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Coupling between the energy gap and single particle energy
ANNALS
OF
36, 42-52
PHYSICS:
Coupling
Between
(1965)
the
Energy Energy* H.
DONALD Quantclm
Gap
and
Single
Particle
E~OBE
Chemistry Group, Uppsala Uppsala, Sweden
University,
The Principle of Compensation of Dangerous Diagrams is applied to second order which, when combined with the quasi-particle self-energy to second order, gives a set of two nonlinear integral equations coupling the energy gap and the single particle energy. These equations are solved for a singlet state shell type interaction. I. INTRODUCTION
Heuley and Wilet’s (1) have recently calculated the energy gap in nuclear matter taking into account corrections to t’he BCS (2) energy gap (3). The higher order corrections were determined from the Principle of Compensation of Dangerous Diagrams (PCDD) formulated by Bogoliubov (4), and were shown to be of importance. In fact, for some potentials the energy gap equation did not have solutions if the corrections were included (1, 5). These calculations cannot be crit’icized on the ground that the PCDD is inadequate, since it has been shown that the PCDD is equivalent’ to maximizing the overlap between the BCS ground state and the true ground state vector (6). In t,he calculations by Henley and Wilets (1, 5) the single particle energy was assumed to be independent of the energy gap and expressible in terms of a constant effective mass. This paper will show that the single particle energy and the energy gap are connected by two coupled iruegral equations through second order. Some insight is also given as to the nat’ure of some of the other approximations made by Henley and Wile&. In Section II the PCDD is given to second order, and t’he dressed energy gap and single particle energy are defined. They arc shown to satisfy a set of two coupled nonlinear integral equations. From the PCDD the coefficients iii the canonical transformation can be determined. However, it is necessary t.o know t.he dressed quasi-particle (QP) energy, and in Sect,ion III the self-energy of the * The work reported Adolf’s 70-Years Fund in part by the Aerospace space Research (OAR),
in this paper has been sponsored in part by the King Gustaf VI for Swedish Culture, Knut and Alice Wallenberg’s Foundation, and Research Laboratories, OAR, through the European Office of AeroUnited States Air Force. 42
EKEKGY
GAP
AND
PAETICLE
ENERGY
COUPLING
43
Q1’ is given to second order. The dressed single particle and hole energies, and energy gap are different from the ones obtained by t’he PCDD, but reduce to t’he same expression if a simplifying assumption is made. In Sect,ion IV the equations for the energy gap and single particle energy are given for a singlet state interaction. In Section V the equations are solved for the energy gap and effective mass in terms of the coul)ling constant for a t,hree dinlensioual “shell” interad-don that only acts near the Fermi surface. Finally in the conclusion some remarks arc made about the al)plication to nuclear matter. II.
PRIXCIPLE
OF
COMPENSATION
OF
DANGEROUS
DIAGRAMS
In t,his sectZion the second order c:orrec%ions to the PCDD will be calculated, and the c~oeffic~ieuts in the canonical transformation, t’he dressed energy gap, and the dressed single particle energy determined. The dangerous diagrams which are colnpensatjed here arc shown in Fig. 1. The diagrams shown in Fig. 1 were considered by Tohuachev and Tiablikov (‘7) for a pairing int’eraction and they concbluded that the second order term (h) was not inq)ortantj for the theory of sul)ercondn~tivity. However, Henley and Wilets have shown that) for nuclear matter the higher order correct ions are indeed important for realist’ic potentials (1, 5). The nlathcnlatical expression for F‘ig. 1 can be derived from t’he general PCDD (6) and is given by the equation’ 4 lb,,(k, -1;)
+ ILe C 96 ho4(12Xk)ha,(123
- lc)A(123L)
= 0
(2.1)
where the sum is over all nunlerals j = 1, 2, 3 where j = (k, , c,) is the n~onlentunl kj and spin mJ . The letter k = (k, c) denotes t’he monlent,um k and the spin g ( f = ul)/down). In Eq. (2.1) the function A is defined as A(1234) = (61 +
E2
+ .$a+ EJ1
(2.2)
where & is the dressed single Ql’ energy which will be discussed later. The reason for ,i occurring in Eq. (2.1) is that there are four QP present in the int~ennediatc alat’e in Fig. 1 (h). When the cxl)ressions for the verlex functions ha*and hod(8) are subst,ituted ht.0 Eq. (2.1) the result can he writ tcrl as 2(w)&
- (PLkZ- L$)&
= 0
(2.3)
The renornmlizcd energy gal) & in Eq. (2.3) is given by & = Ak -
~f(123/~)h(12%)(~~)~
(2.4)
1 The functions he0 , h:~ , and hoa can be found in Appendix A of ref. 10. The typographical error in ha1 has been corrected in ref. 8. The sums over k, are over all values of the momentum, not just the Fermi Sea, because the coherence factors u and ZJ, which occur in the h functions, automatically limit t,he region of integration.
44
KOBE
<+
a_
(4 FIG.
1. The
f(1234)
(b)
principle
where the function
=o
of compensation
of dangerous
diagrams
to second
order
f is defined as
= / (12 / V j - 34) 1’ (UV)I(UV)S + 2 Re (12 1 V 1 - 34)(23 1 V j -114)“~~~~
In Ey. (2.4)
Ak is the “bare”
energy gap which
is given by the equation
Ak = pz F (Ic - k 1 V 1 - 22)(uv)z
(2..5)
(4, 8) (2.6)
Equation (2.6) gives the BCS energy gap equation when the expression for ( UU)~ is used. It will be shown later that Eq. (2.4) gives essentially the same expression as obtained by others (1, 7) for a singlet stat’e interaction. The renormalized single particle (or hole) energy Zk is given by ik = ck - Ji’ xf(123k)A(123k)(~~~ where the “bare”
- I$)
(2.7)
single particle energy ek is given by Ek
=
ek
-
P - F (k2 / V I 2k)vz
w3)
and is just the single particle kinetic energy ek dressed in t’he Hartree-Fock approximation. The chemical potential in Eq. (2.8) is denoted by p. The coefficients in the canonical transformation are related by Uk2 + Vk2 = 1 uk = u-k
(2.9)
vk = - v-k in order for the QP to be fermions. give
Equations
(2.3) and (2.9) can be solved to (2.10)
and
Uk2- Vk2=
gk/d;k’
+
Bk2
(2.11)
ENERGY
GAP
AND
PARTICLE
ENERGY
(b)
(a) FIG.
