Molecular Physics
An International Journal at the Interface Between Chemistry andPhysics

ISSN: 0026-8976 (Print) 1362-3028 (Online) Journal homepage: http://www.tandfonline.com/loi/tmph20

Role of the ward identity and relevance0 0of the G W approximation in normal andsuperconducting states
Yasutami Takada
To cite this article: Yasutami Takada (2016): Role of the ward identity and relevance of00

the G W approximation in normal and superconducting states, Molecular Physics, DOI:
10.1080/00268976.2015.1131860
To link to this article: http://dx.doi.org/10.1080/00268976.2015.1131860

Published online: 08 Jan 2016.

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Download by: [University of Tokyo]

Date: 08 January 2016, At: 21:33

MOLECULAR PHYSICS, http://dx.doi.org/./..

SPECIAL ISSUE IN HONOUR OF ANDREAS SAVIN

Role of the ward identity and relevance of the G0 W0 approximation in normal
and superconducting states
Yasutami Takada
Institute for Solid State Physics, University of Tokyo, Kashiwa, Japan

Downloaded by [University of Tokyo] at 21:33 08 January 2016

ABSTRACTOn the basis of the self-consistent calculation scheme for the electron self-energy  with the useof the three-point vertex function  always satisfying the Ward identity, we find that the obtained
quasi-particle dispersion in the normal state in gapped systems, such as semiconductors, insulators,
and molecules, is well reproduced by that in the one-shot GW (or G0 W0 ) approximation. In calculating
the superconducting transition temperature Tc , we also find a similar situation; the result for Tc in thegauge-invariant self-consistent framework including the effect of  satisfying the Ward identity is different from that in the conventional Eliashberg theory (which amounts to the GW approximation forsuperconductivity) but is close to that in the G0 W0 approximation. Those facts indicate that the G0 W0approximation actually takes proper account of both vertex and high-order self-energy corrections
in a mutually cancelling manner and thus we can understand that the G0 W0 approximation is better
than the fully self-consistent GW one in obtaining some of physical quantities.

1. Introduction
A non-perturbative self-consistent approach to the electron self-energy  was provided in 1965 by Hedin [1] ina closed set of equations, relating  with the one-electron
Green’s function G, the dynamic screened interaction W,
the polarisation function  and the vertex function .
This is a formally exact formulation, but it is difficult toimplement this scheme as it is, because we cannot determine the electron–hole irreducible interaction I,˜ a central quantity in the Bethe–Salpeter equation to calculate , through its original definition using a functional
derivative, I˜ = δ/δG. Thus we are forced to adopt someapproximate treatments such as the GW approximationin which  is taken as unity.
CONTACT Yasutami Takada©  Taylor & Francis

takada@issp.u-tokyo.ac.jp

ARTICLE HISTORY

Received  October Accepted  November KEYWORDS

Superconductivity; vertex
correction; ward identity; GW
approximation; Eliashberg
theory; gauge-invariant
self-consistent (GISC) theory

For more than two decades, successful calculations
have been done for molecules, clusters, semiconductors
and insulators in the one-shot GW (or G0 W0 ) approximation [2–7], but this is usually regarded as a too primitive approximation, mostly because it is, in general, nota conserving approximation in the sense of Baym andKadanoff [8,9]. In contrast, the fully self-consistent GW
approximation obeys the conservation laws related to themacroscopic quantities like the total electron number. In
recent years, this self-consistent calculation has become
feasible, but we are led to a puzzling conclusion that theexperiment on quasi-particle properties in semiconductors and insulators is much better described in the G0 W0approximation than in the GW one [10,11]. In atoms and2

Y. TAKADA

G=
W
W=

V
V

1

∼

-1
0

G=
WI

∼

G

G

W
-1

-1

G -G
-1 -1
G0 -G0

-

∼

W=

1
-1
0

WI

-1

V

-

WI
WI

-1

G -G
-1 -1
G0 -G0

WI

WI

WI

WI

Downloaded by [University of Tokyo] at 21:33 08 January 2016

Figure . Self-consistent iteration loops to determine the electron self-energy (a) in the original GW and (b) in its improved version or
the GW̃ WI , respectively.

