the U(1) gauge invariance of electromagnetism. In a
superconductor the gauge symmetry is completely destroyed and in general we should expect the mass
eigenstates to be mixtures of the states which were the vacuum mass eigenstates [27].To see how this
works we need to replace the gradient Ve”~in (1.118) by the SU(2) X U(1) covariant derivative De~°,
analogous to the electromagnetic covariant derivative given in (3.7). This requires us to determine the
properties of the various bilinear covariants ëdI~e associated with the gap matrix (3.47) under
SU(2) x U(1) gauge transformations, just as we did at the beginning of 3.1. These in turn are fixed by
the transformation properties of the electron field e(x), as in (3.1). In the standard electroweak theory
4
p.
358
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
the left-chiral component eL(x) of the electron field e(x), given by
eL=
(3.59)
2(I—y5)e
is the bottom component of the doublet
EL~(’~’~)
where
~eL
(3.60)
is the electronic neutrino field. Under a general gauge transformation [25]
EL(x)—* exp[—i~gr A (x) + i ~g’JA (x)1 EL(x)
.
(3.61)
where A(x) and A(x) parametrise respectively the SU(2) and U(l) transformations. Since the r’ are
(Pauli) matrices, eL is in general transformed into a mixture of electron and neutrino states. Since there
is no neutrino condensate in a superconductor we should restrict ourselves to the subgroup of
transformations in which eL is transformed only into eL. That is to say, we impose A t = A2 = 0. The right
chiral component
eR =
~(I + y5)e
(3.62)
transforms according to
eR(x)—* exp[ig’ A (x)] eR(x).
Under this restricted group of transformations the SU(2) gauge field
3-*w3+VA3,
w
and the U(1) gauge field B transforms according to
B—*B+VA.
(3.63)
(3.64)
(3.65)
Thus in this case the covariant derivative of the electron field e(x) is
De an Ve — i ~ (gW3 + g’B)eL — igBek
=Ve—~iPe+~iQy
5e
(3.66a)
where
3+ 3g’B)
p
an ~(gW
=
e cosec 20~[(1—4 sin2 0~)Z + 2 sin 20~A]
Q~(gW3—g’B)=ecosec20~Z
(3.66b)
(3.66c)
D. Baulin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
359
with Z and A the neutral vector boson mass eigenstates in vacuum:
B
cos O~W3 — sin O~
sin 9~W3 + cos O~
B
(3.67a)
e=gsin9~=g’cos0~.
(3.68)
Z
=
A=
(3.67b)
and
It follows from (3.66a) that the gauge covariant derivative of the local bilinear ëdle (with I an arbitrary
y-matrix) is
D(ëdle) = (V — iP) (ëd[~e)
+
~iQec{F,y
5}e.
(3.69)
For the case in hand we are concerned with the gap matrix (3.47) which involves the y-matrices
(3.70a)
=
12= n•yy~y5
(3.70b)
13
=
(3.70c)
14= 1
(3.70d)
15=nyy0
(3.70e)
F6=n~y.
(3.70f)
Then (3.69) shows that
D(ë~F1e) (V —
iP) (ë~F1e)+
iQ(ej’4e)
(3.71)
from which it follows that
DLi1(V+iP)Li1—iQLi4,
(3.72a)
and similarly that
DLi2=(V+iP)zi2—iQLi5
(3.72b)
DLi3 = (V + iP) 43
(3.72c)
D44= (V+ iP)Li4— iQLii
(3.72d)
DLi5=(V+iP)45—iQLi2
(3.72e)
D46=(V+iP)Li6.
(3.72f)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
360
Using (3.54) and (3.55) we find
=
(V+ iP) e~t~-
~
4gVgAp~
1
(3.73)
V1- V1, (Li2-~it)].
44,5,t.2 in terms of
e(t) with the result that
Since only e°~
orders we may use (3.57) to express
De”~=
[v+ iP + 8g~A(p.2 + m~V
1—p~V~
iQ] eW.
(3.74)
In the non-relativistic limit this gives
De~’~—*
~
(3.75)
~
The contribution from the combination
is weak since
Q, defined in (3.66) is clearly model-dependent,
but in any case
(3.76)
O(G~m~).
g,~ V0
For this reason we shall henceforth ignore it. When we replace in (1.118) by De”~,the terms quadratic in
P generate mass terms for the vector bosons additional to those generated by the Higgs doublet. In fact
the total mass Lagrangian is given by [24]
2cosec2 20~ v2Z~Z ~48 ~
=
~e
(k
2e~’~I2[(1-4 sin2 0~)Z +2 sin 20~A]2
8T~)
}
(3.77)
where
=
GFV2.
Clearly the mass eigenstates, which diagonalise
order in
Pan~4(kT)2 GFV2~e~t~2,
(3.78)
~M,
are superpositions of Z and A. Expanding to lowest
(3.79)
we find that the eigenstates are
Z=Z+xA
(3.80)
with
m2(.2)=
m~[1—(l—4sin20~)2p],
(3.81)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
361
and
A=A—xZ
(3.82)
[1—(1 —4sin2 0~)2p]
m2(A)= m~4sin220~p
(3.83)
with
where
m~=
4lTa
(GF V~ sin2 20~)t
(3.84)
is the mass eigenvalue in vacuo, and
x
=
2 sin 20~(1—4 sin2 0~)p [I—p (32 sin4 0~
—24 sin2 0..,. + 1)].
(3.85)
As anticipated, both eigenstates have a non-zero mass, since the gauge symmetry is completely broken.
The detection of this gauge boson mixing in a superconductor will derive from interactions of the
bosons with the electrons and nucleons. In terms of the mass eigenstates Z and A, the neutral current
Lagrangian is
=
[ej —2 ex cosec 20~(T3 — sin2 t/~j)] A
.
+
[exj
+ 2e cosec 20~(T3 — sin2 0~j)l Z
(3.86)
where
(3.87)
j=—eye+pyp
is the electromagnetic current, and
T3 =
—
~ey (1 — y~)e + ~j3y(1
—
~y5)p —
~ñy (1
—
y~)n
(3.88)
is the weak isovector current. (We have dropped the neutrino contribution, and others which are
irrelevant to our system.) Since T3 has vector and axial vector pieces, it follows that the interactions of
both mass eigenstates are parity-violating. One effect of these interactions is to modify the form of the
parity-violating interactions between the electrons in the pair, with the result that g.,., and g,~... differ
slightly from the values given in (3.52). However, as already remarked, we do not know how the
presence of a parity-violating component of the order parameter might be detected, let alone any small
deviations from its naively expected form.