2. Second
order
contributions
to the self-energy
of the quasi
If the expressions in Eqs. (2.10) and (2.11) are used id Eqs. set of two coupled equations for the renorrnalized gap & energy & are obtained. However, the equations still involve energy, due t,o the A factor. In the next se&ion, the dressed det,ermined in second order. III.
THE
45
COUPLING
DRESSED
QUASI-PARTICLE
particle
(2.4) and (2.7), a and single particle & , the dressed QP QP energy will be
ENERGY
The self-energy of t’he Bogoliubov QP has been discussed in detail in a previous paper (8), but here only the second order corrections shown in Fig. 2 will be treat,ed. These corrections have previously been considered by Tomasini (9)’ who emphasizes that both terms must be included. The dressed QP energy & is defined as the solution t’o the equation ii = hll(k, k) -
z (k, Lk)
(3.1)
where Z (Ic, &) is the self-energy and t’he function h,,(k, Ic) is the kinet#ic energy of the free QP (8). The self-energy Z (Ic) = Z (k, &) is the sum of the two parts shown in Fig. 2. The contribut,ion front Fig. 2 (a) ii”’ 4 2, (k) where
= -?i
c
144 1hdo(k123) l’A(123k)
(3.2)
h is defined in Eq. (2.2) and occurs because there are four QP Ixesent The contribution front Fig. 2 (b) is4
simultaneously.
&(k)
=
>g
c
18 /
=
(b
ha1(123k) I2 Z( 123/G)
(3.3)
where Z is defined as X(1234)
+
Ez +
ts -
EJ1
(3.4)
* I wish to apologize to Professor Tomasini for omitting a reference to his important paper in my paper of ref. 8. 3 The factor 2 appears in Eq. (3.2) because in ref. 8 the contribution from the diagram shown in Fig. 3 was neglected. When the particles are uncorrelat,ed however, the contribution is a term of the type of Fig. 2 (a). 4 The factor f< is included in Eqs. (3.2) and (3.3) because of over counting in Eq. (1.4) of ref. 8. I want to thank Professor Henley for suggesting this to me.
46
KOBE
FIG.
3. A diagram
which
in second
order
contributes
to the self-energy
in Fig.
2 (a)
The function ,S occurs in Eq. (3.3) because there are at most three quasi particles present in Fig. 2 (b) simultaneously. If the vertex functions hll , h40, and ha1 (10) are substituted into Eqs. (3.1), (3.2)) and (3.3) the following equation is obt,ained for the dressed QP energy ,. f,(P) 2,(h) (3.5) Ek = 2(uv)A + u& es - Vk ek where the dressed energy gap & is given by iik = AA - $6 c
[A( 123k) + Z(123k)]f(
123k)(uv)z
(3.6)
The dressed energy gap & in Eq. (3.6) differs from the renormalized energy gap & of Eq. (2.4) in that A has been replaced with S$( A + z). The function f is the same as in Eq. (2.5). Since the observed energy gap is the minimum energy of a two QP state, it is Eq. (3.6) that should be taken as half the observed energy gap. The coefficients in the canonical transformation are however given by Eqs. (2.10) and (2.11). Since 21.3’is the probability that the QP is a particle, the dressed single particle energy is L(P) Ek
= ck - $5 c
[Z( 123k)uZ2 -
A(123k)z$]f(1231c)
(3.7)
that the QP is a hole is VA’, and so the dressed single hole energy
The probability is
a(h)
El;
=
Q
-
$2
c
[A(123k)u;
- x( 123k)v,*]f(
12%)
(3.8)
It is not surprising that there are different expressions for the dressed particle and hole energies, since the particles and holes do not, have t,he same environment. Now to obtain the renormalized gap Ak and the single particle energy Ek it is necessary to substitute Eq. (3.5) into Eq. (2.2), and Eq. (2.2) into Eqs. (2.4) and (2.7). Then two coupled integral equations are obtained for ik and & which can in principle be solved. The solution should then be substituted into Eq. (3.6) to obtain the dressed energy gap b, which is half the observed gap. The expression for 6, and & can also be substituted into Eq. (3.7) and (3.8) to obtain the
ENERGY
GAP
AND
PARTICLE
ENERGY
47
COUPLING
dressed single particle energy 2:” and single hole energy in?). This procedure is very complicated, but in principle should be followed. It is highly desirable to make some approximations to simplify the two coupled equat,ions in Eqs. (2.4) and (2.7). Henley and Wilet)s (5) have assumed that t4 = e?F = 0 in Eqs. (2.2) and (3.4) so that A = Z.’ This seems like a good approximation in the small gap limit, but if the gap is large this approxinmt)ion can be questioned. However, in Henley and Wilet’s approximation the dressed and renormalized gaps are the same, & = & , and the dressed single particle and -(P) _- Ek .(k) = hole energies are equal to the renormalized single particle energy, Q E/;. Therefore, from Eq. (3.ri) and Eqs. (2.10) and (2.11) we have in this approximation t-1; = dik2 which
is a much simpler function IV.
SINGLET
(3.9)
-t Zk2
of ik and & than the expression STATE
in Eq. (3.5).