molecules, the situation is less clear, but in many cases theG0 W0 approximation gives better results [12–15].
In metals, on the other hand, neither G0 W0 nor GW
works very well [16], requiring us to include  in someway in treating systems possessing gapless excitations.
Some schemes have already been proposed for this purpose [17,18], but they do not satisfy the Ward identity (WI), an exact relation between  and  due togauge invariance representing the local electron-number
conservation [19]. In 2001, based on general consideration on algorithms beyond the Baym–Kadanoff one [20],a scheme was proposed to incorporate  in the GW
approximation with automatically fulfilling the WI [21].
This GW scheme with the use of the information on thelocal-field factor in the electron gas [22] for determining I¯ (see Figure 1(a)) succeeded in obtaining the correct quasi-particle behaviour in simple metals. In 2011,
this scheme was further improved [23] into the GW̃WI
scheme (see Figure 1(b)) so as to avoid the problem of
dielectric catastrophe associated with the divergence ofthe compressibility κ at the electron density specified byrs equal to 5.25 in the electron gas and concomitantly thatof the static  in the long wavelength limit [24,25]. This
scheme is numerically confirmed to provide non-negative
one-electron spectral functions A(p, ω) for the homogeneous electron gas at least for rs ࣘ 8 without any special
treatments to impose on the positivity of A(p, ω), contrary to a recently proposed scheme [26].
In the former part of this paper, we study the quasiparticle properties in the normal state to find that thequasi-particle dispersion self-consistently obtained in the
GW̃WI scheme for semiconductors and insulators isessentially the same as that in the G0 W0 approximation,
implying that G0 W0 is superior to GW in the sense thatfor the systems with gapful excitations, it actually takes
proper account of the mutual cancellation between vertex and high-order self-energy corrections. This observation resolves the above-mentioned long-standing puzzle
on GW versus G0 W0 . Here we emphasise that this cancellation is considered up to infinite order as a whole withspecifying the assumptions needed in the cancellation, in
sharp contrast with the claims of a similar kind in thepast [27–29]; they were inferred from the behaviour of
low-order terms in perturbation expansion for metals.
In the latter part of this paper, we study a similar problem in the superconducting state by the calculation of its
transition temperature Tc in the phonon mechanism inboth the conventional Eliashberg theory [30–34] and thegauge-invariant self-consistent (GISC) method [35,36]
which correspond, respectively, to the GW and GW
schemes for calculating the normal-state properties. Evenfor the case of weakly correlated and weakly coupled
superconductors such as Al, we find that the calculated Tc
in GISC is different from that in the Eliashberg theory, butclose to that in the G0 W0 approximation. We discuss theimplication of this result from various aspects, including
the importance of determining the Coulomb pseudopotential μ∗ [37] from first principles and a proposal of thesuitable functional form for the pairing interaction kernel in the density functional theory for superconductors(SCDFT) [38–41].

2. Normal state
For systems with translation symmetry in which momentum p is a good quantum number, the exact Hedin’s relations can be explicitly written in the following way: theDyson equation relates G(p) with (p) through G(p)−1= G0 (p)−1 − (p) with p, a combined notation of p, spin
σ and fermion Matsubara frequency iωp ࣕ iπT(2p + 1)
at temperature T with an integer p [42]. The bare Green’s
function G0 (p) is written as G0 (p) = (iω p −ε p )−1 with
ε p the bare one-electron dispersion. The Bethe–Salpeter
equation determines (p + q, p) by(p+q, p) = 1 +


p


˜
I(p+q,
p; p +q, p )G(p )× G(p +q)(p +q, p ),(1)

MOLECULAR PHYSICS

where q is a combined notation of q and boson 
Matsubarai2πTqfrequency iωq ࣕ
withq an integer and p represents the sum T ω p p σ  . By using (p + q, p), we
can give (q) and (p) by(q) = −G(p+q)G(p)(p+q, p),(2)

G(p+q)W (q)(p+q, p),(3)p

(p) = −Downloaded by [University of Tokyo] at 21:33 08 January 2016

q


 
respectively, withand W (q) =
q =T
ωq
q
V (q)/[1 + V (q)(q)], where V (q) is the bare Coulombinteraction 4πe2 /q2 .
In [21], the concept of ‘the ratio function’ was introduced to obtain a good approximate functional form for(p + q, p) satisfying the WI. By exploiting this concept,
we have explored an exact functional form for (p + q,p) and succeeded in obtaining the following form:
(p+q, p) =  (a) (p+q, p) (b) (p+q, p),(4)

where  (a) (p + q, p) and  (b) (p + q, p) are, respectively,
defined as
˜ p+q,p(q), (a) (p+q, p) ≡ 1 − I (b) (p+q, p) ≡˜ p+q,p ≡ −
I



