Order of superconductive phase transition
On the occasion of Reinhard Folk’s 60th birthday, I give a brief review of the theoretical progress in understanding the critical properties of superconductors. I point out the theoretical difficulties in finding a second-order transition in the Ginzburg-Landau Model with O(N)-symmetry in 4 − ε D...
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irk-123456789-1193852017-06-07T03:04:50Z Order of superconductive phase transition Kleinert, H. On the occasion of Reinhard Folk’s 60th birthday, I give a brief review of the theoretical progress in understanding the critical properties of superconductors. I point out the theoretical difficulties in finding a second-order transition in the Ginzburg-Landau Model with O(N)-symmetry in 4 − ε Dimensions, and the success in predicting the existence and location of a tricritical point with the help of a dual disorder theory. З нагоди 60-річчя Райнгарда Фолька я приводжу короткий опис теоретичного прогресу в розумінні критичних властивостей надпровідників. Я відзначаю теоретичні труднощі у виявленні фазового переходу другого роду в O(N) симетричній моделі Гінзбурга-Ландау в вимірності 4 − ε, а також успіх в передбаченні його існування і визначенні трикритичної точки за допомогою дуальної невпорядкованої теорії 2005 Article Order of superconductive phase transition / H. Kleinert // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 75–86. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 05.70.Jk, 64.60.Fr, 74.20.-z DOI:10.5488/CMP.8.1.75 http://dspace.nbuv.gov.ua/handle/123456789/119385 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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On the occasion of Reinhard Folk’s 60th birthday, I give a brief review of
the theoretical progress in understanding the critical properties of superconductors.
I point out the theoretical difficulties in finding a second-order
transition in the Ginzburg-Landau Model with O(N)-symmetry in 4 − ε Dimensions,
and the success in predicting the existence and location of a
tricritical point with the help of a dual disorder theory. |
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Article |
| author |
Kleinert, H. |
| spellingShingle |
Kleinert, H. Order of superconductive phase transition Condensed Matter Physics |
| author_facet |
Kleinert, H. |
| author_sort |
Kleinert, H. |
| title |
Order of superconductive phase transition |
| title_short |
Order of superconductive phase transition |
| title_full |
Order of superconductive phase transition |
| title_fullStr |
Order of superconductive phase transition |
| title_full_unstemmed |
Order of superconductive phase transition |
| title_sort |
order of superconductive phase transition |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119385 |
| citation_txt |
Order of superconductive phase transition / H. Kleinert // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 75–86. — Бібліогр.: 40 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kleinerth orderofsuperconductivephasetransition |
| first_indexed |
2025-07-08T15:46:51Z |
| last_indexed |
2025-07-08T15:46:51Z |
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1837094279194869760 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 75–86
Order of superconductive phase
transition
H.Kleinert
Institut für Theoretische Physik,
Freie Universität Berlin,
Arnimallee 14, D–14195 Berlin, Germany
Received December 26, 2004
On the occasion of Reinhard Folk’s 60th birthday, I give a brief review of
the theoretical progress in understanding the critical properties of super-
conductors. I point out the theoretical difficulties in finding a second-order
transition in the Ginzburg-Landau Model with O(N )-symmetry in 4 − ε Di-
mensions, and the success in predicting the existence and location of a
tricritical point with the help of a dual disorder theory.
Key words: superconductivity, phase transitions, renormalization group
PACS: 05.70.Jk, 64.60.Fr, 74.20.-z
1. Introduction
Among the many important contributions made by Reinhard Folk in the field
of critical phenomena is his work on the properties of the superconducting phase
transitions [1,2], where he discussed the effect of two-loop contributions on the renor-
malization flow. To point out its significance, let me briefly recall the historic back-
ground of the problem studied by him with various collaborators, most prominently
with Yurij Holovatch.