INTERACTION
The equations of the last section are valid for any type of interaction. they will be rewritten for a singlet state intera&on (5) (12 1 V 134) = (k, , ka 1 V 1k, , kr),S(l,
2)S(4,
3)
Kow (4.1)
where S(l, 2) = &7,+&T- - 6,,-&,+
(4.2)
If Eq. (4.1) is substituted into Eys. (2.4), (2.5), and (2X), the result is )A( 12%)2(7LU)~ & = AI; - c;rS( 1231c
(4.3)
where j - 22),(UV)~
(4.4)
aud f,(1234)
= - ( (1 - 2 ( V ( - 34), j* (UV)1(UU)~ + Re (1 - 2 1V I - 34),(3 - 2 I I’ I - 14)s*v~ua2
(4.5)
Iu these expressions there is of course no summation over the spin indices, and the numbers j = 1, 2, 3 stand for kj only. Equation (4.3) is the same as that used by Henley and Wilets (1) . If Eq. (4.1) is substituted into Eq. (2.7) for ik and the sum over the spin 5 I would like to thank to me. I am also grateful prior to publication.
Professor to him
Henley for pointing and Professor Wilets
out the nature of this approximtion for sending me copies of their work
48
KOBE
carried out, the result is & = ek - p-2~(k-2lVl-2k),vz + ~fs(123k)A(123Ic)(u:
- vz’)
(4.6)
The equations of Section III can also be written for the singlet state. Equation (3.6) for 8, for the singlet state is the same as Eq. (4.3) except that A should be replaced with >h( A + Z). The Eqs. (3.7) and (3.8) are the same except that f should be replaced with 2fs, and the sum is only over the momentum. In the next section, the interaction will further be restricted to a “shell” type interaction. V. THREE
DIMENSIONAL
“SHELL”
TYPE
INTERACTION
In order to gain some insight into the significance of the coupling between the energy gap and the single particle energy, the two coupled integral equations will be solved for a three dimensional “shell” type interaction. This interaction is defined as (k, , k, I V I k, , k4)s =
(Q--lfSdk + kz - k - k4) o othfrwl;ekj I - IcF 1 < w ?
for
j = 1, 2, 3, 4
(5.1)
where fi is the volume of the system, which will be taken as unity for the rest of the discussion. The constant w is the half width of the shell region around the Fermi momentum kF . In order for the interaction to be just in a shell region the condition w << kF must be satisfied. The Kronecker delta &,(k) is unity for zero argument, and zero otherwise. A. THE
ENERGY
GAP EQUATION
If Eq. (5.1) is substit’uted into Eq. (4.3) for the energy gap, the result can be written as & = i5 + tik
(5.2)
for 1/ k I - kF 1 < w and zero otherwise. The terms on the right side of Eq. (5.2) correspond to the respective terms in Eq. (4.3). It will be assumedhere that j & 1 << A so that 2 can be considered essentially as the energy gap, and 6k is only a small perturbation. If Eqs. (2.10), (3.9), and (5.2) are substitut’ed into Eq. (4.3), and if the sums are transformed to integrals, we obtain 1 = (2r)--3f/
dp(2EJ’Mp)
+J
(5.3)
where the function e1restricts the integral to the shell region and is defined as
em = e(w- j 1P1- k 1)
(5.4)
ENERGY
GAP
AND
PARTICLE
ENERGY
49
COUPLING
where 8 is the step function which is unity for positive argument and zero otherwise. The function J depends on the effectrive mass and energy gap in general, and is defined as
(5.5) -fdp,
P + q -
k, q, W&;;’
F&k
Equation (5.3) must be solved self-consistently for t’he energy gap &. However, first the effective massmust. be determined. B. THE
EFFECTIVE
MASS
The effective massm* is defined as ii = (lc2 - k~)/2wz*
(5.6)
where the dressedsingle part’icle energy is given in Eq. (4.6). If Eq. (5.1) is subsMuted into Eq. (4.6), and t’he chemical potential p is chosen t’o eliminate constant’ terms, the result’ for the effective massrat’io /3is p = d/m
= 1 - Ik
(5.7)
The int.egral Ik is defined as Ik = (27p
11 dp dqh(p, p + q - k, q, Wf.dp, p + q - k 9, k)tI;:r-k
(5.8)
The effective mass of course should be independent of k, but Ik is a slowly varying funct’ion of k. Thus, we could, for example, set k = kr and average over the Fermi surface. However, in order to make the treatment of the effective mass more harmonious with the treatment of the energy gap, another type of averaging will be used. The average value of Ik will be I=
s
dd2&c)-‘h(k)Ik
/I
W2hc-‘Nd
(5.9)
where the weighting factor (2&-l has been used in the shell region, and the denominator is for normalization. Thus the edgesof the shell region are not weighted as much as the Fermi surface. The reason for defining the somewhat, unusual average in Eq. (5.9) is that now Eqs. (5.3) and (5.5) can be used to give I = J/(J
- 1)
(5.10)
so t.hat Eq. (5.7) becomes fl = (1 - J)-’
(5.11)
Because of this simple relat’ionship between the effective mass ratio p and J,
50
KOBE
Eq. (5.3) can easily be solved for the energy gap. After performing and solving for 2\ we obtain
the integral,
i = A”/p sinh (l/g@‘)
(5.12)
where here A0 = w/c&
(5.13)
and the coupling constant g is defined as g = 2(27r-‘frnkF
(5.14)
Equation (5.12) is of the sameform as the BCS solution, but has an extra factor of /3 in the argument of the hyperbolic sine. However, in this treatment ,B t.he effective massratio is a function of the coupling constant g. C. VARIATION
OF THE EFFECTIVE
MASS WITH COUPLING
CONSTANT
If Eq. (5.1) is substituted into Eq. (4.5), and the result is substituted into the int’egral J in Eq. (5.5), we obt’ain
J = Ghr>-“f” jjj
dk dp
&(G
&,+q+J%p,
P + q + k, q, k)
(5.15)
.[(uv),(uu),
~P2uq21’3dp + q + k)&(p)&(q)&(k)
-
It is the terms with p + q + k which cause difficulty in Eq. (5.15), so it will be assumed that w is so small that we can set, 1p + q + k 1 s kF . With t.his assumption, the integral J can be written as J = M~3g3F(cu)
(5.16)
where the function F is defined as F(i)
= jjja
--adJ: dy
ddl
+
Cd.?