(9)

WI (q),¯
1 + I(q)
WI (q)(10)

with WI (q), defined by
WI (q) = −G(p+q)G(p)WI (p+q, p). (11)pThen, we can rewrite (p) in Equation (3) intoG(p+q)W̃ (q)WI (p+q, p),

(12)

q

˜
I(p+q,
p; p + q, p )× G(p )G(p +q)(p +q, p )/(q),G(p+q)−1 − G(p)−1,G0 (p+q)−1 − G0 (p)−1¯
we obtain (p+q, p) = [1 − I(q)(q)]
WI (p+q, p), a
result given in [21], leading to the GW scheme in Figure 1(a).
By putting this form of (p + q, p) into Equation (2),we find that (q) is written as

(p) = −p p+q,p ≡WI (p+q, p) ≡(5)

˜ p+q,p, and a difference in the selfHere an average of I,˜ Ienergy,  p + q, p , are, respectively, introduced by

becomes apparent; the former provides (p + q, p) satisfying the WI irrespective of the choice of I,˜ while thelatter does not.
Physically I˜ takes care of exchange and correlation
effects in (p + q, p) and it is well known that thisphysics can be captured by the local-field factor forthe homogeneous electron gas or by the Jastrow factor
for inhomogeneous systems. In either way, these effects
are well described in terms of a function depending
only on the inter-electron distance, which justifies to˜
assume that I(p+q,p; p +q, p ) depends only on q to¯˜
If this assumption iswrite I(p+q, p; p +q, p ) = I(q).
˜ p+q,p =adopted in our exact framework, we obtain I¯ and  p + q, p = 0. Then, by defining  WI (p + q, p)I(q)by(q) =

G(p + q)−1 − G(p)−1.
G0 (p+q)−1 − G0 (p)−1 −  p+q,p
(6)

3

(7)

˜
I(p+q,
p; p +q, p )[G(p )−G(p +q)],p

(8)
as functionals of G and I.˜ If I˜ is exact,  (a) (p + q, p) is
nothing but (p + q, p) in Equation (1), as can easily be
˜ p+q,p, and  p + q, p isseen from the very definition of Ireduced to (p + q) − (p), leading to  (b) (p + q, p) = 1.
Thus, Equation (4) provides the same (p + q, p) as thatin the Hedin’s exact theory. In reality, the exact I˜ is notknown and we have to employ some approximate I,˜ inwhich an advantage of Equation (4) over Equation (1)¯with W̃ (q) ≡ V (q)/{1 + [V (q) + I(q)]
WI (q)}. Combining these results, we can construct the GW̃WI scheme
as shown in Figure 1(b). This scheme is equivalent to
the GW one in obtaining (p), but computational costs
are much reduced by the calculation of (q) throughEquation (10) via WI (q), because Equation (11) can
be cast into a form convenient for numerical calculations as
WI (q) ≡ WI (q, iωq ) = n(p+q)−n(p)pσiωq −ε p+q +ε p

, (13)

+
where n(p) [= T ω p G(p)eiω p 0 ] is the momentum distribution function in the interacting system. Note that
this expression very much resembles the one for the

4

Y. TAKADA

polarisation function in the random-phase approximation (RPA) 0 (q), which is given by0 (q) = − f (ε p+q )− f (ε p )G0 (p)G0 (p+q) =,iωq −ε p+q +ε p
p
pσ
(14)

where f(ε) is the Fermi distribution function or the
momentum distribution function in the non-interacting
system.
With the use of Equation (9) and introducing ε̃ p byε̃ p = ε p −Downloaded by [University of Tokyo] at 21:33 08 January 2016q

(p) = −W̃ (q),iωq −ε p+q +ε p(15)

 W̃ (q)G(p+q)
qiωq −ε p+q +ε p

.

