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Critical behavior in the linear response of a superconductor to a static magnetic field
ANNALS
OF PHYSICS 65. 35-52 (1971)
Critical Behavior of a Superconductor
in the Linear Response to a Static Magnetic Field
F. DE PA~QUALE Istituto
di Fisica
della Facoltd
di Ingegneria and Gruppo Rome, Ita1.v
Nazionale
Struttura
della Materia,
AND
E. TABET Laboratori
di Fisica,
Istituto Received
Superiore July
di Sanitci.
Rome,
Ital]
29. 1970
The order parameter fluctuations are studied in the framework of the microscopic theory of superconductivity, both for T> T, and for T-C T,. The critical behavior of the local superfluid density ps (k) is investigated and the continuity of pS in the critical region is explicitly shown. Long range correlations in the order parameter-order parameter correlation function T(q) are found below TC The link of the behavior of T(q) with the Goldstone mode induced by the gauge symmetry breakdown and the role played by long range correlations in ensuring continuity in the critical region are discussed. Agreement with the phenomenological theory and with some previous results for T> r, is found.
INTRODUCTION
Inspite of the actual experimental impossibility to get into the temperature region where fluctuations are relevant for a pure bulk superconductor, from the general point of view of the phase transition theory the role played by the fluctuations in the superconductive transition deserves further investigation. It has to be clarified, indeed, the nature of the Goldstone mode and the role that it plays in determining the critical properties of the system. Fluctuations in superconductors were first studied by Batyev’ and Patashinski and Pokrovski’ pursuing on an analogy with superfluid helium theory. It was there pointed out that the order parameter-order parameter correlation function in a Fermi system (T matrix) plays, near T,, the same role of the one-particle boson Green function. In particular it has been recognized that the most important contribution, near T,, to the bosonlike T matrix for superconductors stems from the first (zero frequency) term of the discrete frequencies sum, as it was already suggested by Landau for the Bose case. In the frame-work of the phenomenological theory 35
36
DE PASQUALE
AND
TABET
the analogy with the Bose liquid was exploited by Ferrell and Schmidt3p4 who applied dynamical scaling arguments to superconductors in strict analogy with the helium case. A phenomenological treatment of fluctuations above T, based on the time-dependent Ginsburg-Landau equation has been given by several authors.5 s6 Microscopic calculations by Aslamazov and Larkin’ support the results of the phenomenological theory concerning the behavior of the dynamical conductivity just above T,. These authors introduced a definite microscopic approximation for the response kernel to a weak external field taking fluctuations into account as a first correction to the mean field theory. The role of fluctuations in the analogous problem of liquid helium has been considered, for the static response, in a previous paper8 where the main aspects of the phenomenological theory by Ferrell et al. were recovered. We found indeed that a critical region can be defined on a microscopical basis by looking at the analytical structure of the local superfluid density.’ In this paper we want to investigate, on the basis of the microscopic definition of the local superfluid density for a superconductor, the static critical properties of the superconductive transition. The main difficulty to work out a similar program is the choice of the correct correlation function where the critical properties manifest themselves. The analogy with the Bose liquid suggests to use the Tmatrix. The well known instability of the T matrix as T-+ T, corresponds to the onset of condensation of Cooper pairs. If the analogy with the Bose case can be pursued on, one could expect the appearance of a long-range part in the matrix T(q) near T,. This long range part should be connected with the Goldstone mode resulting from the breakdown of the gauge symmetry. Actually, we find that this long range part of the correlation function exists and that it ensures the continuity of the critical p,(k) across T,. The analytical structure of the static response function p,(k) is the same we found for helium. This allows us to give a microscopic definition of the critical and the hydrodynamical regions and to show the validity of the scaling arguments, i.e., that p,(k) for finite k is continuous across the transition temperature. The above mentioned results seem to confirm that the analytical structure previously discussed in the case of liquid helium is a general aspect of the phase transition theory. However, the hydrodynamical behavior cannot be realistically investigated in the simple model considered. This paper is divided in four parts. We shall first recall the main results of the linear response theory to an e.m. field for a superconducting and a normal metal. The structure of the response kernel is given by making use of gauge invariance arguments. The relation between our model and the approximation of Aslamazov and Larkin for the calculation of the conductivity is also discussed. The structure of the linear response in our model is presented in Section 2.
CRITICAL.
BEHAVIOR
37
OF A SUPERCONDUCTOR
In the third part the explicit form of the matrix T(q) both above and below T, is described. In the last section we evaluate the superfluid density in the critical region. The continuity of p,(k) is shown together with some remarks about the relevance of the Goldstone mode for the critical properties of the system. 1. The linear response kernel K,,” and the gauge invariance requirements on KNy have been extensively discussed by Schrieffer.” The problem is to give a definite approximation for Kpy describing the response current ,j, to an e.m. field jr(q) = e2K,,(q)A,(q),
c = 1, h = 1, m = 1,
(1.1)
where the external field .4,,(q) is coupled to the system by an interaction Hamiltonian
with the usual definitioni of the paramagnetic currentj;(x). From the requirement of gauge invariance and of local charge conservation it follows that the kernel Kpy must satisfy the following constraints: K&v
= 0,
(1.2a)
k,K,,
= 0.
(1.2b)
These constrains are equivalent one to the other. A simple method for constructing Kpy using the temperature dependent Green functions was introduced by Nambu’ i and is extensively explained in Schrieffer’s book.” We only recall that: with
K,, = flpv+ n&,(1 - d,,)
(1.3)
where
is the particle number density. The G and ry are, respectively, the matrix Green functions and vertices introduced by Nambu.’ l We recall that if an approximation is given for the matrix Green function G we can automatically construct the vertices TV and so the kernel Kpy . In terms of the kernel K, j, i, j = 1,2,3, a local superfluid density
can be defined.
p,(k) = i C Kiiko = 0, k) 1
(1.3’)
38
DE PASQUALE
AND
TABET
We shall use in the following this local quantity, recalling that lim p,(k) cc l/At, k-0
where L, is the London penetration depth. The value of Kii at k = 0 is thus directly connected with the gauge-symmetry breakdown. As it was shown in Ref. [8], the general diagrammatic form of nij is the following:
FIG. 1
As we are now interested to investigate the temperature region near T,, we are faced with the problem to take into account, in the renormalized vertex rj, the contribution due to the order parameter fluctuations. Consider first the region T> T,. For this, as it was shown by,’ it is sufficient to work with an effective potential (or T matrix) T(q) given by
.q T(q)= A+ +j- d‘%G(q - p)GWW or, in terms of diagrams G(VP) -=:
+ho4vk 7 CiPl
FIG. 2
where G(q) stands for the free electron Green function and 1 is the coupling constant of the electron-phonon interaction. It is straightforward to verify that T(q) contains only even (bosonlike) frequencies. T(q) corresponds to a simple approximation for the order parameter-order parameter correlation function above T,. Once one considers the effective potential T(q), from the general structure of nij it results that ZZij contains two different kinds of diagrams. The first one is characterized by the fact that the free vertices yi and yj cannot be disconnected by cutting one or more boson (T) Iines. The second includes the diagrams in which yi and Yj can be disconnected by cutting some boson lines. Examples of these two classes of diagrams are shown in Fig. 3.