-[(x2 + 1)(?J2 +
+
1 -
s)(&Fq
1)(x2 + +
(y2 +
+
l)]-““[(x 1)1’2 +
+
?J)l (5.17)
1y2
(x2 +
1y2
+ 11-l
and CLis defined as a = A’/&3
= sinh (l/g@)
(5.18)
from Eq. (5.12). The constant M is related to a geometrical factor M = (27r-32-6 jjj
di dfi d&&.ij
+ $i; + 4-h + 1)
and has a value between 0 and $$j. The unit vector in the p direction is $ and $9 is the element of solid angle. In the strong coupling limit g >> 1 the gap is large and so’OL<< 1. Therefore
ENERGY
GAP
AND
PARTICLE
F( 0) in Ey. (Ti.17) can be evaluated,
ENERGY
51
COUPLING
and the result for the effective mass ratio is
p = [l + P$j@q-‘]-’
(5.19)
Thus, as g + 00, p + 1 and the second order correction does not change the BCS solution in this limit. In the weak coupling limit, g << 1, the funct,iou F(a) can be regarded as a slowly varying function sinre its mgument is extremely large, LY>> 1. Therefore the effective mass ratio is p = [I - Mp3q”F]-’
(5.20)
so that p -+ 1 as g ---f 0 as expected. Thus the secoud order correction is riot, iml)ortant’ in t,his limit. However, it, must, not be concluded t’hat the second order correction is never important. In the intermediate coupling case, it will contribute and cause a deviation from t’he BCS result, as Henley and Wile& have shown (1). For ot,her types of interactions it has been shown to be even more important. Eyuation (5.11) for the effectjive mass is a complicated equation and can best be solved by pert.urbat,ion theory. In general, if J involves the energy gap as it. would for sonle pot.entials, t.he BCS value of the energy gal) can be used to detcrruiue au approximate effective mass. An it,eration procedure can be used to obtaiu a self-consistent result. VI.
CONCLUSION
The previous section outlined a procedure for solving the two coupled iutegral equations satisfied by the energy gap and single particle energy which could be extended to other potentials. However, for other potent’ials, the effective mass alq~roximatjion is not necessarily a good one. As an example, consider the simple Yamaguehi potential used by Henley and Wilets (1) (klk2 1 V ) k,k,),
= li’Ld(Z?r)“X~(;~
/ $ - kz /)w(;i
/ kt - k4 1)
(6.1)
where w(k)
= (k2 + by
(6.2)
aud b aud X are constants. When t)his potential is substituted into Ey. (2.8) for the single particle energy dressed in the Hartree-Pock approximation, and the energy gap is neglected, the result is Ek = f?(k”
- kF2)/2m
+ 2??d[N(k)
- N(kF)]
where N(k)
= -32aX[-
(6.3)
k&e2 + kp’ + 4b2)-’ + (k” + 4b2)-“2 tan-’
kF(k2 + 462)-1’2]
(6.4)
52
KOBE
It can be seen that to replace N(k) with a parabola is not an especially good approximation, especially for large momentum. Since the nuclear energy gap is an especially sensitive function of t,he effe&ive mass, it would be interesting to repeat the calculations of Kennedy, Henley, and Wilets (3) with single particle energies calculated from the potentials used. In this way there would not be any free parameter in the t,heory (except the Fermi momentum), and the energy gap could be determined for normal nuclear density. Since the second order correction t’o the energy gap is important, the second order correction to the single particle energy should also be included. The effect of this coupling using some realistic potentials will be the subject of another paper. ACKNOWLEDGMENTS The early stages of this work were carried out while I had a Fulbright lecture grant in the Republic of China during the 1963 to 1964 academic year. I would like to express my appreciation to the United States Educational Foundation in the Republic of China for the opportunity of receiving the grant. Their concern for the happiness and well being of the grantees went far beyond their official responsibilities. To the faculty, staff, and students at the National Taiwan University, Taiwan Normal University, and National Tsing Hua University, I would like to express my deepest appreciation for their hospitality, friendliness, and help to me during my year in Taiwan. This work was concluded at the Quantum Chemistry Group in Uppsala. I would like to express my sincere gratitude to Professor Per-Olov Lijwdin for the opportunity to come to the Quantum Chemistry Group, for a fellowship to attend the International Summer Institute in Quantum Chemistry, and for his warm hospitality, encouragement and interest in this work. I would like to thank Drs. Fukashi Sasaki and Vedene Smith, Jr. for many stimulating discussions during the course of this work. RECEIVED:
April 12, 1965 REFERENCES
1. E. M. HENLEY AND L. WILETS, Phys. Rev. Letters 11, 326 (1963). 2. J. BARDEEN, L. N. COOPER, aND J. R. SCHRIEFFER, Phys. Rev. 108, 1175 (1957). 3. G. F~NO AND A. TOMASINI, Nuovo Cimento 18, 1247 (1960); G. FANO, M. S.~VOIA, AND A. TOMASINI, Nuovo Cimento 21,854 (1961) ; V. J. EMERY AND A. M. SESSLER, Phys. Rev. 119, 248 (1960); R. KENNEDY, L. WILETS, AND E. M. HENLEY, Phys. Rev. 133, B1131 (1964). 4. N. N. BOGOLIUBOV, Zh. Eksperim. i Teoret. Piz. 34, 58 (1958); Transl. Soviet Phys. -JETP 7, 41 (1958); Nuovo Cimento [lo] 7, 794 (1958). 5. E. M. HENLEY AND L. WILETS, Phys. Rev. 133, B1118 (1964). 6. D. H. KOBE, Preprint No. 145, Quantum Chemistry Group, Uppsala, Sweden (1965); Phys. Rev., in press. 7. V. V. TOLM.~CHEV AND S. V. TISBLIKOV, Zh. Eksperinz. i Teoret. Fiz. 34.66 (1958); Transl. Soviet Phys. -JETP 7, 46 (1958). 8. D. H. KOBE, Ann. Phys. (N.Y.) 28, 400 (1964). 9. A. TOMASINI, Nuovo Cimento [lo] 20,963 (1961). 10. D. H. KOBE AND W. B. CHESTON, Ann. Ph,ys. (N.Y.) 20, 279 (1962).