(16)

¯
On the assumption of I(q)= 0, this equation coincides
with the one for obtaining the asymptotically exact G(p)in a neutral Fermi system such as the one-dimensional
Tomonaga–Luttinger model [43,44] or higher dimensional models with strong forward scatterings [45]. This
coincidence clearly demonstrates the intrinsically nonperturbative nature of our framework.
¯ is at our disposal, but Equation (10)In principle, I(q)¯suggests us to choose I(q)= −G+ (q)V (q) with G+ (q)the local-field factor. Note, however, that the meaningof G+ (q) here is different from the ordinary one that isdefined with respect to 0 (q) instead of WI (q). Fortunately, we already know a good form for G+ (q) with taking account of this difference, which is Gs (q) in [22], satisfying the exact limit due to Niklasson [46] as |q| → ∞.
The self-consistent results for the homogeneous electron¯ are given in [23].
gas up to rs = 8 with this choice of I(q)In the crystalline case, each quantity involved in the
GW̃WI scheme should be represented in the matrix
form with respect to the reciprocal lattice vectors {K}.
For example, G(p) is a matrix composed of the elements{GK,K  (p, iω p )} with p a wave vector in the first Brillouinzone. For some quantities, we need to add the conversion
factors transforming from the plane-wave basis to the
Bloch-function one in considering the matrix elements;for example, 0 K,K  (q) is given as0 K,K  (q) =W0 (q)+ γ (p)G(p)−1 ,−1 −G (p)−1
G
(p+q)
0
0
q
(18)

we can rewrite our scheme into an integral equation todetermine G(p) through(iω p − ε̃ p )G(p) = 1+

where |np is the Bloch function for the nth band
[47,48].
With this understanding, we can apply the GW̃WI
scheme to semiconductors and insulators possessing a
gap in the electronic excitation energies. Then, without
detailed computations, the self-consistently determined
quasi-particle energy E p in this scheme is found to be wellapproximated by that in the G0 W0 approximation, as weshall explain in the following.
¯= 0 forLet us assume that WI (q) = 0 (q) and I(q)the time being. Then, we may rewrite Equation (12) as f (εn p+q )− f (εnp )iωq −εn p+q +εnp
nn pσ
× np|e−i(q+K )·r |n p+qn p+q|ei(q+K )·r |np , (17)with W0 (q) ≡ V (q)/[1 + V (q)0 (q)] and γ (p) adimensionless function, defined by
γ (p) ≡ γ (p, iω p ) = G(p + q)W0 (q)qiωq −ε p+q +ε p

.

(19)

The quasi-particle dispersion E p is determined by thepole of the retarded one-electron Green’s function
GR (p, ω), or GR (p, E p )−1 = 0, amounting to E p = ε p + R (p, E p ), where we obtain the ‘on-shell’ retarded selfenergy as R (p, E p ) = −q

W0 (q),iωq −ε p+q +ε p

(20)

by analytic continuation of (p) in Equation (18). In
deriving Equation (20), we have paid due attention to theconvergence of γ R (p, E p ) in gapful systems. In fact, pro¯
vided that I(q)= 0, ε̃ p and the integral in the right-handside in Equation (16) are, respectively, reduced to E p and
γ (p), leading to the behaviour of GR (p, ω) for ω near E p
as
GR (p, ω) ≈1+γ R (p, E p ).
ω + i0+ − E p(21)

For comparison, let us consider the self-energy in
the G0 W0 approximation, which is given by  0 (p) =−ࢣq G0 (p + q)W0 (q). By analytic continuation iω p →+p +i0 , we obtainW0 (q)iωq −ε p+q +ε pq
1−W0 (q, ε p+q −ε p )2 q
ε p+q − ε p
ε p+q× coth− tanh. (22)2T
2T

0R (p, ε p ) = −MOLECULAR PHYSICS

Because the transition p+q → p involved in Equation (22) is relevant only for the interband transition,|ε p+q −ε p| is always larger than Eg the energy gap. At low
T, the chemical potential μ lies at the centre of the bandgap, indicating that |ε p+q | ≥ Eg /2. These two facts allow us
to safely neglect the contribution from the second sum inEquation (22), as long as T Eg . Thus, we may write E p0
the quasi-article dispersion in the G0 W0 approximationas
E p0 = ε p + 0R (p, ε p ) = ε p −Downloaded by [University of Tokyo] at 21:33 08 January 2016q