Until thirty years ago, the superconductive phase transition was always assumed
to be of second order. All critical exponents had the mean-field values required by
the Ginzburg-Landau model. In 1974, however, it was noted by Halperin, Lubensky,
and Ma [3] (see also [4]) that the renormalization group treatment of the Ginzburg-
Landau model of superconductivity did not produce a critical point. A similar ob-
servation had been published earlier in four-dimensional quantum field theory by
Coleman and Weinberg [5] (working at the same institute). This was interpreted
as an indication that thermal fluctuations which, in general, cause only small devi-
ations of the critical exponents from their mean-field values, would here have the
more dramatic effect of driving the superconductive transition first order. At that
time, the theoretical result could not be tested by experiment. The reason is that in
c© H.Kleinert 75
H.Kleinert
old-fashioned superconductors, the temperature interval around the critical point, in
which fluctuations become important, the so-called Ginzburg interval [6], is extreme-
ly narrow, lying in the nK-regime. This explained the observed mean-field behavior.
The critical regime had always been too small to be resolved. See the textbooks [7,8]
for details.
To probe the Ginzburg interval and confirm the second-order nature of the tran-
sition, one had to rely on Monte-Carlo simulations [9] and a theoretical analogy
with smectic-nematic transitions in liquid crystals [10], which is also described by a
Ginzburg-Landau model, albeit with a different interpretation of the fields [11] (the
so-called Landau-De Gennes model).
The situation has changed since the discovery of high-Tc superconductors, where
the Ginzburg interval is large, up to several K. Recent experiments [12] clearly
show a critical behavior associated with a charged critical point, in which both the
coherence length ξ and the magnetic penetration depth λ grow with the same critical
exponent νλ = νξ ≈ 2/3 [13].
The confusing critical properties of the Ginzburg-Landau model certainly re-
quired further investigation, in particular it appeared necessary to calculate higher
loop approximations. To appreciate the problem, let us first recall what makes the
first-order result of [3] questionable: The simplest argument leading to the first order
is based on a fluctuation-corrected mean-field theory obtained in three dimensions
by integrating out exactly the gauge field at fixed |φ|. The resulting trace-log can be
evaluated exactly, and produces a cubic term ∝ −|φ|3, which makes the transition
first-order. The trouble with this argument is that in the type II regime, fluctuations
will produce vortex lines. At the core of each vortex line, the order field φ vanishes,
which invalidates the assumption of a fixed |φ| in the above calculation.
The first direct theoretical evidence for a second-order phase transition did not
come from the Ginzburg-Landau model but from a dual formulation of it on a lattice
which I set up about 20 years ago [14]. Its properties are discussed in great detail
in the textbook [7]. The dual formulation contains no massless gauge field, and this
avoids the above problem coming from the zeroth in the order field φ. My dual model
predicted definitely a second-order transition in the type-II regime characterized
by a Ginzburg parameter κ > 1/
√
2. This parameter measures the ratio between
magnetic penetration depth and coherence length in the superconductor.
The dual theory contains a different field ψ which is a disorder field . The asso-
ciated Feynman diagrams are direct pictures of the vortex loops, whose density is
|ψ|2. This field has a quartic self-interaction accounting for their short-range repul-
sion, and a gauge-coupling to a massive vector field ~h representing the fluctuating
magnetic induction in the superconducting phase. Due to the mass term, the field ~h
can be integrated out and the assumption of a constant mean field |ψ| presents no
problem, leading to a a Landau-like expansion of the free energy containing terms
|ψ|4, |ψ|6, etc. The |ψ|4 term turned out to be proportional to κ− κt, with [14]
κt ≈
3
√
3
2π
√
1 − 4
9
(
π
3
)4
≡ 0.798√
2
. (1)
76
Order of superconductive phase transition
For κ < κt, vortices attract each other on the average, and the transition is of first
order, whereas for κ > κt, they repel each other and the transition is of second order.
Thus, the mean-field approximation in the disorder theory suggests a second-order
transition in the type II regime of repulsive vortex lines and a first-order transition
in the type I regime of attractive vortex lines.
The point κ = κt is a tricritical point . Its existence has been confirmed by Monte
Carlo simulations on a lattice in [15], but initially at quite a different value κt ≈
0.38/
√
2. Only recently, with the availability of much better simulation techniques,
has my 1982 prediction been confirmed with amazing accuracy [16].
In order to satisfactoryly understand the transition, one should be able to ex-
plain the second order of the transition in the type-II regime without the duality
argument. Thus we must find the effective potential of the Ginzburg-Landau model
as accurately as possible. So far, only the two-loop effective potential is known for
N/2 complex fields in 4− ε dimensions, which I calculated together with my collab-
orator Van den Bossche [17]. The result is given in sections 3 and 4. The case N = 2
concerns the physical situation of the Ginzburg-Landau model of superconductivity.