CRITICAL
BEHAVIOR
.OF A SUPERCONDUCTOR
39
FIG. 3
It can be shown that if a gauge invariant approximation following relation holds: lim Uii(k, 0) E n,,(O, 0) = - 3n.
for nij is choosen the (1.4)
k-0
The main difference between class (a) and class (b) diagrams is that diagrams of class (b) allow intermediate states with only bosonlike particles. It will be shown later that the propagator T(q) for these “particles” for small q has the fOlXl
1 T(q, qo = 0) = A q2+F
’
where A+0 as T-+T,. In the field theory language, as T+T,, the peculiar structure of T(q) gives rise to the appearance of zero mass particles in the intermediate states and thus to a non analytical behavior of the diagrams of class (b). This feature does not appear in diagrams of class (a) due to the odd frequencies associated with fermion lines. On the basis of this argument we can assume that class (a) diagrams are regular in k in the neighbouring of k = 0. From this argument it follows that, due to the rotational symnetry of our system, the first contribution to 3n +lIii(k) = ZIii(k)-IZii(0) from diagrams of class (a) will be at least of O(k’). Consider then class (b) diagrams. The main difficulty arises in connection with the well-known fact that near T, the perturbative technique becomes meaningless. This feature is due mainly to the increase of the role of the fluctuations as T-+T, implying that terms with, say, n and n +l fluctuations can give contributions of the same order of magnitude. This difficulty arises once one tries to investigate the response in the hydrodynamical region (k-0) and could be faced with “strong coupling” techniques (Ferrel14, Migdal13 and Polyakov’4). In this paper we shall limit ourselves to investigate in the critical region (k # 0, T+T,) the continuity through T, of the lowest order term in the fluctuations, calculated from diagram b,. We shall find that pS has the expected linear behaviour, in complete analogy with the case of helium and it is indeed continuous across T,. In the following sections we shall show how a gauge invariant approximation for r, can be found, in order to calculate the contribution to the response kernel
40
DE PASQUALE
AND
TABET
associated with the diagram 6r. It can be noted that the same kind of approximation has been successfully considered by Aslamazov and Larkin’ for the calculation of the dynamical conductivity. 2. A simple method to construct the vertex r, has been described by for the self-energy part C is given, the Schrieffer ’ ‘. Once an approximation vertex r, can be obtained by inserting in ail the topologically distinct ways the free vertex yy in C. We shall begin with considering our system above T,. Let us consider the self-energy at the lowest order in T(q),
(2.1)
the corresponding
vertex r, is given by the following
diagrams:
FIG. 5
We note that the vertex r, is naturally coupled with the vertex T, which is the gauge modification of the T matrix
(2.1’) FIG. 6
Analitically,
we have
CRITICAL
BEHAVIOR
41
OF A SUPERCONDUCTOR
The gauge invariance (G.I.) requirements
r”k=L(P+;)-I,, (Iv-;), Tvk, = 2[++
;)-T(q-
;)I
(2.4)
are verified, as will be shown explicitely in Appendix I. We want here to emphasize that the G.I. requirements (2.4) are verified also if the Eq. (2.3) is decoupled from Eq. (2.2) choosing in (2.3) the free vertex yy and the free Green function G, as we actually do. Diagrams for LIii in our model are described in Fig. 7.
lb)
FIG. 7
As it was explained in the first section, we can limit ourselves to the diagram bz. If we choose free propagators in Eq. (2.3) diagram b, reduces to diagram b, of Fig. 3. The contribution of diagram b, to nii (k) is given by ZIyi, where
(2.5)
The term 3n (see Eq. (1.4)) is given by the sum of all diagrams corresponding to
42
-IIii(0).
DE PASQUALE
AND
TABET
In our model, therefore, p,(k) will be expressed, in the critical region, by P.stk)= k
(2.6)
[nZ(k)-nii(o)l.
We can use the gauge invariance requirement in order to get the explicit expression for TV as k-0 in the critical region, as a function of T. In general, we can express T,(q- k/2, q +k/2) in the following way:
The function a(q, k) is regular as k-+0, due to the absence of long range correlations above T,. Then Ti can be approximated, in the critical region by: Ti N b(q, k)qi and, by making use of the Ward identities (for k, = 0) (2.7)
The following
step is to approximate
We are then left with the following expression for p,(k): 1 p,(k) = z -K’XO)+ By performing
Ekp& (271)j
the angular integration
b(q-
;)-T(q+
;)I,
we obtain
We shall now consider the situation below T,. The response kernel can be easily obtained by taking into account the contribution stemming from the anomalous self-energy parts. The same classification of diagrams made above T, holds. We want to stress that the equation for T(q) is, below T,, drastically modified due to the coupling to the anomalous propagators. It is straightforward
CRITICAL
to show that the following the order parameterorder we have
BEHAVIOR
OF A SUPERCONDUCTOR
43
system of equations holds in our approximation for parameter correlation functions. Diagrammatically,
(2.9)
FIG. 8
We shall solve and where G,, and G,, are the usual B.C.S. propagators”. discuss (2.9) in the following section; we want here only to anticipate that lim T(q, q. = O)- $. q-0
(2.10)
This long range behavior associated with the gauge symmetry breakdown must be taken into account in discussing the structure of response below T,, and must be carefully considered. This becomes clear for instance, if one considers that we have, below T,, a new type of diagrams characterized by the property that two vertices yi are connected only by one T or T line. One example of these diagrams is given in Fig. 9.
FIG. 9
All these diagrams however contain at least a coefficient ]A 1’ where A is the gap parameter. In calculating these contributions the limit k-+0 must be taken with care due to the property (2.10) of the matrix T. We disregard these contributions because we shall always approach to the transition temperature in the critical region, i.e., A-+0, k-0
but $0.
In this limit all the terms in the diagrammatic expansion of p,(k) containing G,, or Tcan be disregarded. We are thus left with the same Eq. (2.8) for p,(k), where T(q) will be given now by the solution of the Eq. (2.9). At this point we want to stress that the limit A-0 cannot be taken directly on the structure of the Eq. (2.9) as it will be discussed in the following section.