W0 (q), (23)iωq −ε p+q +ε pleading us to conclude that E p0 = E p. Note, however, thatthe spectral weight z p [= (1−∂0R (p, ω)/∂ω)−1 |ω=ε p ] isdifferent from 1+γ R (p, E p ).
In the literature, E p0 is sometimes evaluated as E p0 =ε p + z p0R (p, ε p ) and there is a controversy as to whether
this z p should be included or not. As previously discussed
in detail [29], we consider it better not to include z p sothat the vertex corrections beyond the RPA are properly
included, together with higher order self-energy terms in
a mutually cancelling manner. In fact, our present resultof E p0 = E p without this factor z p indicates that this feature of mutual cancellation reaches far up to infinite orderin semiconductors and insulators.
Finally, we comment on the two assumptions as wellas other related issues. (i) The difference between WI
and 0 arises only from that between n(p) and f (ε p ).
In usual semiconductors and insulators, the valenceelectron density is high; for example, rs = 2 for Si. Nown(p) in a metal at such rs does not deviate much fromf (ε p ) except for the states near the Fermi level, as shownin Figure 4 in [23], but those states are absent fromthe outset in these gapful systems. Thus, n(p) is close tof (ε p ), leading to WI ࣈ 0 . (ii) Justification of I¯ = 0
has already been done by numerical studies in [3], inwhich I¯ in our scheme is critically assessed in terms of
Kxc the density derivative of the Kohn–Sham exchange–
correlation potential. From an analytic point of view, it isenough to note that the basic processes to contribute toI¯ are related to the interband electron–hole interactions,in which |q| for principal processes is of the order of |K|,
making V (q) very small and G+ (q) reach its asymptotic
constant. Thus, the effect of I¯ is weak in semiconductors
and insulators. (iii) The inherent problem in the G0 W0approximation is the dependence of the results on thearbitrary starting basis. From our perspective, the dependence originates from the degree of satisfying the above
two assumptions. Thus, if the gap energies (or the vertical ionisation potentials (IPs) for atoms and molecules)
are large enough, the dependence becomes small and5

the results in the G0 W0 approximation from any starting point are close to the experimental values, as seen for
IPs of He and Ne in Table V in [15] and of CO and N2
in Table I in [13]. (iv) We have shown that the G0 W0approximation is good for obtaining the quasi-particle
dispersion for gapful systems, but there is no guarantee
for other physical quantities such as the line shape ofA(p, ω). On the other hand, the framework stipulatedby the set of equations (Equations (2)–(8)) is, in principle, exact and can provide accurate results for the physicalquantities directly derived from G(p). Thus, the physicalrequirements such as the non-negativity of A(p, ω) areautomatically satisfied, as long as we choose an appropri˜
ate approximation to I(p+q,p; p +q, p ) which is a single quantity in the framework to control the accuracy of
the results. We can determine a proper functional form˜
for I(p+q,p; p +q, p ) by the use of the information on
the local-field factor or by perturbation expansion fromeither weak or strong coupling limit, supplemented by the
information obtained by quantum Monte Carlo simulations, if available.

3. Superconducting state
In the conventional phonon mechanism of superconductivity with spin-singlet s-wave Cooper pairing, we can formulate the problem in much the same way as in the normal state, if we employ the Nambu representation [49].
We obtain the rigorous expressions for (q) and (p) as(q) = −Tr τ3 G(p + q)(p + q, p)G(p) , (24)p(p) = −τ3 G(p + q)(p + q, p)W (q),

(25)

q

respectively. They are very similar to Equations (2) and(3), but (p), together with the scalar vertex function (p+ q, p), is now a 2 × 2 matrix. Here, τ i s for i=1, 2, and 3
are the usual 2 × 2 Pauli matrices and W(q) is the effective
electron–electron interaction. For the electron–phonon
coupled system, W(q) is exactly obtained as
W (q) =

V (q) + Vph (q),1 + [V (q) + Vph (q)](q)(26)

where Vph (q) is the bare phonon-mediated electronelectron interaction, given by
Vph (q) =ν|gqν |22qν
,(iωq )2 − qν 2

(27)

6

Y. TAKADA

with bare electron–phonon coupling gqν and bare phonon
energy qν for the νth phonon. In this Nambu representation, the WI is written in the form of

−1



=

N(ε)dε, with N(ε)

piω p − iω p (p , p) − (p − p)·(p , p)
= G(p ) τ3 − τ3 G(p)−1 ,the introduction of the cut-off energy ωc ( εF ) in considering the sum over momentum p in such a way as

≡

(28)



N(0)
0

δ(ε − ε p ) =pwhere (p , p) is the vector vertex function. Without lossof generality, the 2 × 2 matrix (p) can be decomposed
into
(29)