The involved calculations became possible due to recent progress made in evalu-
ation techniques of Feynman diagrams in 4 − ε dimensions. In particular, two-loop
Feynman diagrams with unequal masses of the internal lines have become available
analytically [18,19]. Moreover, the full ε-expansion of the so-called sunset diagram is
now known [20]. This is important for critical phenomena in 4− ε dimensions since,
for instance, a three-loop calculation requires the knowledge of all terms of order ε
of the two-loop diagrams.
Unfortunately, the renormalization flow of the coupling constants in 4−ε dimen-
sions does not improve much over the one-loop calculation of [3] when going to two
loops [21,22]. Apparently, a direct three-dimensional approach is more promising to
lead to a second-order transition for N = 2 [23,24]. The absence of a charged fixed
point for N = 2 in 4 − ε dimensions seems thus to be a specific weakness of the
ε-expansion, although important progress has been made in [1]: There, a [1/1] Padé-
Borel resummation of the two beta functions (associated with the electric charge and
with the self-coupling of the scalar field) has indeed lead to a desired IR stable fixed
point for N = 2. The resummation at such a low order is, however, not very reliable
and the result should be considered as fortuitous.
After all this work it is not even certain that a higher-loop renormalization group
analysis can really lead to a critical point. In my opinion, the perturbation expansion
to low orders knows too little about the vortex fluctuations, which should be included
explicitly into the path integral [25], and which the dual theory takes optimally into
account. In φ4-theory without gauge field, this seems to be unnecessary for reasons
which are not yet understood. The omission is only justified there by the success in
predicting the experimental critical exponents in these systems [8].
We hope that eventually variational perturbation theory will be capable of locati-
ng a fixed point of the Ginzburg-Landau model, thus allowing us to extract physical
values independently of [1]. This theory has been developed and discussed in detail
in [8,26–30] and it has proven to be a powerful tool for determining critical exponents
77
H.Kleinert
in three [28] as well as in 4− ε dimensions [30] of pure φ4−theories. Variational per-
turbation theory is a procedure which allows us to determine resummed quantities
from a strong-coupling limit of divergent expansions in powers of the bare coupling
constant. It no more requires Padé or Padé-Borel resummation. In particular, there
is no freedom in selecting different Padé approximants which yield different results
at higher orders than [1/1]. Variational perturbation theory has yielded the most
precise prediction [26] α ≈ −0.0129±0.0006 for the best-measured critical exponent
α governing the singularity of the specific heat of superfluid helium. To avoid the
smear-out of the critical regime by gravity over 10−6 K and to determine the critical
temperature up to 2nK, the experiment has been performed in the satellite [31] and
gave α = −0.0127 ± 0.0003.
Recently, we have applied variational perturbation theory to a new range of
problems: the determination of amplitude ratios in three dimensions of the O(N)-
model [32]. We based these calculations on expansions of the critical exponents
obtained by a certain regularization method [33], in which analytic regularization
is applied with a formal minimal subtraction in 4 − ε dimensions, but inserting at
the end ε = 1 without invoking further ε-expansions. Our results were consistent
with Padé-Borel methods. Due to its power, we believe that variational perturbation
theory will eventually succeed in producing the desired zeros in the beta functions
to yield an IR-stable charged fixed point.
When calculating the effective potential to be given in sections 3 and 4, it was
useful to confirm the agreement of our renormalization constants in [17] with those
obtained by Reinhard Folk and his collaborator Kolnberger of 1992 [21]. A student
of mine had done this calculation eight years earlier for his M.S. thesis [22], but one
of his coefficients contained an error, which was corrected in Folk’s work.
Let us also mention that the effective potential given in sections 3 and 4 may
be seen as an extension to a two-coupling-constants problem studied by Brézin et
al. [34] (see also [35]), who give the ε-expansion of the equation of state of the
N -components φ4 theory without a gauge field up to two loops [36].