On the other hand Vainshtein and Khriplovich [281have suggested a method whereby the parityviolating interactions between the electrons and the nucleons might be detected. The detection of any
parity-violating effect typically relies upon an observable pseudoscalar, and without using triple
correlations this necessitates the use of at least one spin variable. Vainshtein and Khriplovich (VK) have
proposed the pseudoscalar °N p~,where ~N is a nucleon spin and Pc is an electron momentum. Thus
the technique requires a superconductor with polarised nuclei. The pseudoscalar in question arises from
the coupling of the axial component of the nucleon current to the vector component of the electron
362
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
current. So we have two contributions, one from Z exchange and the other from A exchange. In the
non-relativistic limit the Z contribution to the effective Hamiltonian is
~F
=
/3~)
~
{Pe,
(3.89)
6(r)}
2V2me
where
2 0~)[1—p(32sin4 0~—24sin20.....+ 1)]
(3.90)
—~g~,(l
—4sin
with g~ 1.25 the axial vector constant for nucleon decay, and T3N = + 1 or —1 if the nucleon N = p or
n. The contribution from the (longer-range) A-exchange is a Yukawa interaction
~(A) =
/3
2(A) exp[-m(A) ri}
(3.91)
1(A) UN T3N~jPe~m
2V2m~
4irr
/3
1(Z)=
with m(A) given in (3.83) and
f3
(3.92)
1(A) = —f3~(2).
The contribution to the effective Hamiltonian from Z-exchange
is
obtained from (3.89) by averaging
over the nucleons in the nucleus and over the electrons in the Cooper pair:
~eff(Z) =
(?-F
nKr/3t(Z) {p, ~(r)}
(3.93)
2V2m~
where ~(r) is the nuclear polarisation vector, n is the density of nuclei, the factor K allows for
differences between the electron current in the vicinity of the nucleus and its average value in the
crystal, and
~
~T3NIUNI)~
(3.94)
with the sum running over all nucleons in the nucleus. The form of the effective interaction is similar to
the first order electromagnetic interaction
~em =
{p, 2eA(r)}
(3.95)
which derives from the usual minimal coupling (3.7) of the pair. In our case we have already modified
the minimal coupling to include interactions with the electroweak gauge fields as given in (3.75). So the
addition of (3.93) has to first order the effect of further modifying the covariant derivative
De”~—~
[D_
~?‘~ nKo’13t(Z) ~(r)] e”~.
2V2m~
(3.96)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
363
This further modification of the minimal coupling causes deviations from the quantization of flux [281
which would otherwise occur when a superconducting loop is placed in an external magnetic field. To
see how this arises recall that the source current j, where
VnB=j,
(3.97)
is zero in the interior of a superconductor, because B vanishes by virtue of the Meissner effect. Now
j=3~I3A
(3.98)
and the dependence of ~ upon the vector potential is via the covariant derivatives associated with the
gradient free energy density
~grad
=
yIDe”12.
(3.99)
Integrating j around a closed contour inside the superconducting loop thus gives
0
=
j ‘dl =
2ie’y
dI [(De~))*e°~
— e~)*De”~].
.
(3.100)
We may assume the magnitude e°iof the order parameter is constant inside the superconducting loop,
since its value is fixed by the bulk temperature. However the phase ~(r) of e°~
will in general have
spatial dependence. Thus using the covariant derivative (3.74) we have that
~ dI’(P+AQ)=—~dIVx=2irm
(3.101)
where m=O,l,2,...,and
p.2X
2+m2)Vo—p~V
0
(3.102)
g~ (p.
1
Stokes’ theorem converts the left-hand side into a surface integral and since the Z-field propagates only
A
8gVgA
very short distances from the magnets producing the external field B we have that
VAP=2eB
(3.103a)
VAQ=0.
(3.103b)
Then (3.101) yields the flux quantization condition
~P”rJB.dS=mcoo
(3.104a)
where
ç00=IT/e=h/2e
(3.1041,)
364
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
is the fluxoid. When the modified covariant derivative (3.96) is used in (3.100) instead of (3.74) the
nuclear polarization ~(r)
causes deviations from flux quantization
—
m~o=
gF
4 v 2 eme
nK~/3t(Z)~
~ d!.
(3.105)
This is almost identical with the result of Vainshtein and Khriplovich essentially because the extra terms
in the covariant derivative (3.74), over and above the usual covariant derivative (3.7), do not propagate.
The only difference (so far) is that f31(Z), given in (3.90), differs by terms of order p from that of VK,
who used only Z-exchange. However we must also include the contribution (3.91) from A-exchange. In
general an exact
moreapproximate
difficult than
Z-exchange
of the
Yukawa
2(A)treatment
(r2) 1 weis may
the for
Yukawa
functionbecause
by a delta
function,
andcoupling.
because
However
if
m
of (3.92) the contribution from A-exchange will precisely cancel that from Z-exchange.
Now when T = T~,the order parameter e°~
vanishes so p and m(A) also vanish, using (3.79), (3.83). It
follows that when T is just below T~essentially the full VK effect should be seen with small
modifications of order p. However, deep inside the superconducting phase (when t — I) the contribution from A-exchange is maximized and there is the possibility of a partial or complete restoration
of flux quantization if m2(A)(r2) ~ 1. We estimate that typically
~‘
(r)
-~
10~cm,
(3.106)
whereas for a non-relativistic superconductor, using (3.83), (3.14), (3.12) we find
m(A)nr~—106 cmt .
(3.107)
Thus in this case the contribution from A-exchange is at most a few percent of the Z-exchange, and
essentially the full VK effect will be seen at all temperatures~the only effect of gauge boson mixing is
small deviations of order p. However for an ultra-relativistic superconductor we find
m(A)ur~109(p.Ime)cmt
(3.108)
with p. ~ me. So in this case we predict a full restoration of flux quantization deep inside the
superconducting phase, because of the complete cancellation between Z and A exchange.
3.3. p-wave superconductivity [261
The observation of the VK effect and its quenching deep inside the superconducting phase requires a
superconductor with the (possibly unlikely) concurrence of two properties. In the first place, since it is
required that the material has a sizeable nuclear polarization, the superconductivity must withstand the
magnetic field necessary to produce the polarization. This latter field (at 102 K) is of order 300 kG
while the (zero temperature) critical field is typically of order 3 G. It may be possible to surmount this
obstacle by the use of Chevrel superconductors [29]which do have critical fields of order 300 kG. In the
second place, to obtain a sizeable quenching an ultra-relativistic superconductor is required, as we saw
in the previous section. Actually the effect of ultra-relativistic corrections is to increase B~,as is clear
from (3.19), sO it might be that the concurrence required is not too unlikely. Nevertheless we know of
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
365
no candidate materials at the present and the question arises as to whether other systems might not be
more amenable to the detection of parity-violation. One possibility, which is interesting in its own right,
is to use p-wave superconductors.