44
DE PASQUALE
AND
TABET
Such a procedure, in fact, would destroy the long range behavior (2.10) and this is equivalent to neglect, as T-T,, the breakdown of gauge symmetry which is a basic feature that must be verified up to T,. 3. We shall now proceed to calculate the T matrix structure for q # 0 both above and below T,. Above T,, Eq. (2.9) reads: m=m
+
(3.1)
FIG. 10
where the wavy line represents T(q). The solution of (3.1) is given by: T(q)
or
=
A--
__
(2n)3
T 1n
d3kG(q - k)G(k)T(q) -1
1 T(q) = A(l+’ 3 T15 (274 n
,
(3.2)
T = absolute temperature.
As it was already discussed before we shall limit ourselves to the zero frequency term in T(q). T(q) will be evaluated inserting in (3.2) the free Green functions. The angular integration in the integral j d3kG(q)G(q-k) can be easily performed with the result: In where ,H is the chemical potential and o, = (2n +l)rrT. Performing a series expansion in terms of q and taking into account that, as usually, the region of integration in k-space is limited by the condition (Q-P/
~1:
- “fi-
k&k-k,)(So,
where kF stands for the Fermi momentum,
s
d3kG(q- W(k)
z VW,
1 ; &
(wn = Debyefrequency), we get - f
q=k; 1;
(02 5=)= ,
n
(3.3)
where r = k,(k- kF) and we have neglected terms O(r2) with respect to ks (recall that kg w wD). Moreover, linear terms in 5 vanish due to the symmetry of the limits of
CRITICAL
BEHAVIOR
OF A SUPERCONDUCTOR
45
integration. The first term in (3.3), by insertion in (3.2), gives rise to the wellknown phenomenon of instability of the Fermi gas for A
where [ is the Riemann function. Inserting these results in (3.2) we get for T(q) the equation (3.4)
where In y is Euler’s constant. Near T,, (3.4) can be written as T(q)=
-Fu F
where a,=L.
1 Oq2+aY0
(3.5)
247~~T2 T-T, 7{(3)k; ’ ’ = T,
The formula (3.5) for T(q) is in agreement with Schmidt’s5 calculations, based on the Ginsburg-Landau equation. The result (3.5) confirms what has been said in Section 1 about the long range feature of correlations as T-+Tz. In the complex q plane T(q) exhibits two poles +iJlelcc,, which approach to the real axis as T-t T,’ . This analytical structure is completely analogous to what has been found’ for the case of a Bose gas near the Bose-Einstein condensation temperature. As for the case TX T,, we have to solve the complete system (2.9) of equations. If we define the following quantities:
(3.6)
in which G, and G12 stand for the BCS Green functions:
G12
=
-j$, ”
tk = ;k2-p; k
Ek” = t;+A’
ROTATING
489
COSMOLOGIES
= -uifdf(P+p)-l+m),
(2.21)
so the direction of the fluid momentum flux does not tumble, to lowest order. The ,.d term in Eq. (2.21) gives uiuiccR2 when ,p = p/3. If we investigate the lowest order p-dependent terms in froo we have f&II
=
-
f(P
+
(2.22)
Pk”jBijR-2
+(terms whose variation in /I is of higher order). In Eq. (2.22) one inserts the constant direction for ui, and either Jp +p)cc Re4 and ui cc R (when the model is radiation dominated) or /(p +p) ccR- 3 and ui = const (when the baryonmatter dominates). Only the explicit Bijin Eq. (2.22) is varied [3]. We note first that Eq. (2.22) is of third order in smallness, so should be consistently ignored, as we have, to the order we are working. Second, if we were to include it, the ,T,,, term would give a third-order correction to the potentials affecting the fl motion via a term which shifts the location of the zero of the effective potential. In that case, since the neutrinos do decouple at the zero of V,, there will no longer be straight line motion, and the momentum flux direction would change to this order. There would, of course, be terms in V, giving tumbling to this order also. We will ignore the matter terms subsequently, in view of their smallness. At present, a,T,,,/@ = O(Z;\,&,) = O(p2), so the neglect of these terms is justified. From Eqs. (2.20),
(2.23) Notice that GoiGoioc(~3)2 because of the straight line motion. The general definition of p2 in terms of &,,, ge, gLILis given in Eq. [7. (8.34)]. To the order we are working, neglecting the pgV, terms, the Lagrangian is +=
R3{&&
$(&~dl)2+(p2-P:)2+
i
%I}.
(2.24)
Y=3
It describes the Sdimensional motion of a time-varying mass in a timevarying harmonic-oscillator potential. To apply this Lagrangian to a model of the real universe, we recall the following facts. In a RW cosmology neutrinos would have decoupled at a temperature of approximately 10” K [9]. Hence if the anisotropy was small, ]/?I 2 1, ]p”]? 1, I/?]? 1, and 16,1? 1, the universe since the temperature of 10” K could be described by our small anisotropy model. Such a model would have a RW expansion law, with the source term
CRITICAL
BEHAVIOR
47
OF A SUPERCONDUCTOR
where G(k) stands for the one particle Green function for T> T,. The second term is equal to the q* term in the expansion of T(q) above T,. As for S,-S, adding and subtracting 2A2 it can be written as &-S,
1 1 o$+Ek” o>;+E,2-k’
= S,+S2+2A2
(3.11)
We shall approximate the coefficient of A2 using the free particle energy in the integral. It is then straightforward to show that s,-s2
kFA2 1 7L2 (rely
= s,+s,+
7 8 a3L
~
-
(3.12)
by making, in the k integration, the same kind of approximations used for T(q) for T> T,. By making use of (3.7) (3.10) and (3.12) we obtain the expression for T(q) for TS T,: 2
27r2a, TM
=
-
q2+$ F
7
2
F
(3.12)
q2 q’+$(
F >
with a,, previously defined. From Eq. (3.13) it is clearly seen that T(q) below T, exhibits also a short range part with a range determined by the gap parameter. The same structure can be found in the one particle Green function for the Bose case, where the fraction of condensed particles gives the range of the correlations. 4. We are now in the condition to evaluate the local superfluid density in the critical region, both below and above T,. Above T, one makes use of (2.8) and (3.5) obtaining the following formula for p,(k, 7’) in the critical region:
,dk T)=
1 s+ 1
(4.1)
The integrand in (4.1) is an even function of q, so that the integral can be extended from - co to +co. The analytical dependence of ps from k is induced by the presence of the two poles If: idI E1a0 in the integrand, which pinch the integration path as T+T,.