2. Ginzburg-Landau model
The Lagrangian density contains N/2 complex scalar fields φ coupled to the
abelian fields Aµ and reads, with a covariant gauge fixing,
L = |Dφ|2 +m2φ2 +
g
3!
|φ|4 +
1
4
F 2
µν +
1
2α
(∂µAµ)2, (2)
where Dµ = ∂µ − ieAµ is the covariant derivative, Fµν is the usual field-strength
tensor, and α is a gauge parameter. In principle, there are also ghost fields which,
however, decouple in the symmetric phase and remain massless. Working in dimen-
sional regularization they do not contribute to the energy due to Veltman’s rule
∫
dDp pα = 0 for all α [8]. The effective potential will be obtained using the so-called
background-field method of DeWitt [38]. We shift the scalar field by an unknown
constant Φ: φ → Φ + φ. This generates new vertices. To simplify the calculation,
78
Order of superconductive phase transition
we shall use throughout the Landau gauge α → 0. This reduces the number of
Feynman diagrams and, since α = 0 enforces ∂µAµ ≡ 0 at the Lagrangian level,
removes a possible mixing of Aµφ and Aµφ
† terms, thus decoupling scalar and gauge
propagators. It is further advantageous to use real fields, defining
φ =
1√
2
(φ1 + iφ2), Φ =
1√
2
(Φ1 + iΦ2). (3)
Then the Lagrangian has the expansion around the background field:
L = L0 +
1
2
φ[GTPT +GLP L]φ+
1
2
AµD
TP T
µνAν +
g
4!
[
(φ2)2 + 4Φφ(φ2)
]
+ e2A2Φφ +
1
2
e2A2
µφ
2 + eAµ(φ2∂µφ1 − φ1∂µφ2), (4)
where φ and Φ are now N components real fields written as two-dimensional iso-
vectors φ = (φ1, φ2) and Φ = (Φ1,Φ2). The notation is:
L0 =
1
2
m2Φ2 +
g
4!
Φ4, (5)
GT ≡ −∂2 +m2
T = −∂2 +m2 +
g
3!
Φ2, GL ≡ −∂2 +m2
L = −∂2 +m2 +
g
2
Φ2, (6)
DT ≡ −∂2 +m2
γ = −∂2 + e2Φ2, (7)
PT
ij = δij −
ΦiΦj
Φ2
, P L
ij =
ΦiΦj
Φ2
, P T
µν = δµν −
∂µ∂ν
∂2
, (8)
with GT, GL being the transverse and longitudinal inverse propagators of the scalar
field, and DT
µν is the inverse transverse propagator of the photon field. The transver-
sality of the latter is due to the Landau gauge. Note that there is no term eAµ∂µ(Φ2φ1−
Φ1φ2) which would mix vector and scalar propagators.
So far, all quantities (fields, coupling constants, and masses) are bare quantities.
Up to the second order in the loop expansion, divergences show up as poles in ε up
to the order 1/ε2. They have to be removed to have a finite limit ε → 0. This is
achieved by the renormalization constants
φ = φr
√
Zφ, Aµ = Aµr
√
ZA,
m2 = m2
r
Zm2
Zφ
, g = grµ
εZg
Z2
φ
, e = erµ
ε/2 Ze
Zφ
√
ZA
=
er√
ZA
µε/2, (9)
where in the last equation, the relation Ze = Zφ has been used, which is a con-
sequence of a Ward identity. Intuitively, it comes from the requirement Dµφ →
√
ZφDµφr for the covariant derivative. In the above equations, the bare quantities are
on the left-hand-side and the renormalized ones are on the right-hand-side, indicated
by the subscript “r”. Note that the vacuum energy requires a special treatment [37].
We now state the result for the effective potential derived in [17] up to two loops.
It has the general form
V = V (0) +
h̄
(4π)2
[
V (1,0) + εV (1,ε)
]
+
[
h̄
(4π)2
]2
V (2). (10)
The expansion terms will be given in the following section.
79
H.Kleinert
3. Renormalized effective potential to zero- and one-loop order
The zero-loop effective potential is trivial:
V (l = 0) =
1
2
m2Φ2 +
g
4!
µεΦ4 +
m4
gµε
. (11)
The renormalized one-loop potential is simply a combination of the previous bare
zero-order and the trace-log terms:
V (l = 1) =
1
2
m2Φ2Z
(1)
m2 +
g
4!