We have already encountered one form of p-wave superconductivity (in passing) in section 3.2. The
combination ~ defined in (3.55b) is predominantly 7 = 0. In a p-wave superconductor this
combination might order before e~t),in which case the gap matrix Li is given by
+mV
Li =[(p.2+m2)V1—p~Vo]1 jPFVo1+/2~”ln~Y7o
1ny+
4~V~APF2~[(
g~p.(
2~
o)
Vo)p.y5
(3.109)
2~
+pFVIfl~77075
mVoyoys]} p.e~
where e~2~
is obtained by minimizing (1.118) with d replaced by ~
4-4
[_n o.+
In the non-relativistic limit this gives
e~
(3.110)
so that, as before, the parity-violation is weak and of order vF/c, and we know of no experiment which
might be sensitive to its existence. Proceeding as in section 3.2 we find that the covariant derivative for
this form of p-wave superconductivity is given by
De~2~=
[v+iP+ g~[(p.2+m2) V
2VI
(3.111)
iQ]e(2)
1p
which gives
De’2~~(V+
iP) e~21
(3.112)
in the non-relativistic limit. Thus in this J” = 0 dominant phase the Z—y mixing is precisely the same as
that found in the s-wave case with the results given in (3.80)—(3.83).
However, the above form is not the most general, nor even the most likely to arise, as is well-known
from studies of superfluid 3He. In general a p-wave spin triplet pair will have a gap of the form
Li
=
(3.113)
d,~o~’nt.
So neglecting the parity violation the above form of J”
=
0 pairing is described by
(3.114)
~
Our experience of 3He suggests that we should consider the following special cases:
(i) B-phase
d~,= d e~xR,~m
where d and x are real and RNi is a rotation matrix. (This includes (3.114) as a special case.)
(3.115)
366
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
(ii) Planar phase
d~1= d e~cIkw”
where d and x are real and
(iii) A-phase
(3. 116)
w is a real unit vector.
d,~1= df3,..
(ati + ia21)
(3.117)
where d is real, J3 is a real unit vector, and a~,a2 are real orthonormal vectors.
(iv) A1-phase
d,~,= d~(f311..+ i/32~)(at1 + ia21)
(3.118)
where d is real, ~ fl2 are real orthonormal vectors, and so are a1, a2.
In these more general cases the derivative terms in the free energy have a more complicated form
than that given in (1.118), which describes only the jP = 0 case. It is easy to see [30] that the gradient
free energy has the general structure.
2 +~KTlVA dMI2}
~grad
where d,.
,~
is
{~KLIV~
dM1
(3.119)
the (complex) vector with components dMj, and in weak coupling approximation
K —1KL~80~.4m(kT)2
—
3120
(i.e. KT is ~ times the coefficient of jVdj2 in (1.118)). As before, when we replace the derivative V by the
covarient derivative D given in (3.112), we generate gauge boson mass terms additional to those
generated by the Higgs doublet in the standard model. In this case however the additional mass terms
are anisotropic when KL is different from KT. In lowest order the mass eigenstates all have the form
Z~,= Za + XaA,,
= A,,, — XaZa}
A,,,
no summation
(3.121)
and the corresponding eigenvalues have the form
m2(.~,,)=
m~{1+ K,, V2 GFd2(I —4 sin2 0~)2}
(3.l22a)
m2(A,,,)
4e2K,,d2
(3. l22b)
=
with K,, in general anisotropic, and we find that in all cases
x,,
=
K,, \/2GFd22sin2 0~(1—4sin20~).
(3.123)
Of the four phases defined in (3.115) to (3.118) only the B-phase has no anisotropy, since
RMIRMJ
=
41.
(3.124)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
367
In this case the mixing and the mass eigenvalues are given by
(3.125)
K~= KL+2KT.
As the name implies, the planar phase_has a planar symmetry (in the plane perpendicular to w). In
this plane the mass eigenstates; denoted Z~f0,A~are given by
K~P~
= KL+ KT.
(3.126)
However in the direction parallel to w the masses and mixing are given by
Kr~ 2KT.
(3.127)
=
As expected, the masses and the mixing are isotropic when KL = KT.
In the remaining two phases the anisotropy is quite independent of the vectors ~ 132 or 13. In both of
these cases there is also a symmetry in the plane defined by at and a
2 (perpendicular to at n a2). In this
plane
K~±”~~
= K~±”>
= ~(KL+ KT),
(3.128)
while in the direction parallel to a1 A a2 we find
K~=Kr~=KT.
(3.129)
Again there is isotropy when KL = KT, as expected.
One way to look for these anisotropic effects would be to look for deviation from the magnetic flux
quantization such as was discussed in the s-wave case. The more complicated form of the gradient free
energy density (3.119) yields a more complicated form for the current j when V is replaced by
D = V + iP. Then the analogue of (3.100) is
dl j(KL— KT) ~ [(D
.dM)*
dM
— d~(D dM)] —
KT~
d~DdM}
=
0.
(3.130)
For the B-phase order parameter defined in (3.115), this gives flux quantization but, as we have already
noted, the B-phase has no anisotropy. In the remaining phases the anisotropy generally destroys the flux
quantization. For example, in the planar phase (3.116) the above equation yields
CP— mço0= ~+~T~(VX+2eA).ww
~d1.
(3.131)
In this case the flux will be quantized in the special cases that the loop is perpendicular to a uniform w,
or if we can prepare a loop with w parallel to dI everywhere. In the A and A1 phases flux quantization
can also be achieved in the special case that the spatial variation of the order parameter is constrained
to a variation of its phase, i.e.
t’~d~
(3.132)
dM
=
e
368
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
with d~constant. However all of these special cases will need to be prepared with considerable
accuracy if the deviations from flux quantization due to electroweak effects are to be discernible.
4. Pairing in quark matter
4.1. The order parameterfor superfluid quark matter
The relativistic techniques developed in the preceding sections have been useful in exploring the
consequences of a theory such as electroweak theory which is formulated relativistically. The superconducting systems in which this theory has been applied are either non-relativistic or marginally
relativistic. However, the quark matter which is the subject of this section is certainly relativistic and
possibly ultra-relativistic. Thus the techniques we have described are indispensable in its treatment.
We have already encountered superfluid neutron star matter in section 2. The density of the neutron
matter is of the order of i0’~g cm3, which corresponds to a distance between neutrons of about 3
fermis. In the interiors of large neutron stars it is quite conceivable that the inter-neutron distance
approaches I fermi, in which case the neutrons are overlapping. At such densities clearly the quarks
inside different neutrons can “see” each other and we must view the system as a quark fluid rather than
a neutron fluid [31].
This raises the question of whether the quarks might not form a superfluid [32] just
as the neutrons do at sufficiently low temperature. Since a neutron contains two d quarks and one u
quark, one might naively suppose that the quark fluid had twice as many d quarks as u quarks. However
this is not so [33].