48
DE
PASQUALE
AND
TABET
We shall then calculate the integral in (4.1) by an analytical continuation of the integrand in the complex q plane, by introducing the complex function
which has a cut from -k/2 to +k/2. The contour of integration, shown in Fig. 11,
FIG.
11
has been choosen so that the contribution regular in k (integral along C’) can be separated from the contribution associated with the residues of L(z) for z = +i&a,. We shall not calculate the contribution O(k’) to pS, coming from the integral along C’, which can depend on the detailed structure of T(q) for q B k, where the approximate expression for T(q) given above is not valid. The contribution from the pole +iJlela,, to (4.1) gives
p,(k T) = T
i Re<2rci Res L(z = +imo)> C
CTa, = ___ k,
where the limit T+T,
rc=k 8 +O(k’),
+O(k’) (4.2)
k-+0 has been taken with the constraint
In the opposite limit, that is k/2,/J E(a,,+0 we get only the regular [O(k2)] contribution. As in the case of helium8 we find, in a quite natural way, that the length 5 = (2JlelaJ ’ separates two regions kt =P 1 (critical region) and kt ez 1 (hydrodynamical region), where pS has a different behavior as a function of k. The calculation for T< T, proceeds in the same way, with the significant
CRITICAL
BEHAVIOR
49
OF A SUPERCONDUCTOR
difference that a q = 0 pole appears in the integrand for pS (see (3.13). In this case, we have
so that: dk
CTa, T) = k F
1 -
(
2
z = id$ kF
+niResL(z
= O)>+O(k’)
>
CTcr, n2k = 8 + O(k2), k,
(4.3)
so that continuity in the critical region is verified. We want to emphasize, as it was already noted for the boson case, that the continuity is ensured by the presence of the long range term in the order parameter-order parameter correlation function T(q). The correlation length 5’ for T< T, is now given by
and, inserting for A the expansion valid near T,, that isl2 112 ’
we get &j’ =
(-2”s q2 (274”
d4pp . clC2(P)C(q - p).
It is not difficult to show (see Appendix II) by making use of the same kind of approximations described in the third section, that one gets (4.4)
50
DE PAsQUALE AND TABET
Inserting (4.4) in (4.3) and (4.2) and using the definition of CI~, we finally obtain
which is identical to the expression found in Ref. [8] for the critical superfluid density of a Bose system. The analogy between the Bose Einstein condensation of a boson system and the Cooper phenomenon has given the suggestion for this work. We want the stress that the analogy in the critical behavior of the two models we have developed is strict enough to give the same formula for p,(k), included the numerical coefficients.
CONCLUSIONS
The analysis worked out in this paper refers only to the static critical properties in a simple model for the treatment of fluctuations in a superconductor. We think that one gets some insight either in those aspects of the critical behavior of a superconductor which are common to other systems (e.g., the continuity of p,(k) across T,) either in some typical features of the superconductive transition. We refer here to the question of the critical mode and of the relationship between this mode and the long range behavior induced by the breakdown of the gauge symmetry. A deeper understanding of the Goldstone boson associated with the matrix T(q,q,) requires, of course, the solution of the dynamical problem which we hope to carry out in another paper. However, we have already got a striking evidence for the existence of a Goldstone mode looking at the long range behavior of the order parameter-order parameter correlation function. A further support has been obtained to the hypothesis that some aspects of the behavior of a many body system near the phase transition point seem to be quite independent both from the detailed structure of the interparticle potential and from statistics. APPENDIX
I
We want here to show explicitly the validity of Eq. (2.4). By making use of the definition of the vertex TV, we have
51
CRITICAL BEHAVIOR OF A SUPERCONDUCTOR
+&j-d%&(m-;)G(q-r&T,. From the Ward identities it follows that k,T, = G-‘(rn+
;)-G-‘(m-
;).
Therefore, we get
k,T,=$$j-d4m+;)G(q-m)++ ;)- $$hmG(m+ ;)
qq-m,,(q+ ;)+&jd%G(--
;)G(q-m)k.T,.
64.1)
Now, by recalling that
with some obvious change of the integration k,T, = 2[T(q+
variables in (A.l), we obtain
;)-T(q-
$1.
64.2)
It should be noted that the factor 2 in (A.2) comes from the two possible ways of inserting the gauge field along the fermion loop as can be seen from Eq. (2.1’). a*+ APPENDIX
II: CALCULATION
OF THE CONSTANT
C
From the definition of C, it follows that
Performing the integration
over the angle between p and q, we get l%+Ep+Ep-P P4
ln p-iiO,--Ep-cq+pq p-io,-~EP-~q-pq
(B.1) 1G’(P)
52
DE PASQUALE
By a series expansion of the logarithm C-L
AND
TABET
in (B.l), the expression for C becomes
T q2
and, by introducing
-1 cw2
n
soep’dp&
(i.
n
,“p’,
P
(i.
n
-;
P
+/q
the variable l and after some obvious simplifications
where as usually we have neglected terms O([‘) in respect to ks. Now, +W
4
J71(3), T J-1 (o,Z+{~)~ = (xT)~ 8
so that we finally obtain (for TN T,)
which is the result (4.4).
RERERENCES
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
E. G. BATYEV, A. Z. PATASHINSKI, AND V. L. POKR~~KII, Sov. Phys. JETP 19 (1964), 1412. A. Z. PATASHINSKII AND V. L. POKROVSKII, Sov. Phys. JETP 19 (1964). 677. R. A. FERRELL AND H. SCHMIDT, Phys. Letters 25A (1967), 544. R. A. FERRELL, “International Symposium on Contemporary Physics,” Trieste, 1968. H. SCHMIDT, Zeit Phys. 216 (1968), 336. E. ABRAHAM~ AND J. W. F. Woo, Phys. Letters 27A (1968), 117. L. G. ASLAMAZOV AND A. I. LARKIN, Phys. Letters 26A (1968), 238. F. DE PA.SQUALE AND E. TABET, Phys. Letters 29A (1969), 197. F. DE PA~QUALE AND E. TABET, Ann. Phys. New York 51 (1969). 223. J. R. SCHRIEFFER, “Theory of Superconductivity,” pp. 207, Benjamin, New York, 1964.
Y. Nambu, Phys. Rev. 117 (1960). 648. A. ABRIKOSOV, L. P. GORKOV, AND L. DZYALOSHINSKI, “Methods of Quantum in Statistical Physics,” Prentice Hall, Englewood, New Jersey, 1963. A. MIGDAL, Zh. Eksper. Teor. Fiz. (USSR) 55 (1968), 5, 1964. M. POLYAKOV, Zh. Eksper. Teor. Fiz. (USSR) 55 (1968), 3, 1026.