µεΦ4Z(1)
g +
m4
gµε
Z(1)
v +
Γ(1−D/2)
(4π)D/2
1
D
µ−ε
×
(N−1)(m2
T)2
(
µ2
m2
T
)ε/2
+(m2
L)2
(
µ2
m2
L
)ε/2
+(D−1)(m2
γ)
2
(
µ2
m2
γ
)ε/2
. (12)
The constants Z
(1)
j are chosen to remove the ε-poles atD = 4 in the Euler Γ(1−D/2)
function. They are
Z
(1)
m2 = g
(N + 2)
3ε
, (13)
gZ(1)
g =
g2(N + 8) + 108e4
3ε
, (14)
Z(1)
v = g
N
2ε
. (15)
From this we obtain in the Landau gauge
V (0) =
1
2
m2Φ2 +
g
4!
µεΦ4 +
m4
gµε
, (16)
V (1,0) =
µ−ε
8
{
(N − 1)m4
T
[
−3 + 2ln(m2
T)
]
+m4
L
[
−3 + 2ln(m2
L)
]
+m4
γ
[
−5 + 6ln
(
m2
γ
)]}
, (17)
V (1,ε) = −µ
−ε
96
(
(N − 1)m4
T
{
21 − 18ln
(
m2
T
)
+ 6
[
ln
(
m2
T
)]2
+ π2
}
+m4
L
{
21 − 18ln
(
m2
L
)
+ 6
[
ln
(
m2
L
)]2
+ π2
}
+ 3m4
γ
{
9 − 10ln
(
m2
T
)
+ 6
[
ln
(
m2
T
)]2
+ π2
}
)
. (18)
A factor (4π)2 is absorbed in the definition of h̄ in (10). The function ln(m2) is
defined by
ln(m2) ≡ ln
(
m2
µ2
)
+ γ − ln(4π). (19)
80
Order of superconductive phase transition
4. Two-loop contribution
The cancellation of the poles at the two-loop order gives renormalization coef-
ficients Z
(2)
m2 , Z(2)
v which depend on ln(mL), ln(mT) and Z
(1)
φ and a renormalization
coefficient Z(2)
g which depends on ln(mL), ln(mT), ln(mγ) and Z
(1)
φ , Z
(1)
A . The results
are
Z
(2)
m2 =
(N+2)
9ε2
[
g2(N+5) − 18ge2 + 54e4
]
− 1
6ε
[
g2(N+2) − 8ge2(N+2) − 6e4(5N+1)
]
, (20)
gZ(2)
g =
1
9ε2
[
g3(N+8)2 − 18g2e2(N+8) + 108ge4(N+8) + 108e6(N+18)
]
− 1
9ε
[
g3(5N+22) − 12g2e2(N+5) − 18ge4(5N+13) + 18e6(7N+90)
]
, (21)
Z(2)
v = g
N
6ε2
[
−18e2 + g(N+2)
]
+ ge2N
2
ε
, (22)
Z
(1)
φ = e2
6
ε
, (23)
Z
(1)
A = −e2N
3ε
, (24)
where a factor 1/(4π)2 is again absorbed in the definition of h̄. Then the renormalized
two-loop potential in (10) is
V (2) = µ−ε
2
∑
i=0
2
∑
j=0
2
∑
k=0
V
(2)
i,j,k
[
ln
(
m2
T
)]i [
ln
(
m2
L
)]j [
ln
(
m2
γ
)]k
, (25)
where
V
(2)
0,0,0 =
1
432gm4
γ
[
18m2
γ
(
g2m2
γ
[
(N−1)(N +5)m4
T+18m2
Lm2
T−(2N +13)m4
L
]
+ge2{42(m2
L−m2
T)2(m2
L+m2
T)−6m2
γ
[
(19N−31)m4
T+24m2
Tm2
L+7m4
L
]
−18m4
γ
[
(N−1)m2
T+m2
L
]
+(19N−36)m6
γ
}−108e4m2
γ
(m2
L−m2
T)
[
(N−1)m2
T+3m2
L
]
)
+ 6gΦ2
{
7g2m4
γ
[
2(N−1)m2
T+(N +8)m2
L
]
+18e4(7m6
L−7m4
Lm2
γ
+11m2
Lm4
γ
+54m6
γ
)
}
+ π2
(
g2m4
γ
[
2(N−1)m2
T+(N +8)m2
L
] [
3(m2
T−m2
L)+gΦ2
]
+18ge2m2
γ
{
(m2
L−m2
T)2(m2
L+m2
T)+3m4
γ
[
(N−1)m2
T+m2
L
]