The numbers tend to equalize and in addition there are a (different large) number of s
quarks. Since the Fermi energies of the I = 1/2 quarks (u and d) are equal we may entertain the
possibilities of uu, ud and dd pairs, and of ss pairs, but not of us or ds pairs. At the densities under
discussion the Fermi momentum is of order PF 400 MeV/c. This necessitates a relativistic treatment of
s quarks no matter whether the current quark mass (150 MeV) or the constituent quark mass (540 MeV)
is used. For the u and d quarks a relativistic treatment is necessary if constituent masses (m,, md
300 MeV) are used, and an ultra-relativistic treatment can be used if current quark masses (m~
50 MeV, md 70 MeV) are appropriate. The precise form of the pairing is determined by the channel
having the most attractive quark—quark scattering amplitude. The interaction between quarks is
believed to be described by QCD, in which the colour of any quark is coupled to a massless gluon field,
independently of the quark flavour (u, d, s,. . .). Since QCD is asymptotically free [34],
the running fine
structure constant as(q2) decreases as q2 increases, and for quarks separated by one fermi or less the
scattering amplitudes may be computed reliably using only the single gluon exchange contributions
shown in fig. 1.
The dependence of the scattering amplitudes on flavour and colour is of course trivial, and only the
dependence upon the angular momentum J requires detailed investigation. For pairing of s quarks
there is clearly only one (symmetric) flavour channel. For the I = 1/2 quarks (u, d) the amplitudes may
be decomposed into flavour symmetric channels having I = 1 (uu, ud + du, dd) or a flavour antisymmetric channel, having I = 0 (ud du). The colour labels of the quarks enter the amplitudes via the 3 x 3
colour matrices t,, (a = 1,. . . ,8) which appear at each interaction vertex in fig. 1. Since
—
(ta)ki (ta)ij
=
~,(OkI4i
+ OkjOli) — ~(8kIOIl
8kJ3Ii)
(4.1)
it is apparent that amplitude may be diagonalised using the colour symmetric channel (6) and the colour
369
D. Bamlin and A. Love, Superfluidity and superconductivity in relativistic fermion Systems
antisymmetric channel (~),and that these are associated with the pre-factors given in (1.20). The
helicity amplitudes needed have already been written down in (1.18), and the above discussion shows
that for the ss and I = 1 flavour channels the upper sign should be used in the colour ~ channel and the
lower sign for the colour 3 channel. For the remaining I = 0 (ud du) channel the upper sign should be
used in the ~ channel and the lower sign for the 6 channel. To proceed further we require further
information on the quantities VjE.M defined in (1.19) which in turn requires knowledge of the propagator
functions Dm.M, defined in (1.17). Since the precise form of the screening of the “electric” colour force
by the medium is not known, we model it by giving the electric components D°°
of the gluon propagator
1E~So
a mass (
—
D°°=(q2+fl~1
(4.2)
where q is the momentum transfer. Since q
DE(COS
0)
(1
=
— cos 0
+
=
p’
—
P2
and
I~iI
=
AE)t
P21
PF,
this gives from (1.17)
(4.3)
where
A~=fl~/2p~.
(4.4)
Similarly we take
DM (cos 0)
=
(1— cos 0
+ AMY1
(4.5)
to model the non-perturbative effects which we assume cut off the infrared divergence of the
“magnetic” propagator. It then follows from (1.19) using Rodrigues formula, that
(4.6a)
V~>0
V~>V~+
1
(4.6b)
with p = E, M. We can also show, using the recurrence relation
z(2J+ 1)Pj(z)= (J+ 1)P~+1(z)+JP~.1(z)
(4.7)
that all of the expressions between {} in the helicity amplitudes (1.18) are positive. It follows that only
the colour 3 channel can ever be attractive, since from (1.20) —y is then positive. Restricting ourselves
to this (antisymmetric) colour channel we find then that the (symmetric) flavour channels (ss and I = 1)
have non-zero amplitudes f~(J even) and fog, fit, f i’~,f~(J odd), while the antisymmetric I = 0 channel
(ud du) has non-zero amplitudes f~(J odd) and fo’, ft’i, fi~,f~(J even). To find precisely which
pairing is preferred in each channel we must identify which helicity amplitude is largest. The three
amplitudes fir, ft’2, f~2are defined [11] in terms of a pair of spin triplet states
—
lJ,1)anlJ,)+~IJ,_~,_~)
r,,~_ 1
,.~
i~
1
J~L/ç7~ J,~/+~7~
(4.8a)
r11\
J,
2,2/,
.
370
D. Bamlin and A. Love, Superfluidmty and superconductivity in relativistic fermion systems
which have J = / ± 1, by
f~~(J,iITIJ,j)=f~
(i,j=l,2).
(4.9)
Thus for these amplitudes we need only consider the largest eigenvalue of the matrix
~J
It +
—
—
Ii
~
2~J It
L ~J
1/ J
~ 122 + RI 22
4~J\2 L 4/
I ti)
~
~J
,cJ
f,~,namely
\21t/2
12)
and which has parity P = (— 1)~~’.
The two remaining amplitudes f~and f~(spin singlet and triplet,
respectively) both have J = I and hence have parity P = (~1)~.
Consider first the symmetric flavour channels (ss and I = 1). We can show that fi decreases
monotonically with J increasing, and since J must be even and non-zero, fi is the largest amplitude with J
even. It can also be shown that
A~>f~’2>f~1>f~ (Jl).
(4.11)
and that A~Idecreases monotonically with J. Thus, since J must be odd, A ~ is the largest helicity
amplitude with J odd. Finally we can show that
A~>f~ (Jl)
(4.12)
SO
(4.13)
and it follows that the dominant amplitude is A and that the pairing in this channel will therefore have
J~= l~.
Next consider the antisymmetric (isoscalar) flavour channel (ud — du). The monotonic property of f/
means that f~
is the most attractive amplitude with J odd, and with J even we only have to consider
f~,f’~since A~÷
is monotonic. We can show that
‘~
(4.14)
(4.15)
so f~
is the largest amplitude with J even. Since we can also show that
(4.16)
follows that pairing in this channel has J~= 0’ and is controlled byf~.It also follows from (4.14) that
pairing in the u—d system will favour the isoscalar channel rather than the isovector channel.
The conclusion of the foregoing analysis [71is that ss pairing is expected to have a gap matrix
transforming as a colour ~ and having J” = l~,while in the u—d system the pairing is expected to occur
in the isoscalar channel, with a gap matrix also transforming as a colour ~ but with j’° = 0~.The latter
case has already been studied in section 1, and the complication induced by colour is minimal. Since Li
it
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
371
transforms as colour ~, each component 4~,42, 43 in (1.50) is antisymmetric in its colour labels and
proceeding as in sections 1.4 and 1.5 we find on this occasion that
4(n)11 = qp[p.2( V~+3V~)+p~(V~—V~)+m2(V~—3V~)]~
x {p.(V~+3V~)y
5 pF(V~— V~)n 77075 — m(Vo~— 3V~’)y0y5}d~
—
.
(4.17)
where the overall gap scale d~is now found by minimizing
d*d
— d
“
p
+
4(k
2 “ “~+ 48i~(k~T~)2
~
7~(3)p.j~d*d
7~3~’F Vd*.Vd
(418
. )
,,.