OF
36, 42-52
PHYSICS:
Coupling
Between
(1965)
the
Energy Energy* H.
DONALD Quantclm
Gap
and
Single
Particle
E~OBE
Chemistry Group, Uppsala Uppsala, Sweden
University,
The Principle of Compensation of Dangerous Diagrams is applied to second order which, when combined with the quasi-particle self-energy to second order, gives a set of two nonlinear integral equations coupling the energy gap and the single particle energy. These equations are solved for a singlet state shell type interaction. I. INTRODUCTION
Heuley and Wilet’s (1) have recently calculated the energy gap in nuclear matter taking into account corrections to t’he BCS (2) energy gap (3). The higher order corrections were determined from the Principle of Compensation of Dangerous Diagrams (PCDD) formulated by Bogoliubov (4), and were shown to be of importance. In fact, for some potentials the energy gap equation did not have solutions if the corrections were included (1, 5). These calculations cannot be crit’icized on the ground that the PCDD is inadequate, since it has been shown that the PCDD is equivalent’ to maximizing the overlap between the BCS ground state and the true ground state vector (6). In t,he calculations by Henley and Wilets (1, 5) the single particle energy was assumed to be independent of the energy gap and expressible in terms of a constant effective mass. This paper will show that the single particle energy and the energy gap are connected by two coupled iruegral equations through second order. Some insight is also given as to the nat’ure of some of the other approximations made by Henley and Wile&. In Section II the PCDD is given to second order, and t’he dressed energy gap and single particle energy are defined. They arc shown to satisfy a set of two coupled nonlinear integral equations. From the PCDD the coefficients iii the canonical transformation can be determined. However, it is necessary t.o know t.he dressed quasi-particle (QP) energy, and in Sect,ion III the self-energy of the * The work reported Adolf’s 70-Years Fund in part by the Aerospace space Research (OAR),
in this paper has been sponsored in part by the King Gustaf VI for Swedish Culture, Knut and Alice Wallenberg’s Foundation, and Research Laboratories, OAR, through the European Office of AeroUnited States Air Force. 42
EKEKGY
GAP
AND
PAETICLE
ENERGY
COUPLING
43
Q1’ is given to second order. The dressed single particle and hole energies, and energy gap are different from the ones obtained by t’he PCDD, but reduce to t’he same expression if a simplifying assumption is made. In Sect,ion IV the equations for the energy gap and single particle energy are given for a singlet state interaction. In Section V the equations are solved for the energy gap and effective mass in terms of the coul)ling constant for a t,hree dinlensioual “shell” interad-don that only acts near the Fermi surface. Finally in the conclusion some remarks arc made about the al)plication to nuclear matter. II.
PRIXCIPLE
OF
COMPENSATION
OF
DANGEROUS
DIAGRAMS
In t,his sectZion the second order c:orrec%ions to the PCDD will be calculated, and the c~oeffic~ieuts in the canonical transformation, t’he dressed energy gap, and the dressed single particle energy determined. The dangerous diagrams which are colnpensatjed here arc shown in Fig. 1. The diagrams shown in Fig. 1 were considered by Tohuachev and Tiablikov (‘7) for a pairing int’eraction and they concbluded that the second order term (h) was not inq)ortantj for the theory of sul)ercondn~tivity. However, Henley and Wilets have shown that) for nuclear matter the higher order correct ions are indeed important for realist’ic potentials (1, 5). The nlathcnlatical expression for F‘ig. 1 can be derived from t’he general PCDD (6) and is given by the equation’ 4 lb,,(k, -1;)
+ ILe C 96 ho4(12Xk)ha,(123
- lc)A(123L)
= 0
(2.1)
where the sum is over all nunlerals j = 1, 2, 3 where j = (k, , c,) is the n~onlentunl kj and spin mJ . The letter k = (k, c) denotes t’he monlent,um k and the spin g ( f = ul)/down). In Eq. (2.1) the function A is defined as A(1234) = (61 +
E2
+ .$a+ EJ1
(2.2)
where & is the dressed single Ql’ energy which will be discussed later. The reason for ,i occurring in Eq. (2.1) is that there are four QP present in the int~ennediatc alat’e in Fig. 1 (h). When the cxl)ressions for the verlex functions ha*and hod(8) are subst,ituted ht.0 Eq. (2.1) the result can he writ tcrl as 2(w)&
- (PLkZ- L$)&
= 0
(2.3)
The renornmlizcd energy gal) & in Eq. (2.3) is given by & = Ak -
~f(123/~)h(12%)(~~)~
(2.4)
1 The functions he0 , h:~ , and hoa can be found in Appendix A of ref. 10. The typographical error in ha1 has been corrected in ref. 8. The sums over k, are over all values of the momentum, not just the Fermi Sea, because the coherence factors u and ZJ, which occur in the h functions, automatically limit t,he region of integration.
44
KOBE
<+
a_
(4 FIG.
1. The
f(1234)
(b)
principle
where the function
=o
of compensation
of dangerous
diagrams
to second
order
f is defined as
= / (12 / V j - 34) 1’ (UV)I(UV)S + 2 Re (12 1 V 1 - 34)(23 1 V j -114)“~~~~
In Ey. (2.4)
Ak is the “bare”
energy gap which
is given by the equation
Ak = pz F (Ic - k 1 V 1 - 22)(uv)z
(2..5)
(4, 8) (2.6)
Equation (2.6) gives the BCS energy gap equation when the expression for ( UU)~ is used. It will be shown later that Eq. (2.4) gives essentially the same expression as obtained by others (1, 7) for a singlet stat’e interaction. The renormalized single particle (or hole) energy Zk is given by ik = ck - Ji’ xf(123k)A(123k)(~~~ where the “bare”
- I$)
(2.7)
single particle energy ek is given by Ek
=
ek
-
P - F (k2 / V I 2k)vz
w3)
and is just the single particle kinetic energy ek dressed in t’he Hartree-Fock approximation. The chemical potential in Eq. (2.8) is denoted by p. The coefficients in the canonical transformation are related by Uk2 + Vk2 = 1 uk = u-k
(2.9)
vk = - v-k in order for the QP to be fermions. give
Equations
(2.3) and (2.9) can be solved to (2.10)
and
Uk2- Vk2=
gk/d;k’
+
Bk2
(2.11)
ENERGY
GAP
AND
PARTICLE
ENERGY
(b)
(a) FIG.