W (iωq ) =with τ 0 the unit matrix. Then, we may rewrite Equation (28) into the form asμ∗ − λ(ωq )N(0)∞with λ(ωq ) ≡= [iω p Z(p ) − iω p Z(p)]τ3 − [ε̃(p ) − ε̃(p)]τ0+ φ(p )+φ(p) iτ2 ,(30)with ε̃(p) defined by ε̃(p) ≡ ε p + χ (p).
Now let us assume that the average phonon energy 
is much smaller than the Fermi energy εF of conduction
electrons, as is usually the case for superconductors in thephonon mechanism. In this situation, it is physically relevant to separate the purely Coulombic part from W(q) in
Equation (26) in such a way as
1
V (q)
+
1 + V (q)(q)[1 + V (q)(q)]2
Vph (q)×.
1 + Vph (q)(q)/[1 + V (q)(q)]

(31)

2
ωq 2 + 2

.

2
λ

∞

d α 2 F ().

(34)

0

The definition of the Eliashberg function and its calculated results on various superconductors are available
in the literature [50], but in recent years we can obtain
the results rather easily by using the packages for firstprinciples calculations such as Quantum Espresso [51]. In
Figure 2, we show an example of the calculated α 2 F(),which is obtained for fcc Al with the mesh of 48 × 48 ×48 for p and 16 × 16 × 16 for q in the first Brilouinzone.
Incidentally, under the condition of the momentumindependent interaction in Equation (33) and the0.8

In Equation (31), the second term represents the fully
screened phonon-mediated interaction, which is supposed to play a central role in bringing about superconductivity. Because this interaction works only in theenergy range of about , the purely Coulombic term,
which extends over the range of εF , is usually renormalised and reduced into the Coulomb pseudopotentialμ∗ which is supposed to work only in the range of about
 in the gap equation to determine Tc [37]. The actual
value for μ∗ will be determined phenomenologically inorder to reproduce the experimental value of Tc . Uponthese assumptions, W(q) is not zero only in the vicinityof the Fermi level. This situation can well be treated by(33)

Note that λ = λ(0) is the usual non-dimensional
electron–phonon coupling constant, and  is given by =W (q) =d α 2 F ()0(iω p − iω p )(p , p) − (p − p) · (p , p)(32)

where N(ε) is the electronic density of states per
one spin.
Traditionally, a further simplification is made byassuming that W(q) is independent of q. Then, W(q) =W(iωq ) is written in terms of α 2 F(), the so-called
Eliashberg function as

fcc Alλ=0.409
0.6 〈Ω〉 =27.51 meV
0.7

α2 F (Ω)

Downloaded by [University of Tokyo] at 21:33 08 January 2016

(p) = [1 − Z(p)]iω p τ0 + φ(p)τ1 + χ (p)τ3 ,for |ε| < ωc ,otherwise,

0.5
0.4
0.3
0.2
0.1
0

10

20
30
Ω (meV)

Figure . The Eliashberg function for fcc Al.

40

MOLECULAR PHYSICS

Downloaded by [University of Tokyo] at 21:33 08 January 2016(iω p − iω p )(iω p , iω p ) = [iω p Z(iω p )− iω p Z(iω p )]τ3 + φ(iω p )+φ(iω p ) iτ2 .

(35)This relation is derived by setting p = p in Equation (30) and indicates that (iω p , iω p ) contains the τ 2 component. As originally discussed by Nambu [49], thiscomponent is related to the phase-collective Nambu–
Goldstone mode. In superconductors, the energy of thismode is the plasmon energy ωpl , which is the energy scale
far beyond . Thus, we will omit this contribution to
the vertex function in the present treatment in which all
physical quantities beyond the energy scale  will be
neglected with the hope that due effects will be renormalised and included into the definition of μ∗ . There arealso discussions on the τ 1 -component in the vertex function, which is related to the amplitude-collective Nambu–
Goldstone mode [52,53], but it is known that this contribution does not change Tc [54], assuring us of neglecting it altogether in the calculation of Tc . Thus we obtain
(iω p , iω p ) as(iω p , iω p ) =iω p Z(iω p ) − iω p Z(iω p )τ3 ,iω p − iω p