Field
Theory
OF PHYSICS 65. 35-52 (1971)
Critical Behavior of a Superconductor
in the Linear Response to a Static Magnetic Field
F. DE PA~QUALE Istituto
di Fisica
della Facoltd
di Ingegneria and Gruppo Rome, Ita1.v
Nazionale
Struttura
della Materia,
AND
E. TABET Laboratori
di Fisica,
Istituto Received
Superiore July
di Sanitci.
Rome,
Ital]
29. 1970
The order parameter fluctuations are studied in the framework of the microscopic theory of superconductivity, both for T> T, and for T-C T,. The critical behavior of the local superfluid density ps (k) is investigated and the continuity of pS in the critical region is explicitly shown. Long range correlations in the order parameter-order parameter correlation function T(q) are found below TC The link of the behavior of T(q) with the Goldstone mode induced by the gauge symmetry breakdown and the role played by long range correlations in ensuring continuity in the critical region are discussed. Agreement with the phenomenological theory and with some previous results for T> r, is found.
INTRODUCTION
Inspite of the actual experimental impossibility to get into the temperature region where fluctuations are relevant for a pure bulk superconductor, from the general point of view of the phase transition theory the role played by the fluctuations in the superconductive transition deserves further investigation. It has to be clarified, indeed, the nature of the Goldstone mode and the role that it plays in determining the critical properties of the system. Fluctuations in superconductors were first studied by Batyev’ and Patashinski and Pokrovski’ pursuing on an analogy with superfluid helium theory. It was there pointed out that the order parameter-order parameter correlation function in a Fermi system (T matrix) plays, near T,, the same role of the one-particle boson Green function. In particular it has been recognized that the most important contribution, near T,, to the bosonlike T matrix for superconductors stems from the first (zero frequency) term of the discrete frequencies sum, as it was already suggested by Landau for the Bose case. In the frame-work of the phenomenological theory 35
36
DE PASQUALE
AND
TABET
the analogy with the Bose liquid was exploited by Ferrell and Schmidt3p4 who applied dynamical scaling arguments to superconductors in strict analogy with the helium case. A phenomenological treatment of fluctuations above T, based on the time-dependent Ginsburg-Landau equation has been given by several authors.5 s6 Microscopic calculations by Aslamazov and Larkin’ support the results of the phenomenological theory concerning the behavior of the dynamical conductivity just above T,. These authors introduced a definite microscopic approximation for the response kernel to a weak external field taking fluctuations into account as a first correction to the mean field theory. The role of fluctuations in the analogous problem of liquid helium has been considered, for the static response, in a previous paper8 where the main aspects of the phenomenological theory by Ferrell et al. were recovered. We found indeed that a critical region can be defined on a microscopical basis by looking at the analytical structure of the local superfluid density.’ In this paper we want to investigate, on the basis of the microscopic definition of the local superfluid density for a superconductor, the static critical properties of the superconductive transition. The main difficulty to work out a similar program is the choice of the correct correlation function where the critical properties manifest themselves. The analogy with the Bose liquid suggests to use the Tmatrix. The well known instability of the T matrix as T-+ T, corresponds to the onset of condensation of Cooper pairs. If the analogy with the Bose case can be pursued on, one could expect the appearance of a long-range part in the matrix T(q) near T,. This long range part should be connected with the Goldstone mode resulting from the breakdown of the gauge symmetry. Actually, we find that this long range part of the correlation function exists and that it ensures the continuity of the critical p,(k) across T,. The analytical structure of the static response function p,(k) is the same we found for helium. This allows us to give a microscopic definition of the critical and the hydrodynamical regions and to show the validity of the scaling arguments, i.e., that p,(k) for finite k is continuous across the transition temperature. The above mentioned results seem to confirm that the analytical structure previously discussed in the case of liquid helium is a general aspect of the phase transition theory. However, the hydrodynamical behavior cannot be realistically investigated in the simple model considered. This paper is divided in four parts. We shall first recall the main results of the linear response theory to an e.m. field for a superconducting and a normal metal. The structure of the response kernel is given by making use of gauge invariance arguments. The relation between our model and the approximation of Aslamazov and Larkin for the calculation of the conductivity is also discussed. The structure of the linear response in our model is presented in Section 2.
CRITICAL.
BEHAVIOR
37
OF A SUPERCONDUCTOR
In the third part the explicit form of the matrix T(q) both above and below T, is described. In the last section we evaluate the superfluid density in the critical region. The continuity of p,(k) is shown together with some remarks about the relevance of the Goldstone mode for the critical properties of the system. 1. The linear response kernel K,,” and the gauge invariance requirements on KNy have been extensively discussed by Schrieffer.” The problem is to give a definite approximation for Kpy describing the response current ,j, to an e.m. field jr(q) = e2K,,(q)A,(q),
c = 1, h = 1, m = 1,
(1.1)
where the external field .4,,(q) is coupled to the system by an interaction Hamiltonian
with the usual definitioni of the paramagnetic currentj;(x). From the requirement of gauge invariance and of local charge conservation it follows that the kernel Kpy must satisfy the following constraints: K&v
= 0,
(1.2a)
k,K,,
= 0.
(1.2b)
These constrains are equivalent one to the other. A simple method for constructing Kpy using the temperature dependent Green functions was introduced by Nambu’ i and is extensively explained in Schrieffer’s book.” We only recall that: with
K,, = flpv+ n&,(1 - d,,)
(1.3)
where
is the particle number density. The G and ry are, respectively, the matrix Green functions and vertices introduced by Nambu.’ l We recall that if an approximation is given for the matrix Green function G we can automatically construct the vertices TV and so the kernel Kpy . In terms of the kernel K, j, i, j = 1,2,3, a local superfluid density
can be defined.
p,(k) = i C Kiiko = 0, k) 1
(1.3’)
38
DE PASQUALE
AND
TABET
We shall use in the following this local quantity, recalling that lim p,(k) cc l/At, k-0
where L, is the London penetration depth. The value of Kii at k = 0 is thus directly connected with the gauge-symmetry breakdown. As it was shown in Ref. [8], the general diagrammatic form of nij is the following:
FIG. 1
As we are now interested to investigate the temperature region near T,, we are faced with the problem to take into account, in the renormalized vertex rj, the contribution due to the order parameter fluctuations. Consider first the region T> T,. For this, as it was shown by,’ it is sufficient to work with an effective potential (or T matrix) T(q) given by
.q T(q)= A+ +j- d‘%G(q - p)GWW or, in terms of diagrams G(VP) -=:
+ho4vk 7 CiPl
FIG. 2
where G(q) stands for the free electron Green function and 1 is the coupling constant of the electron-phonon interaction. It is straightforward to verify that T(q) contains only even (bosonlike) frequencies. T(q) corresponds to a simple approximation for the order parameter-order parameter correlation function above T,. Once one considers the effective potential T(q), from the general structure of nij it results that ZZij contains two different kinds of diagrams. The first one is characterized by the fact that the free vertices yi and yj cannot be disconnected by cutting one or more boson (T) Iines. The second includes the diagrams in which yi and Yj can be disconnected by cutting some boson lines. Examples of these two classes of diagrams are shown in Fig. 3.