2. Second
order
contributions
to the self-energy
of the quasi
If the expressions in Eqs. (2.10) and (2.11) are used id Eqs. set of two coupled equations for the renorrnalized gap & energy & are obtained. However, the equations still involve energy, due t,o the A factor. In the next se&ion, the dressed det,ermined in second order. III.
THE
45
COUPLING
DRESSED
QUASI-PARTICLE
particle
(2.4) and (2.7), a and single particle & , the dressed QP QP energy will be
ENERGY
The self-energy of t’he Bogoliubov QP has been discussed in detail in a previous paper (8), but here only the second order corrections shown in Fig. 2 will be treat,ed. These corrections have previously been considered by Tomasini (9)’ who emphasizes that both terms must be included. The dressed QP energy & is defined as the solution t’o the equation ii = hll(k, k) -
z (k, Lk)
(3.1)
where Z (Ic, &) is the self-energy and t’he function h,,(k, Ic) is the kinet#ic energy of the free QP (8). The self-energy Z (Ic) = Z (k, &) is the sum of the two parts shown in Fig. 2. The contribut,ion front Fig. 2 (a) ii”’ 4 2, (k) where
= -?i
c
144 1hdo(k123) l’A(123k)
(3.2)
h is defined in Eq. (2.2) and occurs because there are four QP Ixesent The contribution front Fig. 2 (b) is4
simultaneously.
&(k)
=
>g
c
18 /
=
(b
ha1(123k) I2 Z( 123/G)
(3.3)
where Z is defined as X(1234)
+
Ez +
ts -
EJ1
(3.4)
* I wish to apologize to Professor Tomasini for omitting a reference to his important paper in my paper of ref. 8. 3 The factor 2 appears in Eq. (3.2) because in ref. 8 the contribution from the diagram shown in Fig. 3 was neglected. When the particles are uncorrelat,ed however, the contribution is a term of the type of Fig. 2 (a). 4 The factor f< is included in Eqs. (3.2) and (3.3) because of over counting in Eq. (1.4) of ref. 8. I want to thank Professor Henley for suggesting this to me.
46
KOBE
FIG.
3. A diagram
which
in second
order
contributes
to the self-energy
in Fig.
2 (a)
The function ,S occurs in Eq. (3.3) because there are at most three quasi particles present in Fig. 2 (b) simultaneously. If the vertex functions hll , h40, and ha1 (10) are substituted into Eqs. (3.1), (3.2)) and (3.3) the following equation is obt,ained for the dressed QP energy ,. f,(P) 2,(h) (3.5) Ek = 2(uv)A + u& es - Vk ek where the dressed energy gap & is given by iik = AA - $6 c
[A( 123k) + Z(123k)]f(
123k)(uv)z
(3.6)
The dressed energy gap & in Eq. (3.6) differs from the renormalized energy gap & of Eq. (2.4) in that A has been replaced with S$( A + z). The function f is the same as in Eq. (2.5). Since the observed energy gap is the minimum energy of a two QP state, it is Eq. (3.6) that should be taken as half the observed energy gap. The coefficients in the canonical transformation are however given by Eqs. (2.10) and (2.11). Since 21.3’is the probability that the QP is a particle, the dressed single particle energy is L(P) Ek
= ck - $5 c
[Z( 123k)uZ2 -
A(123k)z$]f(1231c)
(3.7)
that the QP is a hole is VA’, and so the dressed single hole energy
The probability is
a(h)
El;
=
Q
-
$2
c
[A(123k)u;
- x( 123k)v,*]f(
12%)
(3.8)
It is not surprising that there are different expressions for the dressed particle and hole energies, since the particles and holes do not, have t,he same environment. Now to obtain the renormalized gap Ak and the single particle energy Ek it is necessary to substitute Eq. (3.5) into Eq. (2.2), and Eq. (2.2) into Eqs. (2.4) and (2.7). Then two coupled integral equations are obtained for ik and & which can in principle be solved. The solution should then be substituted into Eq. (3.6) to obtain the dressed energy gap b, which is half the observed gap. The expression for 6, and & can also be substituted into Eq. (3.7) and (3.8) to obtain the
ENERGY
GAP
AND
PARTICLE
ENERGY
47
COUPLING
dressed single particle energy 2:” and single hole energy in?). This procedure is very complicated, but in principle should be followed. It is highly desirable to make some approximations to simplify the two coupled equat,ions in Eqs. (2.4) and (2.7). Henley and Wilet)s (5) have assumed that t4 = e?F = 0 in Eqs. (2.2) and (3.4) so that A = Z.’ This seems like a good approximation in the small gap limit, but if the gap is large this approxinmt)ion can be questioned. However, in Henley and Wilet’s approximation the dressed and renormalized gaps are the same, & = & , and the dressed single particle and -(P) _- Ek .(k) = hole energies are equal to the renormalized single particle energy, Q E/;. Therefore, from Eq. (3.ri) and Eqs. (2.10) and (2.11) we have in this approximation t-1; = dik2 which
is a much simpler function IV.
SINGLET
(3.9)
-t Zk2
of ik and & than the expression STATE
in Eq. (3.5).