− 9m6
γ
}
+18e4
{
−9m4
γ
(m2
L−m2
T)
[
(N−1)m2
T+3m2
L
]
+gΦ2(2m6
L−m4
Lm2
γ
+5m2
Lm4
γ
+10m6
γ
)
}
)
+ 216(4π)4ge2
[
m2
γ
(m2
L−m2
T)2I(0)(mL, mT, 0)+e2Φ2(m2
γ
−m2
L)2I(0)(mγ , mL, 0)
]
+ 54ge2
(
m2
γ
{(N−2)(m2
γ
−4m2
T)m2
γ
ξ(mγ , mT, mT)
+2
[
m4
T−2m2
T(m2
γ
+m2
L)+(m2
γ
−m2
L)2
]
ξ(mγ , mT, mL)}
81
H.Kleinert
+e2Φ2(m4
L−4m2
Lm2
γ
+12m4
γ
)ξ(mγ , mγ , mL)
)
+ 6g3Φ2m4
γ
[(N−1)ξ(mT, mT, mL)+3ξ(mL, mL, mL)]
]
, (26)
V
(2)
1,0,0 =− m2
T
12gm2
γ
(
g2(N−1)m2
γ
[−m2
L+(N +3)m2
T+2gΦ2]
+ 6ge2
{
3(m2
L−m2
T)2−m2
γ
[2m2
L+m2
T(7N−9)]+5(N−1)m4
γ
}
−54e4(N−1)m2
γ
(m2
L−m2
T)
)
, (27)
V
(2)
2,0,0 =
1
72gm2
γ
(
g2(N−1)m2
γ
{
−m2
L(6m2
T+gΦ2)+m2
T
[
3m2
T(N +3)+4gΦ2
]}
−9ge2
[
−2
(
m2
L−m2
T
)2
m2
T+m2
γ
(N−2)
(
m4
γ
+6m4
T
)
−6(2N−3)m4
γ
m2
T
]
−162e4(N−1)m2
γ
(m2
L−m2
T)m2
T
)
, (28)
V
(2)
0,1,0 =− m2
L
12gm4
γ
(
g2m4
γ
[−(N +5)m2
L+(2N +7)m2
T+(N +8)gΦ2]
+6ge2m2
γ
{
3(m2
L−m2
T)2−m2
γ
[5m2
L+2m2
T]+5m4
γ
}
−6e4
{
m4
γ
[
27(m2
L−m2
T)−16gΦ2
]
+3gΦ2m2
L(2m2
γ
−m2
L)
}
)
, (29)
V
(2)
0,2,0 =
m2
L
72gm4
γ
(
g2m4
γ
[−3(N +5)m2
L+3(N +8)m2
T+(N +17)gΦ2]
+18ge2m2
γ
[
(m2
L−m2
T)2+3m4
γ
]
−9e4
{
2m4
γ
[
27(m2
L−m2
T)−7gΦ2
]
+gΦ2m2
L(4m2
γ
−3m2
L)
}
)
, (30)
V
(2)
0,0,1 =
e2
6m2
γ
{
−9e2m4
LΦ2−3m2
γ
[
m4
L+m4
T−2m2
L(m2
T+4e2Φ2)
]
+12m4
γ
[
m2
L+(N−1)m2
T−6e2Φ2
]
−m6
γ
(4N−9)
}
, (31)
V
(2)
0,0,2 =
e2
8m4
γ
[
−18m8
γ
+e2Φ2
(
−m6
L+8m4
Lm2
γ
−22m2
Lm4
γ
+40m6
γ
)]
, (32)
V
(2)
1,1,0 =
1
36m2
γ
{
g(N−1)m2
Lm2
γ
(3m2
T+gΦ2)
−9e2
[
−(m2
L−m2
T)2(m2
L+m2
T)+3m2
γ
(m4
L+m4
T)−3m4
γ
(m2
L+m2
T)+m6
γ
]
}
, (33)
V
(2)
1,0,1 =
e2
4m2
γ
[
−(m2
L−m2
T)3+3m2
γ
(m4
L−m4
T)−3m4
γ
(m2
L−m2
T)+(N−1)m6
γ
]
, (34)
V
(2)
0,1,1 =
e2
4m4
γ
{
e2m6
LΦ2+m2
γ
[
m6
L−m4
L(3m2
T+4e2Φ2)+3m2
Lm4
T−m6
T
]
+m4
γ
[
m2
L(−3m2
L+14e2Φ2)+3m4
T
]
+3m6
γ
(m2
L−m2
T)+m8
γ
}
. (35)
The function I(0)(m1, m2, 0) in (26) denotes a non-diverging piece, without the co-
efficient (µε)2, of
(µε)2(4π)4I(m1, m2, 0)=− 2
ε2
(m2
1+m2
2)−
2
ε
[
3
2
(
m2
1+m2
2
)
−L1(m1, m2, 0)
]
− 1
2
{
L2(m1, m2, 0)−6L1(m1, m2, 0)+2m2
1ln
(
m2
1
)
ln
(
m2
2
)
+
[
ln
(
m2
1−m2
2
)]2 (
m2
1−m2
2
)
82
Order of superconductive phase transition
−2ln
(
m2
1−m2
2
)
ln
(
m2
2
) (
m2
1−m2
2
)
+2
(
m2
1−m2
2
)
Li2
(
m2
2
m2
2−m2
1
)
+
(
m2
1+m2
2
)
[7+ζ(2)]
+
π2
3
(
m2
1−m2
2
)
}
, (36)
where