161T
8T~)
(Note the overall normalization of (4.18) differs from (1.118) because of colour.) The critical temperature T~is given from (1.58) and (1.56) by
kBTC
exp[—(4fg)~],
= ~
(4.19)
where f~
is the helicity amplitude defined in (1.18). As expected the phase transition is controlled by the
partial wave amplitude f~.Clearly this conclusion and the form (4.17) of the gap 4 depend sensitively
on the assumed properties (4.2) and (4.5) of the colour interaction. For example, if the 7 = 0
amplitude ~ was dominant instead of jg a similar analysis gives
4(n),,,
=
jjp[p~(VoE+ 3V~)+p.2(V~—V~)+m2(V~+Vr)]~
—
x{pF(V~+ 3V~’)I p.(V~—V~)n•yyo~m(V~+V~)n y}d,,
.
(4.20)
but the overall scale is still found by minimizing (4.18) except that on this occasion
kBT~=~oexp[—(4f~
1)~].
(4.21)
Thus the scale of the gap when J = 0 is relatively model independent, once the critical temperature is
known, and with a suitable choice of axes we may take
d~
=
L 7 8t~(3)(ITkBTC?j
I
]1/2
~m
(4.22)
(if d~is constant). In this phase the free energy density has the value
2
42
— —
(4.23)
7 ~(3)p.pp (kBT~)
for both 7 = 0~and 0-. (Notice, in any case, that (4.17) and (4.20) coincide in the limit m = 0, as do
(4.18) and (4.21), because of chiral invariance.)
The pairing of s quarks is considerably more involved because the higher angular momentum (J = 1)
permits a much more complicated gap. Recall from (4.13) that in this case the dominant helicity
372
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
amplitude is A~.which has 7
=
1~.The most general gap matrix then has the form
4(n) = 4(~.y + 4(2). nn y + j4(3) A fl )‘y~+ 4(4). nI
+4w. yy
6~nn~
‘fyo+4~7 fl’y
8~A
fl .yy~y~
0+4’~
0+4~
where each 4~ (p
(4(P)\
“
1’;
=
.
1,. .
=
,
.
(4.24)
8) is independent of n and belongs to the colour 3 representation. Thus
(r’)
p
EL)
As in section 2, the gap equations for the individual matrices 4~j~)
are highly model dependent.
However, the gap equations for the order parameters d~°~
(0 = 1, 2), where
d(~=p~44)_p.(4(S)+4(ó))_
m(4t)+
(4.25a)
4(2))
d~2~
= PF4 (3) + p.4 0) + m4 (5)
(4.25b)
may be expressed purely in terms of the quark—quark scattering helicity amplitudes. These too satisfy
an equation of the form (2.10)
~
=
F°”°
(ad~
+ bD~)
(4.26)
where a, b are as given in (1.65), (1.66), and F°”is now the matrix of J”
by
2
Q
=
l~helicity amplitudes given
~1
/
F=°-~--~
It’
p.p~(—1/\/2)f~
427
VLIt2
2
fk
The quantities D~°~
(0 = 1, 2) which are cubic in d~,are now given by
p.
S
p
2
—
(1) —
m
p
(I)
•
m
m
—
p
m
m
m
p
2)*d(2)
m
p
—
(1)*
(t) ~
(t) (1)~(1)(2)*
(1)* (2)
~. + (1).
(1)* (2)*
(1) (t) —
(1) . (2)
d(
(2)~~(2)*
4d~2~
—d(t)
p
m
m
m
m
p
m
m
p
p
—
(1)
(1)*
m
m
+4d~d~*d~
+ d~. d~*d~4d~d~d~*
+ d~ d~*d~,
5p. 2D~2~
d(2)*d(2)
d(2)d(2)*
p = 3d~2~~
p
m
m — 2d~2~
p
mm
— I (1)~ (2) (1)*
mm
2p
-~-2dm
m
—1
(2) •
2m
(0*
m
p(2) ~.I
2p (t) •
.
.
+~
(0* (1) ~
m
‘~
m
p
~p
(2)*
m
(0
m
‘~
‘.p
.
d(2)*d(2)
m
p
— ‘d~2~•
d(t)*d~)
m
1
(t)*
(t)
(1)
mm(2)* j 12m(1)
.
(2)
2p
(1)•
m
(4.28a)
m
2p
(2) •
•
(t)
mm
(0*
(2)*
(1)
m
p~
As before (4.26) may be diagonalised using a matrix almost identical to (2.14):
s1
(~
:)
=
(1 +
~2 +
Vi
+ z2)t/2
(l+V1+z2
1 +V1+Z2)
(4.29)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
373
where now z is given in terms of the J = 1 helicity amplitudes
z
=2f~2/(f~2—fh).
(4.30)
Then
t
=
diag(A°’,A~2~)
(4.31)
SFS
with
A~1~=
~~-[f~
±f22±(fiI
-f~
2)Vi+
Z2],
(4.32)
p.PF
and the quantities
e~°~
S°’~’d~
(4.33)
S°~’D~
(4.34)
satisfy (2.21), (2.22) and (2.23) with2~
A~°~
as given
with
the
higherabove.
critical temperature T~C°~
is the one which orders first,
The
order
parameter
eW
or
e~
and from (2.23) we see that this corresponds to the largest eigenvalue ~ Now,~from(4.11) it follows
that
fh—f~
2<0
(4.35)
which means that
2]t”2}
2{f~+ f~2~
=4A~
A~ =
[(f~2
— f~t)~+ 4(fh)
(4.36)
using the notation (4.10). Thus we expect e~2~
to order before et~,and in its gap equation (2.21) we may
set e~’~
to zero. However, even if e~”orders before ~ both possibilities may be subsumed into the
single formula
2)t [p(ep e~em+ em e~e~)+
qep• eme,]
te
1,
=
(4.37a)
b(5p.
where
p
=
(x4 + 8x2y2 + 6y4) (x2 + 2y2)t
q= (x4— 12x2y2—4y4)(x2+2y2)t
(4.37b)
(4.37c)
and for p’~,q(O) we use x
9, y~as defined in (4.29). The (normalization of) the free energy corresponding
374
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
to (4.37) may be found using (1.113) with the result that the (bulk) free energy is given by
(3p.2)t 2(x2 + 2y2)
=
jj ~
t
tr e~e+
/32 (tr
e~e)2+
/33
tr(e~e)(e+e)* + /34 tr(e+e)2}
(4.38)
with
/321P = /3
3/q =
/3~/p= ~
(lrkB Tap. )_2
~(3)
(4.39)
and e~1is a 3 X 3 matrix with p the colour ~ index and i the (J = i) angular momentum label.