INTERACTION
The equations of the last section are valid for any type of interaction. they will be rewritten for a singlet state intera&on (5) (12 1 V 134) = (k, , ka 1 V 1k, , kr),S(l,
2)S(4,
3)
Kow (4.1)
where S(l, 2) = &7,+&T- - 6,,-&,+
(4.2)
If Eq. (4.1) is substituted into Eys. (2.4), (2.5), and (2X), the result is )A( 12%)2(7LU)~ & = AI; - c;rS( 1231c
(4.3)
where j - 22),(UV)~
(4.4)
aud f,(1234)
= - ( (1 - 2 ( V ( - 34), j* (UV)1(UU)~ + Re (1 - 2 1V I - 34),(3 - 2 I I’ I - 14)s*v~ua2
(4.5)
Iu these expressions there is of course no summation over the spin indices, and the numbers j = 1, 2, 3 stand for kj only. Equation (4.3) is the same as that used by Henley and Wilets (1) . If Eq. (4.1) is substituted into Eq. (2.7) for ik and the sum over the spin 5 I would like to thank to me. I am also grateful prior to publication.
Professor to him
Henley for pointing and Professor Wilets
out the nature of this approximtion for sending me copies of their work
48
KOBE
carried out, the result is & = ek - p-2~(k-2lVl-2k),vz + ~fs(123k)A(123Ic)(u:
- vz’)
(4.6)
The equations of Section III can also be written for the singlet state. Equation (3.6) for 8, for the singlet state is the same as Eq. (4.3) except that A should be replaced with >h( A + Z). The Eqs. (3.7) and (3.8) are the same except that f should be replaced with 2fs, and the sum is only over the momentum. In the next section, the interaction will further be restricted to a “shell” type interaction. V. THREE
DIMENSIONAL
“SHELL”
TYPE
INTERACTION
In order to gain some insight into the significance of the coupling between the energy gap and the single particle energy, the two coupled integral equations will be solved for a three dimensional “shell” type interaction. This interaction is defined as (k, , k, I V I k, , k4)s =
(Q--lfSdk + kz - k - k4) o othfrwl;ekj I - IcF 1 < w ?
for
j = 1, 2, 3, 4
(5.1)
where fi is the volume of the system, which will be taken as unity for the rest of the discussion. The constant w is the half width of the shell region around the Fermi momentum kF . In order for the interaction to be just in a shell region the condition w << kF must be satisfied. The Kronecker delta &,(k) is unity for zero argument, and zero otherwise. A. THE
ENERGY
GAP EQUATION
If Eq. (5.1) is substit’uted into Eq. (4.3) for the energy gap, the result can be written as & = i5 + tik
(5.2)
for 1/ k I - kF 1 < w and zero otherwise. The terms on the right side of Eq. (5.2) correspond to the respective terms in Eq. (4.3). It will be assumedhere that j & 1 << A so that 2 can be considered essentially as the energy gap, and 6k is only a small perturbation. If Eqs. (2.10), (3.9), and (5.2) are substitut’ed into Eq. (4.3), and if the sums are transformed to integrals, we obtain 1 = (2r)--3f/
dp(2EJ’Mp)
+J
(5.3)
where the function e1restricts the integral to the shell region and is defined as
em = e(w- j 1P1- k 1)
(5.4)
ENERGY
GAP
AND
PARTICLE
ENERGY
49
COUPLING
where 8 is the step function which is unity for positive argument and zero otherwise. The function J depends on the effectrive mass and energy gap in general, and is defined as
(5.5) -fdp,
P + q -
k, q, W&;;’
F&k
Equation (5.3) must be solved self-consistently for t’he energy gap &. However, first the effective massmust. be determined. B. THE
EFFECTIVE
MASS
The effective massm* is defined as ii = (lc2 - k~)/2wz*
(5.6)
where the dressedsingle part’icle energy is given in Eq. (4.6). If Eq. (5.1) is subsMuted into Eq. (4.6), and t’he chemical potential p is chosen t’o eliminate constant’ terms, the result’ for the effective massrat’io /3is p = d/m
= 1 - Ik
(5.7)
The int.egral Ik is defined as Ik = (27p
11 dp dqh(p, p + q - k, q, Wf.dp, p + q - k 9, k)tI;:r-k
(5.8)
The effective mass of course should be independent of k, but Ik is a slowly varying funct’ion of k. Thus, we could, for example, set k = kr and average over the Fermi surface. However, in order to make the treatment of the effective mass more harmonious with the treatment of the energy gap, another type of averaging will be used. The average value of Ik will be I=
s
dd2&c)-‘h(k)Ik
/I
W2hc-‘Nd
(5.9)
where the weighting factor (2&-l has been used in the shell region, and the denominator is for normalization. Thus the edgesof the shell region are not weighted as much as the Fermi surface. The reason for defining the somewhat, unusual average in Eq. (5.9) is that now Eqs. (5.3) and (5.5) can be used to give I = J/(J
- 1)
(5.10)
so t.hat Eq. (5.7) becomes fl = (1 - J)-’
(5.11)
Because of this simple relat’ionship between the effective mass ratio p and J,
50
KOBE
Eq. (5.3) can easily be solved for the energy gap. After performing and solving for 2\ we obtain
the integral,
i = A”/p sinh (l/g@‘)
(5.12)
where here A0 = w/c&
(5.13)
and the coupling constant g is defined as g = 2(27r-‘frnkF
(5.14)
Equation (5.12) is of the sameform as the BCS solution, but has an extra factor of /3 in the argument of the hyperbolic sine. However, in this treatment ,B t.he effective massratio is a function of the coupling constant g. C. VARIATION
OF THE EFFECTIVE
MASS WITH COUPLING
CONSTANT
If Eq. (5.1) is substituted into Eq. (4.5), and the result is substituted into the int’egral J in Eq. (5.5), we obt’ain
J = Ghr>-“f” jjj
dk dp
&(G
&,+q+J%p,
P + q + k, q, k)
(5.15)
.[(uv),(uu),
~P2uq21’3dp + q + k)&(p)&(q)&(k)
-
It is the terms with p + q + k which cause difficulty in Eq. (5.15), so it will be assumed that w is so small that we can set, 1p + q + k 1 s kF . With t.his assumption, the integral J can be written as J = M~3g3F(cu)
(5.16)
where the function F is defined as F(i)
= jjja
--adJ: dy
ddl
+
Cd.?