L1(m1, m2, m3) =m2
1ln
(
m2
1
)
+m2
2ln
(
m2
2
)
+m2
3ln
(
m2
3
)
, (37)
L2(m1, m2, m3) =m2
1
[
ln
(
m2
1
)]2
+m2
2
[
ln
(
m2
2
)]2
+m2
3
[
ln
(
m2
3
)]2
, (38)
The function ξ(m1, m2, m3) in (26) is defined by
ξ(m1, m2, m3) ≡ 4
(
2m2
1m
2
2 + 2m2
1m
2
3 + 2m2
2m
2
3 −m4
1 −m4
2 −m4
3
)1/2
×
[
L(θ1) + L(θ2) + L(θ3) −
π
2
ln(2)
]
. (39)
where L(t) is the Lobachevsky function, defined by
L(t) = −
∫ t
0
dx ln cosx, (40)
and the angles are given by
θj = arctan
[
(m2
1 +m2
2 +m2
3) − 2m2
j
(2m2
1m
2
2 + 2m2
1m
2
3 + 2m2
2m
2
3 −m4
1 −m4
2 −m4
3)
1/2
]
. (41)
5. Conclusion
As should have become clear from this short review, the superconductive phase
transition still poses many interesting problems. In particular, the inclusion of vor-
tex loop fluctuations into the ordinary ε-expansion seems necessary to stabilize the
renormalization flow in the type-II regime. I hope that after his 60th birthday, Rein-
hard Folk will have many more active years to contribute to our understanding of
this theoretically and experimentally very important phase transition.
Acknowledgements
I am grateful to Dr. F.S.Nogueira for many interesting discussions and to the
European Network COSLAB for partial support.
83
H.Kleinert
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1/3. See [23,24].
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84
Order of superconductive phase transition
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85
H.Kleinert
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Порядок фазового переходу в надпровідний стан
Г.Кляйнарт
Інститут теоретичної фізики,
Вільний університет Берліну,
Арнімалє 14б D–14195 Берлін, Німеччина
Отримано 26 грудня 2004 р.
З нагоди 60-річчя Райнгарда Фолька я приводжу короткий опис
теоретичного прогресу в розумінні критичних властивостей над-
провідників. Я відзначаю теоретичні труднощі у виявленні фазового
переходу другого роду в O(N ) симетричній моделі Гінзбурга-Ландау
в вимірності 4 − ε, а також успіх в передбаченні його існування і
визначенні трикритичної точки за допомогою дуальної невпорядко-
ваної теорії.
Ключові слова: надпровідність, фазові переходи,
ренормалізаційна група
PACS: 05.70.Jk, 64.60.Fr, 74.20.-z
86
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