We have to minimize (4.38) with respect to variations not only of the overall scale of the matrix e but
also with respect to variations of the shape of e. This is a difficult problem to solve in full generality, so
we restrict ourselves to the so-called “inert” phases [35]. These are the matrices which separately
minimize the invariants
R3 =
(4.40a)
(4.40b)
tr(e~e) (e+e)*
2
R4 = tr(e~e)
associated with /33 and ~ subject to the constraint
tr(e~e)= constant =
N2
(4.41)
which fixes the overall scale of e. A full analysis of this for the case of superfluid 3He may be found in
Barton and Moore 1351. In our case because the rows label colour and the columns label spatial
variables, the free energy is invariant under the transformation
e
—*
UeR
(4.42)
where U is a unitary matrix and R is a rotation. For this reason we need only consider the following
representatives of four phases. (Our nomenclature is by analogy with superfluid 3He, although in that
case the rows and column labels refer to spin and orbital angular momentum.)
(i) B-phase: e~
1= ~
(N real)
(4.43)
with
2)t 2
=
(x2 +
2y2)
tN2 +
(/32+
~f3~+~/34) N4}.
(4.44)
(3p.
(ii) Planar phase: e~
1=
—
N(O~1
0p3
o~~)
(4.45)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion Systems
375
with
~PIanar
= (3,~2)1
2 (x2 + 2y2) ~
(iii) Polar phase: e~
tN~+ (132 + ~/33 + ~f34)N4}.
—
1 N(&,,~
2)’
2 (x2 + 2y2)
T~
(4.46)
— ~p2&2)
(4.47)
tN2 + (/32 + /33 + /34 N4}.
(4.48)
~PoIar = (3p.
(iv) A-phase: e~,=
Nô~
1(41+ iö,2)
2)~’2 (x2 + 2y2)
=
(3p.
j~
(4.49)
tN2 + (132 + /34 N4}.
(4.50)
It is clear that the planar phase never has the lowest energy and the actual values of /33 and /34
determine which of the remaining three phases is the best minimum. Fig. 5 summarizes the results. In
our case the (weak coupling) values of /33 and /34 are given in (4.39) and (4.37b, c). It follows from these
that
(4.51)
/33+/340
(with equality when x2 = y2) and that
2134— 133>0.
(4.52)
(4.29) implies that
x~> y~
(4.53)
while x~= y~is achieved only when z = 2V2, which is the strict non-relativistic limit. Thus neither case
I3)~
/
4
j
/a8~+/t=3i~1.
A-phase
/
B-phase
Polar-phase
Fig. 5. Phase diagram for J
=
I superfluid quark matter.
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
376
saturates the inequality. We conclude [7] that in both cases the superfluid orders in the B-phase, and
that with a suitable choice of axes the order parameter is given by
d 1
112
e~1={_t~~(3(32+(33+/34)_1}
~
(2x2 +
}
2y2)~t 6,~
7 ~(3)(irk8T~~)
=
the values of (32,3,4 given
the B-phase has the value
using
in
(4.55)
(4.39) and (4.37b, c). Using (4.44) we find that the free energy density in
(4.56)
x2+2y2
8p.
d
3(32+/33±/34
- 7~(3)p.pF(kBTC),
=
(4.54)
(4.57)
just as in (4.23).
Finally we may compute the gradient free energy using (1.95), and we find
3~grad=
(3p.2)~2 (x2 + 2y2) {yI(ö
1e~)(491e~J)
+ y2(ô~e,)(ô1e~1)}
(4.58)
where
2 dnlde
y1/r
= 72/S =
(4.59)
~-~j~(3)p~ (kBTCITp.)
with
2y2)~t
r
=
(x2 + 4y2) (x2 +
s
=
2(x2 — y2) (x2 + 2y2)~t.
Thus the total free energy functional is given by
(4.60a)
(4.60b)
~
+
~igrad
with ~ as in (4.38) and S~gradas in (4.58).
4.2. Superconductivity in quark matter [8]
The treatment of superfluid quark matter discussed in the previous section parallels the treatment of
superfluid neutron matter given in section 2. However an important difference between the two systems
is that quarks, unlike neutrons, have non-zero electric charge. Thus the quark superfluid will also be
superconducting, and it is this aspect which we shall now explore.
Consider first ud-pairing. We saw in section 4.1 that this is characterized by an order parameter d~
with 7 = 0~and p the colour ~ label. The coupling to the electromagnetic field is obtained by replacing
the ordinary derivative Vd~in (4.17) by the covariant derivative
Dd~=
—
Vd~ iqAd~
(4.61)
D. Bailin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
377
where
(4.62)
q=~e
is the charge of the ud pair. Then the Gibbs free energy density of the superfluid in an external
magnetic field is, analogously to (3.12),
=
+ y(Vd~
2 + iqAd~).(Vd,,
S~,+ add,, + ~/i~(d~d~)
— iqAd~)+
—k--- (B
— p.oII)2,
(4.63)
where now a, /3, y are exactly twice the values given in (3.12b, c, d), because of colour. The critical
external field B~necessary to destroy the superconductivity is given, from (3.16), by balancing the
condensation energy in the superconducting phase against the energy of the applied field. Thus
B~= 8p.pF (kBTCt)2/7 ~(3),
(4.64)
a factor two larger than (3.19) because of colour. The second critical field B~
2is given by (3.22) with the
electric charge of the pair now q = —e/3 rather than —2e. Thus
K — B~ y2q2
7 ~(3)(e2/4ir)
3
(/~‘J’\2
\ p. ) \.pp)
(.~\5
2 2—
= J~_
=
648w
(4 65)
a factor 18 larger than the expression on the right of (3.23). Thus
K =
404(kBT~/p.)(j2/pF)’.
With current quark masses we may take m 4 EF
(4.66)
p..
Then the superconductivity is type I (K
kBT~/p.<1.75 X 10~.
<1/V~)
if
(4.67)
The number on the left-hand side is notoriously difficult to estimate, since it depends sensitively on the
details of the model, as is clear from (4.19). However we find k
8T~/p. 10~which means that the
superconductivity is on the borderline between type I and type II. In the core of a neutron star, which is
where we may hope to find quark matter, it makes little difference whether the superconductivity is
marginally type I or type II. Because of the closeness of K to 1/V2, both of the critical fields are of the
same order of magnitude as B~in (4.64). Taking p. PF 400 MeV/c and kB Tip. iO~,(4.64) gives
the zero temperature critical field
6G (t=1).
(4.68)
B~=10’
The typical magnetic field in a pulsar is probably only about 1012 G, so irrespective of whether the
superconductivity is type I or type II, the phase transition will occur into a phase with total flux
expulsion (rather than into an Abrikosov vortex phase).
We turn next to the pairing of s quarks. In this case the (7 = 1~)order parameter is a 3 x 3 matrix e~,
with p the colour ~ label and i the angular momentum label, as we saw in section 4.1. The critical
magnetic field necessary to destroy the (type I) superconductivity is obtained as before by balancing the
-=
~—
—.