-[(x2 + 1)(?J2 +
+
1 -
s)(&Fq
1)(x2 + +
(y2 +
+
l)]-““[(x 1)1’2 +
+
?J)l (5.17)
1y2
(x2 +
1y2
+ 11-l
and CLis defined as a = A’/&3
= sinh (l/g@)
(5.18)
from Eq. (5.12). The constant M is related to a geometrical factor M = (27r-32-6 jjj
di dfi d&&.ij
+ $i; + 4-h + 1)
and has a value between 0 and $$j. The unit vector in the p direction is $ and $9 is the element of solid angle. In the strong coupling limit g >> 1 the gap is large and so’OL<< 1. Therefore
ENERGY
GAP
AND
PARTICLE
F( 0) in Ey. (Ti.17) can be evaluated,
ENERGY
51
COUPLING
and the result for the effective mass ratio is
p = [l + P$j@q-‘]-’
(5.19)
Thus, as g + 00, p + 1 and the second order correction does not change the BCS solution in this limit. In the weak coupling limit, g << 1, the funct,iou F(a) can be regarded as a slowly varying function sinre its mgument is extremely large, LY>> 1. Therefore the effective mass ratio is p = [I - Mp3q”F]-’
(5.20)
so that p -+ 1 as g ---f 0 as expected. Thus the secoud order correction is riot, iml)ortant’ in t,his limit. However, it, must, not be concluded t’hat the second order correction is never important. In the intermediate coupling case, it will contribute and cause a deviation from t’he BCS result, as Henley and Wile& have shown (1). For ot,her types of interactions it has been shown to be even more important. Eyuation (5.11) for the effectjive mass is a complicated equation and can best be solved by pert.urbat,ion theory. In general, if J involves the energy gap as it. would for sonle pot.entials, t.he BCS value of the energy gal) can be used to detcrruiue au approximate effective mass. An it,eration procedure can be used to obtaiu a self-consistent result. VI.
CONCLUSION
The previous section outlined a procedure for solving the two coupled iutegral equations satisfied by the energy gap and single particle energy which could be extended to other potentials. However, for other potent’ials, the effective mass alq~roximatjion is not necessarily a good one. As an example, consider the simple Yamaguehi potential used by Henley and Wilets (1) (klk2 1 V ) k,k,),
= li’Ld(Z?r)“X~(;~
/ $ - kz /)w(;i
/ kt - k4 1)
(6.1)
where w(k)
= (k2 + by
(6.2)
aud b aud X are constants. When t)his potential is substituted into Ey. (2.8) for the single particle energy dressed in the Hartree-Pock approximation, and the energy gap is neglected, the result is Ek = f?(k”
- kF2)/2m
+ 2??d[N(k)
- N(kF)]
where N(k)
= -32aX[-
(6.3)
k&e2 + kp’ + 4b2)-’ + (k” + 4b2)-“2 tan-’
kF(k2 + 462)-1’2]
(6.4)
52
KOBE
It can be seen that to replace N(k) with a parabola is not an especially good approximation, especially for large momentum. Since the nuclear energy gap is an especially sensitive function of t,he effe&ive mass, it would be interesting to repeat the calculations of Kennedy, Henley, and Wilets (3) with single particle energies calculated from the potentials used. In this way there would not be any free parameter in the t,heory (except the Fermi momentum), and the energy gap could be determined for normal nuclear density. Since the second order correction t’o the energy gap is important, the second order correction to the single particle energy should also be included. The effect of this coupling using some realistic potentials will be the subject of another paper. ACKNOWLEDGMENTS The early stages of this work were carried out while I had a Fulbright lecture grant in the Republic of China during the 1963 to 1964 academic year. I would like to express my appreciation to the United States Educational Foundation in the Republic of China for the opportunity of receiving the grant. Their concern for the happiness and well being of the grantees went far beyond their official responsibilities. To the faculty, staff, and students at the National Taiwan University, Taiwan Normal University, and National Tsing Hua University, I would like to express my deepest appreciation for their hospitality, friendliness, and help to me during my year in Taiwan. This work was concluded at the Quantum Chemistry Group in Uppsala. I would like to express my sincere gratitude to Professor Per-Olov Lijwdin for the opportunity to come to the Quantum Chemistry Group, for a fellowship to attend the International Summer Institute in Quantum Chemistry, and for his warm hospitality, encouragement and interest in this work. I would like to thank Drs. Fukashi Sasaki and Vedene Smith, Jr. for many stimulating discussions during the course of this work. RECEIVED:
April 12, 1965 REFERENCES
1. E. M. HENLEY AND L. WILETS, Phys. Rev. Letters 11, 326 (1963). 2. J. BARDEEN, L. N. COOPER, aND J. R. SCHRIEFFER, Phys. Rev. 108, 1175 (1957). 3. G. F~NO AND A. TOMASINI, Nuovo Cimento 18, 1247 (1960); G. FANO, M. S.~VOIA, AND A. TOMASINI, Nuovo Cimento 21,854 (1961) ; V. J. EMERY AND A. M. SESSLER, Phys. Rev. 119, 248 (1960); R. KENNEDY, L. WILETS, AND E. M. HENLEY, Phys. Rev. 133, B1131 (1964). 4. N. N. BOGOLIUBOV, Zh. Eksperim. i Teoret. Piz. 34, 58 (1958); Transl. Soviet Phys. -JETP 7, 41 (1958); Nuovo Cimento [lo] 7, 794 (1958). 5. E. M. HENLEY AND L. WILETS, Phys. Rev. 133, B1118 (1964). 6. D. H. KOBE, Preprint No. 145, Quantum Chemistry Group, Uppsala, Sweden (1965); Phys. Rev., in press. 7. V. V. TOLM.~CHEV AND S. V. TISBLIKOV, Zh. Eksperinz. i Teoret. Fiz. 34.66 (1958); Transl. Soviet Phys. -JETP 7, 46 (1958). 8. D. H. KOBE, Ann. Phys. (N.Y.) 28, 400 (1964). 9. A. TOMASINI, Nuovo Cimento [lo] 20,963 (1961). 10. D. H. KOBE AND W. B. CHESTON, Ann. Ph,ys. (N.Y.) 20, 279 (1962).