CRITICAL
BEHAVIOR
.OF A SUPERCONDUCTOR
39
FIG. 3
It can be shown that if a gauge invariant approximation following relation holds: lim Uii(k, 0) E n,,(O, 0) = - 3n.
for nij is choosen the (1.4)
k-0
The main difference between class (a) and class (b) diagrams is that diagrams of class (b) allow intermediate states with only bosonlike particles. It will be shown later that the propagator T(q) for these “particles” for small q has the fOlXl
1 T(q, qo = 0) = A q2+F
’
where A+0 as T-+T,. In the field theory language, as T+T,, the peculiar structure of T(q) gives rise to the appearance of zero mass particles in the intermediate states and thus to a non analytical behavior of the diagrams of class (b). This feature does not appear in diagrams of class (a) due to the odd frequencies associated with fermion lines. On the basis of this argument we can assume that class (a) diagrams are regular in k in the neighbouring of k = 0. From this argument it follows that, due to the rotational symnetry of our system, the first contribution to 3n +lIii(k) = ZIii(k)-IZii(0) from diagrams of class (a) will be at least of O(k’). Consider then class (b) diagrams. The main difficulty arises in connection with the well-known fact that near T, the perturbative technique becomes meaningless. This feature is due mainly to the increase of the role of the fluctuations as T-+T, implying that terms with, say, n and n +l fluctuations can give contributions of the same order of magnitude. This difficulty arises once one tries to investigate the response in the hydrodynamical region (k-0) and could be faced with “strong coupling” techniques (Ferrel14, Migdal13 and Polyakov’4). In this paper we shall limit ourselves to investigate in the critical region (k # 0, T+T,) the continuity through T, of the lowest order term in the fluctuations, calculated from diagram b,. We shall find that pS has the expected linear behaviour, in complete analogy with the case of helium and it is indeed continuous across T,. In the following sections we shall show how a gauge invariant approximation for r, can be found, in order to calculate the contribution to the response kernel
40
DE PASQUALE
AND
TABET
associated with the diagram 6r. It can be noted that the same kind of approximation has been successfully considered by Aslamazov and Larkin’ for the calculation of the dynamical conductivity. 2. A simple method to construct the vertex r, has been described by for the self-energy part C is given, the Schrieffer ’ ‘. Once an approximation vertex r, can be obtained by inserting in ail the topologically distinct ways the free vertex yy in C. We shall begin with considering our system above T,. Let us consider the self-energy at the lowest order in T(q),
(2.1)
the corresponding
vertex r, is given by the following
diagrams:
FIG. 5
We note that the vertex r, is naturally coupled with the vertex T, which is the gauge modification of the T matrix
(2.1’) FIG. 6
Analitically,
we have
CRITICAL
BEHAVIOR
41
OF A SUPERCONDUCTOR
The gauge invariance (G.I.) requirements
r”k=L(P+;)-I,, (Iv-;), Tvk, = 2[++
;)-T(q-
;)I
(2.4)
are verified, as will be shown explicitely in Appendix I. We want here to emphasize that the G.I. requirements (2.4) are verified also if the Eq. (2.3) is decoupled from Eq. (2.2) choosing in (2.3) the free vertex yy and the free Green function G, as we actually do. Diagrams for LIii in our model are described in Fig. 7.
lb)
FIG. 7
As it was explained in the first section, we can limit ourselves to the diagram bz. If we choose free propagators in Eq. (2.3) diagram b, reduces to diagram b, of Fig. 3. The contribution of diagram b, to nii (k) is given by ZIyi, where
(2.5)
The term 3n (see Eq. (1.4)) is given by the sum of all diagrams corresponding to
42
-IIii(0).
DE PASQUALE
AND
TABET
In our model, therefore, p,(k) will be expressed, in the critical region, by P.stk)= k
(2.6)
[nZ(k)-nii(o)l.
We can use the gauge invariance requirement in order to get the explicit expression for TV as k-0 in the critical region, as a function of T. In general, we can express T,(q- k/2, q +k/2) in the following way:
The function a(q, k) is regular as k-+0, due to the absence of long range correlations above T,. Then Ti can be approximated, in the critical region by: Ti N b(q, k)qi and, by making use of the Ward identities (for k, = 0) (2.7)
The following
step is to approximate
We are then left with the following expression for p,(k): 1 p,(k) = z -K’XO)+ By performing
Ekp& (271)j
the angular integration
b(q-
;)-T(q+
;)I,
we obtain
We shall now consider the situation below T,. The response kernel can be easily obtained by taking into account the contribution stemming from the anomalous self-energy parts. The same classification of diagrams made above T, holds. We want to stress that the equation for T(q) is, below T,, drastically modified due to the coupling to the anomalous propagators. It is straightforward
CRITICAL
to show that the following the order parameterorder we have
BEHAVIOR
OF A SUPERCONDUCTOR
43
system of equations holds in our approximation for parameter correlation functions. Diagrammatically,
(2.9)
FIG. 8
We shall solve and where G,, and G,, are the usual B.C.S. propagators”. discuss (2.9) in the following section; we want here only to anticipate that lim T(q, q. = O)- $. q-0
(2.10)
This long range behavior associated with the gauge symmetry breakdown must be taken into account in discussing the structure of response below T,, and must be carefully considered. This becomes clear for instance, if one considers that we have, below T,, a new type of diagrams characterized by the property that two vertices yi are connected only by one T or T line. One example of these diagrams is given in Fig. 9.
FIG. 9
All these diagrams however contain at least a coefficient ]A 1’ where A is the gap parameter. In calculating these contributions the limit k-+0 must be taken with care due to the property (2.10) of the matrix T. We disregard these contributions because we shall always approach to the transition temperature in the critical region, i.e., A-+0, k-0
but $0.
In this limit all the terms in the diagrammatic expansion of p,(k) containing G,, or Tcan be disregarded. We are thus left with the same Eq. (2.8) for p,(k), where T(q) will be given now by the solution of the Eq. (2.9). At this point we want to stress that the limit A-0 cannot be taken directly on the structure of the Eq. (2.9) as it will be discussed in the following section.