378
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
condensation energy (4.56) against the magnetic field energy as in (3.16). Thus
fldnt’\2 x2+2y2
B2—
~
469
3/32+$3+/34
=
8/2pF(kB T~t)2/7C(3)
(4.70)
where the last equality follows using the values of /32.3,4 given in (4.39) and (4.37b, c). We note that (4.70)
is precisely the same expression as for the J = 0 case, although the value of B~may differ slightly
because the mass m is no longer negligible. Taking PF 400 MeV/c, as before, and m~c 300 MeV/c,
gives p. -~EF 500 MeV. Thus in the present case B~is increased by a factor (5/4)1/4 Thus B~again
satisfies (4.68).
The calculation of the second critical field proceeds as before. We first include the coupling to the
electromagnetic field by the minimal substitution of the covariant derivation De~
1into the free energy in
place of the ordinary derivative Ve~1.On this occasion
-~
-=
De~1= Ve~1— iqAe~1
(4.71)
q = —~e
(4.72)
with
for an s-quark pair. As in section 3.1 the field B~2is determined by dropping the terms in the free
energy which are quartic in e, and arranging that the minimum of the e-dependent part of the free
energy
2~2 (x2 + 2y2) {aee~,+ y~(8~
±iqA
=
(3p.
1) e,~(8~— iqA,) e~1+
72(8, +
iqA,) e1 (8~,— iqA1) e~1}
(4.73)
is
zero. In (4.73)
(4.74)
as in (4.38), and Yi,
72
are given in (4.59) and (4.60). With the B field in the z-direction, we may take
A=Bxy
(4.75)
and then e~1needs only x-dependence. Taking e~, to be in the B-phase, as predicted in section 4.1, we
write
(4.76)
e~1=~=N(x)c~~~
where N(x) is now in general complex. Then
2)t 2 (x2 + 2y2){aN*N + (yt ±b’
We =
(3p.
2B2x2N*N]}
2)
[NI*NF ±q
(4.77)
D. Baiin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
379
which gives
a + (71+
~y4qB~2 = 0.
(4.78)
Then
2(y
2(x2 + 2y2)
2 K2_~_
— B~
— q /L2(3p+L83+$)
1 + ~y2)
—
162ir3
(j~1’\~2(~\5
— 7 ~(3)(e2/41r) \. p. 1 \p~)
79
a factor of 4 less than (4.65), because the charge of the s quark pair is now given by (4.72) rather than
(4.62). In the present case the factor (p./pF)5 enhances the right-hand side by a factor of 3 or so, so
overall K is very close to (0.87 times) the value in the J = 0 case. Thus the superconductivity is again on
the borderline between type I and type II. Thus both critical fields are of order 10160, as in (4.68), so
that in a pulsar the phase transition will again be to a phase with total flux expulsion.
It is an open question whether the superconductivity of the quark matter in the interior of a neutron
star will have observable consequences. In the case of proton superconductivity in a neutron star, Baym
et al. [36] have noted that the time scale for total flux expulsion is about 108 years. In our case the
conductivity of the quark matter core may be reduced by up to four orders of magnitude, because of the
low electron density [12] and the strong quark—quark scattering amplitude. Thus the characteristic time
for the phase transition to achieve the total flux expulsion which we predict may be as little as 10~years.
If a suitable signal for the flux expulsion can be devised, it might be possible to detect neutron stars at
various stages of the process.
4.3.
Colour superconductivity in quark matter [34]
If we assume that quark matter exists in the core of (some) neutron stars, then as the star cools we
expect there to be a phase transition to the superfluid state discussed in sections 4.1 and 4.2 at some
(unknown) temperature. We saw in the previous section that a signal of this phase transition might be
provided by the superconductivity of the medium, if the flux expulsion can be observed. It follows from
(3.14) and (3.17) that the jump in the order parameter d at the (first order) phase transition in an
external field H is given by
d12=Vp.
0113H.
(4.80)
Thus
IdIIIdIT~o= (B/B~)ta= 10_2,
(4.81)
since we have argued that B 1012 G and B~ 1016 G, from (4.68).
However this (ordinary) superconductivity may not be the source of the largest effect upon the order
parameter. The quark superfluid also couples to the colour vector gluons, since the order parameter
belongs to the non-trivial 3 representation of colour, and this coupling is much stronger than the
380
D. Baulin and A. Love, Superfluidity and superconductivity in relativistic fermion systems
electromagnetic coupling to the photon field. The assumed confinement of colour probably precludes
the creation of an applied colour magnetic field, so there is no colour analogue of the ambient 1012 G
field to which we have just alluded. However there may still be a first order phase transition caused by
fluctuations of the gluon field, just as fluctuations of the electromagnetic field make the phase transition
in an ordinary superconductor first order in zero field, as we saw in section 3.1.
We consider the case of ud pairs which are described by a
= 0~(colour ~) order parameter d~
and
the free energy density given in (4.18). For the purposes of the present discussion we may ignore the
electromagnetic interaction of the pair and retain only its coupling to the colour gauge fields. This is
obtained by replacing the derivatives Vd~by the covariant derivative
Dd~=
Vd~— igALl(tad)~
where g is the QCD coupling constant,
(4.82)
A”
(a
=
1,.. . 8) are the gluon fields and
,
(4.83)
are the ~ representation of (colour) SU(3). Then the free energy density is
=
ad~d,,+ ~/3(d~d~)2±y(Dd)~. (Dd),,
+
~G~JG?J
(4.84)
where, as in (4.63), a, /3, y have exactly twice the values given in (3.12b, c, d),
Ga”,
3
=
1AJ” — I9JA~— gfO.bCA~Ay
(4.85)
with f°~the structure constants of SU(3).
We now mimic the treatment [22,23] given in section 3.1 as far as possible. First we ignore the spatial
variation of d~in the critical region. As we saw 2 inis (3.32),
thisproportional
is justified iftoV2K
1. Inbethe
present
inversely
g2, as4 can
seen
from
context
this
is
justified
by
the
observation
that
K
(3.23). Thus, although the ordinary superconductivity V2K is of order unity, for colour superconductivity it is a factor (a/as)u2 — 0.15 smaller, since
cEA/ad
=
10~,
a~=
g2I4ir —~ Thus from (3.32)
~.
(4.86)
which means that we may certainly ignore the effect of fluctuations in d~and retain only those in A.
Taking d~constant then gives
F= F
1(d~)+ F2[d~, A”]
(4.87)
where
J
3x [ad~d~+ ~f3(d~d~)2]
(4.88)
F~(d~)
=
d
and
F
2[d~, Aa]=
Jd3x ~
(4.89)
D. Baum and A. Love, Superfluidity and superconductivity in relativisticfermion systems
381
with
M?J’(x, y) = ~[2yg2(t”d)~(t6d)~&.,
+
~
—
V~~)5”] ~(x — y),
(4.90)
and the
indicates the terms of ~G~G?,which are cubic and quartic in A”. In the Abelian case these
terms did not arise and we were able to perform the functional integral (3.35) exactly. In the present
case, we drop the cubic and quartic terms, and this amounts to a single-loop approximation a Ia
Coleman—Weinberg [38].In practice we have to add a gauge fixing term (1/2~)(V.A”~ to (4.84), and the
associated Fadeev—Popov ghost term. However, in the Landau gauge (~= 0) the ghosts do not couple to
4, and we may perform the functional integral as before. Then the effective free energy
••~
3x ~
F~~= F
1+
(4.91)
Jd
where I is the unit 8 x 8 matrix and D is the 8 x 8 matrix with elements
D0~=
dtt~~tbd.