44
DE PASQUALE
AND
TABET
Such a procedure, in fact, would destroy the long range behavior (2.10) and this is equivalent to neglect, as T-T,, the breakdown of gauge symmetry which is a basic feature that must be verified up to T,. 3. We shall now proceed to calculate the T matrix structure for q # 0 both above and below T,. Above T,, Eq. (2.9) reads: m=m
+
(3.1)
FIG. 10
where the wavy line represents T(q). The solution of (3.1) is given by: T(q)
or
=
A--
__
(2n)3
T 1n
d3kG(q - k)G(k)T(q) -1
1 T(q) = A(l+’ 3 T15 (274 n
,
(3.2)
T = absolute temperature.
As it was already discussed before we shall limit ourselves to the zero frequency term in T(q). T(q) will be evaluated inserting in (3.2) the free Green functions. The angular integration in the integral j d3kG(q)G(q-k) can be easily performed with the result: In where ,H is the chemical potential and o, = (2n +l)rrT. Performing a series expansion in terms of q and taking into account that, as usually, the region of integration in k-space is limited by the condition (Q-P/
~1:
- “fi-
k&k-k,)(So,
where kF stands for the Fermi momentum,
s
d3kG(q- W(k)
z VW,
1 ; &
(wn = Debyefrequency), we get - f
q=k; 1;
(02 5=)= ,
n
(3.3)
where r = k,(k- kF) and we have neglected terms O(r2) with respect to ks (recall that kg w wD). Moreover, linear terms in 5 vanish due to the symmetry of the limits of
CRITICAL
BEHAVIOR
OF A SUPERCONDUCTOR
45
integration. The first term in (3.3), by insertion in (3.2), gives rise to the wellknown phenomenon of instability of the Fermi gas for A
where [ is the Riemann function. Inserting these results in (3.2) we get for T(q) the equation (3.4)
where In y is Euler’s constant. Near T,, (3.4) can be written as T(q)=
-Fu F
where a,=L.
1 Oq2+aY0
(3.5)
247~~T2 T-T, 7{(3)k; ’ ’ = T,
The formula (3.5) for T(q) is in agreement with Schmidt’s5 calculations, based on the Ginsburg-Landau equation. The result (3.5) confirms what has been said in Section 1 about the long range feature of correlations as T-+Tz. In the complex q plane T(q) exhibits two poles +iJlelcc,, which approach to the real axis as T-t T,’ . This analytical structure is completely analogous to what has been found’ for the case of a Bose gas near the Bose-Einstein condensation temperature. As for the case TX T,, we have to solve the complete system (2.9) of equations. If we define the following quantities:
(3.6)
in which G, and G12 stand for the BCS Green functions:
G12
=
-j$, ”
tk = ;k2-p; k
Ek” = t;+A’
ROTATING
489
COSMOLOGIES
= -uifdf(P+p)-l+m),
(2.21)
so the direction of the fluid momentum flux does not tumble, to lowest order. The ,.d term in Eq. (2.21) gives uiuiccR2 when ,p = p/3. If we investigate the lowest order p-dependent terms in froo we have f&II
=
-
f(P
+
(2.22)
Pk”jBijR-2
+(terms whose variation in /I is of higher order). In Eq. (2.22) one inserts the constant direction for ui, and either Jp +p)cc Re4 and ui cc R (when the model is radiation dominated) or /(p +p) ccR- 3 and ui = const (when the baryonmatter dominates). Only the explicit Bijin Eq. (2.22) is varied [3]. We note first that Eq. (2.22) is of third order in smallness, so should be consistently ignored, as we have, to the order we are working. Second, if we were to include it, the ,T,,, term would give a third-order correction to the potentials affecting the fl motion via a term which shifts the location of the zero of the effective potential. In that case, since the neutrinos do decouple at the zero of V,, there will no longer be straight line motion, and the momentum flux direction would change to this order. There would, of course, be terms in V, giving tumbling to this order also. We will ignore the matter terms subsequently, in view of their smallness. At present, a,T,,,/@ = O(Z;\,&,) = O(p2), so the neglect of these terms is justified. From Eqs. (2.20),
(2.23) Notice that GoiGoioc(~3)2 because of the straight line motion. The general definition of p2 in terms of &,,, ge, gLILis given in Eq. [7. (8.34)]. To the order we are working, neglecting the pgV, terms, the Lagrangian is +=
R3{&&
$(&~dl)2+(p2-P:)2+
i
%I}.
(2.24)
Y=3
It describes the Sdimensional motion of a time-varying mass in a timevarying harmonic-oscillator potential. To apply this Lagrangian to a model of the real universe, we recall the following facts. In a RW cosmology neutrinos would have decoupled at a temperature of approximately 10” K [9]. Hence if the anisotropy was small, ]/?I 2 1, ]p”]? 1, I/?]? 1, and 16,1? 1, the universe since the temperature of 10” K could be described by our small anisotropy model. Such a model would have a RW expansion law, with the source term
CRITICAL
BEHAVIOR
47
OF A SUPERCONDUCTOR
where G(k) stands for the one particle Green function for T> T,. The second term is equal to the q* term in the expansion of T(q) above T,. As for S,-S, adding and subtracting 2A2 it can be written as &-S,
1 1 o$+Ek” o>;+E,2-k’
= S,+S2+2A2
(3.11)
We shall approximate the coefficient of A2 using the free particle energy in the integral. It is then straightforward to show that s,-s2
kFA2 1 7L2 (rely
= s,+s,+
7 8 a3L
~
-
(3.12)
by making, in the k integration, the same kind of approximations used for T(q) for T> T,. By making use of (3.7) (3.10) and (3.12) we obtain the expression for T(q) for TS T,: 2
27r2a, TM
=
-
q2+$ F
7
2
F
(3.12)
q2 q’+$(
F >
with a,, previously defined. From Eq. (3.13) it is clearly seen that T(q) below T, exhibits also a short range part with a range determined by the gap parameter. The same structure can be found in the one particle Green function for the Bose case, where the fraction of condensed particles gives the range of the correlations. 4. We are now in the condition to evaluate the local superfluid density in the critical region, both below and above T,. Above T, one makes use of (2.8) and (3.5) obtaining the following formula for p,(k, 7’) in the critical region:
,dk T)=
1 s+ 1
(4.1)
The integrand in (4.1) is an even function of q, so that the integral can be extended from - co to +co. The analytical dependence of ps from k is induced by the presence of the two poles If: idI E1a0 in the integrand, which pinch the integration path as T+T,.
48
DE
PASQUALE
AND
TABET
We shall then calculate the integral in (4.1) by an analytical continuation of the integrand in the complex q plane, by introducing the complex function
which has a cut from -k/2 to +k/2. The contour of integration, shown in Fig. 11,
FIG.
11
has been choosen so that the contribution regular in k (integral along C’) can be separated from the contribution associated with the residues of L(z) for z = +i&a,. We shall not calculate the contribution O(k’) to pS, coming from the integral along C’, which can depend on the detailed structure of T(q) for q B k, where the approximate expression for T(q) given above is not valid. The contribution from the pole +iJlela,, to (4.1) gives
p,(k T) = T
i Re<2rci Res L(z = +imo)> C
CTa, = ___ k,
where the limit T+T,
rc=k 8 +O(k’),
+O(k’) (4.2)
k-+0 has been taken with the constraint
In the opposite limit, that is k/2,/J E(a,,+0 we get only the regular [O(k2)] contribution. As in the case of helium8 we find, in a quite natural way, that the length 5 = (2JlelaJ ’ separates two regions kt =P 1 (critical region) and kt ez 1 (hydrodynamical region), where pS has a different behavior as a function of k. The calculation for T< T, proceeds in the same way, with the significant
CRITICAL
BEHAVIOR
49
OF A SUPERCONDUCTOR
difference that a q = 0 pole appears in the integrand for pS (see (3.13). In this case, we have
so that: dk
CTa, T) = k F
1 -
(
2
z = id$ kF
+niResL(z
= O)>+O(k’)
>
CTcr, n2k = 8 + O(k2), k,
(4.3)
so that continuity in the critical region is verified. We want to emphasize, as it was already noted for the boson case, that the continuity is ensured by the presence of the long range term in the order parameter-order parameter correlation function T(q). The correlation length 5’ for T< T, is now given by
and, inserting for A the expansion valid near T,, that isl2 112 ’
we get &j’ =
(-2”s q2 (274”
d4pp . clC2(P)C(q - p).
It is not difficult to show (see Appendix II) by making use of the same kind of approximations described in the third section, that one gets (4.4)
50
DE PAsQUALE AND TABET
Inserting (4.4) in (4.3) and (4.2) and using the definition of CI~, we finally obtain
which is identical to the expression found in Ref. [8] for the critical superfluid density of a Bose system. The analogy between the Bose Einstein condensation of a boson system and the Cooper phenomenon has given the suggestion for this work. We want the stress that the analogy in the critical behavior of the two models we have developed is strict enough to give the same formula for p,(k), included the numerical coefficients.
CONCLUSIONS
The analysis worked out in this paper refers only to the static critical properties in a simple model for the treatment of fluctuations in a superconductor. We think that one gets some insight either in those aspects of the critical behavior of a superconductor which are common to other systems (e.g., the continuity of p,(k) across T,) either in some typical features of the superconductive transition. We refer here to the question of the critical mode and of the relationship between this mode and the long range behavior induced by the breakdown of the gauge symmetry. A deeper understanding of the Goldstone boson associated with the matrix T(q,q,) requires, of course, the solution of the dynamical problem which we hope to carry out in another paper. However, we have already got a striking evidence for the existence of a Goldstone mode looking at the long range behavior of the order parameter-order parameter correlation function. A further support has been obtained to the hypothesis that some aspects of the behavior of a many body system near the phase transition point seem to be quite independent both from the detailed structure of the interparticle potential and from statistics. APPENDIX
I
We want here to show explicitly the validity of Eq. (2.4). By making use of the definition of the vertex TV, we have
51
CRITICAL BEHAVIOR OF A SUPERCONDUCTOR
+&j-d%&(m-;)G(q-r&T,. From the Ward identities it follows that k,T, = G-‘(rn+
;)-G-‘(m-
;).
Therefore, we get
k,T,=$$j-d4m+;)G(q-m)++ ;)- $$hmG(m+ ;)
qq-m,,(q+ ;)+&jd%G(--
;)G(q-m)k.T,.
64.1)
Now, by recalling that
with some obvious change of the integration k,T, = 2[T(q+
variables in (A.l), we obtain
;)-T(q-
$1.
64.2)
It should be noted that the factor 2 in (A.2) comes from the two possible ways of inserting the gauge field along the fermion loop as can be seen from Eq. (2.1’). a*+ APPENDIX
II: CALCULATION
OF THE CONSTANT
C
From the definition of C, it follows that
Performing the integration
over the angle between p and q, we get l%+Ep+Ep-P P4
ln p-iiO,--Ep-cq+pq p-io,-~EP-~q-pq
(B.1) 1G’(P)
52
DE PASQUALE
By a series expansion of the logarithm C-L
AND
TABET
in (B.l), the expression for C becomes
T q2
and, by introducing
-1 cw2
n
soep’dp&
(i.
n
,“p’,
P
(i.
n
-;
P
+/q
the variable l and after some obvious simplifications
where as usually we have neglected terms O([‘) in respect to ks. Now, +W
4
J71(3), T J-1 (o,Z+{~)~ = (xT)~ 8
so that we finally obtain (for TN T,)
which is the result (4.4).
RERERENCES
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
E. G. BATYEV, A. Z. PATASHINSKI, AND V. L. POKR~~KII, Sov. Phys. JETP 19 (1964), 1412. A. Z. PATASHINSKII AND V. L. POKROVSKII, Sov. Phys. JETP 19 (1964). 677. R. A. FERRELL AND H. SCHMIDT, Phys. Letters 25A (1967), 544. R. A. FERRELL, “International Symposium on Contemporary Physics,” Trieste, 1968. H. SCHMIDT, Zeit Phys. 216 (1968), 336. E. ABRAHAM~ AND J. W. F. Woo, Phys. Letters 27A (1968), 117. L. G. ASLAMAZOV AND A. I. LARKIN, Phys. Letters 26A (1968), 238. F. DE PA.SQUALE AND E. TABET, Phys. Letters 29A (1969), 197. F. DE PA~QUALE AND E. TABET, Ann. Phys. New York 51 (1969). 223. J. R. SCHRIEFFER, “Theory of Superconductivity,” pp. 207, Benjamin, New York, 1964.
Y. Nambu, Phys. Rev. 117 (1960). 648. A. ABRIKOSOV, L. P. GORKOV, AND L. DZYALOSHINSKI, “Methods of Quantum in Statistical Physics,” Prentice Hall, Englewood, New Jersey, 1963. A. MIGDAL, Zh. Eksper. Teor. Fiz. (USSR) 55 (1968), 5, 1964. M. POLYAKOV, Zh. Eksper. Teor. Fiz. (USSR) 55 (1968), 3, 1026.
Field
Theory