(4.92)
The result is that
~eff =
a(d~d~)
+ ~/~(d~d~)2
—
(2yg~)~
tr D312
(4.93)
where
a
=~(T—
i~)/i~
with i’~differing from
we find
tr D312
=
(4.94)
because of a temperature-independent renormalization, as before. Using (4.83)
T~
(~=
+ —~=)(d~d43”2.
(4.95)
Writing
(4.96)
d~d~=ldI2
we have
=
& d12 —
~ d~3+
~/3 l’1I~
(4.97)
(as in (3.37)) but now a and ~ have twice the values of section 3.1, and
e ~
=
(~3~
+ 3i~J3)
(2yg2)3~
(4.98)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
382
Thus, as before, the fluctuations induce cubic terms in the free energy which lead to a first order phase
transition. Since the form of (4.97) is identical to (3.37), all of the analysis used in section 3.1 can be
translated immediately and we need only note the numerical differences. All of the quantities
characterizing the phase transition are parameterised by the quantity e defined in (3.43):
=
~a~/2~2I~.0.
(4.99)
In this case we find
—
—
For p~
5
4861T
49 ~(3)2
(g2/4’n)3
p.,
=
7x
g2/4ir =
fk
2[p. \7
~ 0T~\
p. 1 “PF’
a~-~~,
and kB Ta/p.
[1
~~V2
±
1
(4 100)
~
3V3)
10~,as we have supposed,
102
(4.101)
and the transition is strongly first order. The importance of the fluctuations in the present circumstance
derives from two sources. First, we are dealing with a relativistic system, which, as anticipated in section
3.1, has the effect of decreasing and thereby magnifying the effect of the fluctuations. Second, because
the coupling is now a~rather than a this too decreases . With the value (4.101) we find from (3.45) that
jdjT~/ldIT,o=2.8
(4.102)
considerably greater than the jump (4.81) caused by the ordinary superconductivity of the quark matter.
Similarly the ratio of the latent heat to the zero-temperature condensation energy is
I ~ i~SI~e
8(T
=
0)1
=
0.54
(4.103)
from (3.46), again showing that the gluon field fluctuations have made the transition strongly first order.
~eff
I dl
a
Fig. 6. Quark matter free energy when (a) T = T~and (b) T
T~.
D. Baum and A. Love, Superfluidity and superconductivity in relativisticfermion systems
383
In practice the very slow nucleation rate will prevent the phase transition occurring at T = i~,because
of the free energy_barrier to the ordered state (see fig. 6a). Instead the transition will be delayed to a
temperature T T~when the barrier is very small (fig. 6b). It is easy to see that
IdlT~tiIdIr~o
= 2 [1+ (1 + ~~)1I2]_1 = 4.2
(4.104)
and that the ratio of the latent heat to zero temperature condensation energy is a factor 9 times that
given in (3.46), namely 0.34.
Appendix A. Angular integrals
For a function D(n, n’), where
V,
=
J~
P,(n, n’) D(n,
vi
and n’ are unit vectors, define
(Al)
vi’)
where the angular integral is over vi. Then, neglecting V, for 1
order parameters with J 2, we have
J
~
D(n,
vi’)
=
4, consistently with considering only
(A2)
Vo
J~D(n,n’)n= V1n’
(A3)
J~D(n, n’)n,n,
(A4)
=
V2nn+~(Vo—V2)c5~
J~
D(n, vi’)
flifljflk =
J~
D(n,
fltfljflkflt
8ktfl ifl~+ Sjkflfl~
vi’)
J~
D(n, vi’) flifljflkfltflm
J
D(n,
~
(AS)
V3n~n~n~
+ ~(V1
— V3) (ô~n~
+ ~5ikflJ+ 5jkfl~)
vi’)
~V2(~,n~ +
+ t5j,fl iflk + ôitflfflk +
fljfljflkfllflmfln
öjkfl ,fl ~)+ (~V
~V3(511n~n~n,’,,
+9 terms)
0— rV2) ~
+ (~rVi— ~gV3)
~V2(&,Sk,n,~n~+44 terms)
+
+
~,k15J,
+
S,$5,k)
(&fl5k:n,~+ 14 terms)
(thy0 — ~V2)
(~ök,&,~+
(A6)
(A7)
14 terms)
(A8)
J
~
D(n, vi’)
~
~V3
(ô
kifl,,flflfl,+ 104 terms)
+ (~Vi —
th V3) (6ijtSktSmnfl, + 104 terms).
(A9)
D. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion systems
384
Appendix B.
J =
2 projections
The following projections of the J = 2 parts of covariants can be useful. The notations are those of
eqs. (2.2)—(2.7). We suppose that a0, b,1 and
compact notation
c1
are traceless, symmetric 3 X 3 matrices, and we adopt the
a1~a0n1.
(BI)
The results are (with V~as in (Al))
J ~ D(n, n’) a n b n c y -~th V3 (4abc ±4bac ±40acb ± lOb tr ac ±lOa tr bc
— 8c tr ab)0 Y~,, ±~rVi(4abc + 4bac + 2c tr ab),1S11
J 4q12 D(n, n’) a b c y —~~iV3(3abc ±3bac — 2c tr ab)0Y~1+ ~V1(abc + bac + c tr ab)11~,1
•
.
(B2)
(B3)
2D(n,n’)a bc •nn y—*(2bac ±2abc±c trab),
J~
1 (~V3~1
±~V1~1)
(B4)
J~D(n,n’)a.nb.nc.nn.v—÷(8abc+8bac+8acb+2ctrab±2btrac
2a tr bc),1 (~V3Y~,±~
2D(n,n’)(n •a A b)(n A c.y)—* (abc— bac+3btrac—3a ~
+
(B5)
(B6)
J~
i
J ~~D(n,
n’)a . b(c
Avi
yy~)..*(3cab ±3cba
+
Sc tr ab)~
1V2X,1
2 — 4a tr ab
J
i ~ D(n, n’) (n a A b) a y-y5--*(7b tr a
.fdi2
ij -4--—D(n, n )n . an b (c An~
2D(n, n’)n a(n
iJ4J
1 ~V2X~1
— 8acb — 2b tr ac — 2a tr bc ±7c tr ab)0 1~V2X,1
n’) n a (b AC y’y5)—~(2acb — 2abc + b tr ac — c tr ab)~j?rV2X0
(l0abc ±l0bac
if ~~D(n,
— 6a2b),
. b A
C)n
. yy~—~O.
(B7)
(B8)
(B9)
(BlO)
(Bil)
V. Baum and A. Love, Superfluidity and superconductivity in relativistic fermion
systems
385
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