(36)

for ω p = ω p . At ω p = ω p, we cannot use the WI todetermine it, but on general considerations, it is givenby [1 − δ(p)/δμ]τ 3 with the normal-state self-energy
(p) and μ the Fermi level, which is different fromlimω p →ω p (iω p , iω p ) in Equation (36). In this paper,
however, we shall use this limit, i.e. (iωp , iωp ) =[Z(iωp ) + ωp δZ(iωp )/δωp ]τ 3 , partly because no appropriate information on δ(p)/δμ is available and partly
because we know that this difference, whichappears only
in a single term in the infinite sum of T ω p , becomesvery important only when the small polaronic effect islarge [55].
By summarising all the above assumptions and considerations, we can derive a couple of equations, one
for Z(iωp ) and the other for φ(iωp ), to determine Tc byretaining only up to the linear order in φ(p) in Equation (25) in the following way:2T λ(ω p −ω p )(iω p , iω p )ωp ω 
p


ωc
−1
× tan
,
(37)
ω p Z(iω p )2T  (iω p )(iω p ) =[λ(ω p −ω p )−μ∗ ]Z(ω p ) ω  ω p
p


ωc
−1
× (iω p , iω p ) tan, (38)
ω p Z(iω p )Z(iω p ) = 1 +where (iωp ) is defined by φ(iωp )/Z(iωp ). We can define
Tc by the highest temperature at which non-zero (iωp )can be found. We shall call Tc obtained from theseequations a result in the GISC scheme. In the Eliashberg theory, Tc is determined by the solution of Equations (37) and (38) with taking (iω p , iω p ) = 1. Usually, the Eliashberg function α 2 F() is not renormalisedin the process of self-consistently determining Z(iωp ). In
this sense, the Eliashberg theory might be better regarded
as the ‘GW0 ’ approximation, rather than the GW one. If
we do not solve Equation (37) for Z(iωp ) but simply takeZ(iωp ) = 1 in the solution of Equation (38), the obtained
Tc may be considered as a result in the G0 W0 approximation.
By using the α 2 F() function in Figure 2, we have calculated Tc for fcc Al in three different schemes and shown
the results as a function of μ∗ in the Eliashberg theory(solid curve), the GISC (dashed curve), and the G0 W0(dotted curve) in Figure 3. Note that this is a typical example for the weakly correlated and weakly coupled superconductors. The cut-off energy ωc is increased up to 5in order to obtain the convergent results for Tc . As canbe seen in Figure 3, Tc in the most sophisticated GISC
scheme is very different from that in the Eliashberg theory but very close to that in the simplest scheme of the20

fcc Al15

Tc (K)

electron–hole symmetric situation as suggested in Equation (32), the level-shift function χ(p) in Equation (29)
is easily found to be zero. The renormalisation function
Z(p) and the gap function φ(p) are not zero but independent of the momentum variable p. Then, we can assume
that the scalar vertex function is also independent of
momentum variables and it must obey the relation, given
by

7

G0W0
GISC

10

GW(Eliashberg Theory)

50Exp. (1.18K)

0.05

0.10

0.15

μ∗

0.20

0.25

0.30

Figure . Superconducting transition temperature Tc for fcc Al asa function of the Coulomb pseudopotential μ∗ in three diﬀerent
calculation schemes in comparison with the experimental Tc of. K.

Y. TAKADA(a) 1.4

Z (iωp )1.3

fcc Al
T =1.18KGISC
GW(Eliashberg)(b) 1.0

1.2

GW(Eliashberg)μ∗ =0.144

0.5

GISC

μ∗ =0.219

0.0

1.1

-0.5
G0W0

1.0
0

fcc Al T =1.18K

Δ (iωp ) /Δ (iπT )82040 60 80 100
ω p (meV)(p) = −
μ∗ =0.236

-1.0
0

20

40 60 80 100
ω p (meV)G0 W0 , indicating that the vertex corrections included in
the GISC is mostly cancelled by the self-energy renormalisation corrections.
Because the experimental value of Tc for fcc Al is 1.18
K, we can determine the value of μ∗ for each scheme;
they are 0.144, 0.219 and 0.236 for the Eliashberg, the
GISC and the G0 W0 , respectively. Then, we can compare the results for Z(iωp ) and (iωp )/ (iπT) at T =1.18 K, as shown in Figure 4. Although the Eliashberg
and the GISC provide more or less the same Z(iωp ), theresults for (iωp )/ (iπT) are very different. We see thatas far as the superconducting properties are concerned,
the G0 W0 gives about the same quality of results as
the GISC.
Four comments are in order. (i) According to theMigdal’s theorem [56], one might expect that the Eliashberg theory gives approximately the same Tc as the GISC
for such weakly coupled conventional superconductors
because of the irrelevance of the vertex corrections. A
closer inspection of the Migdal’s proof reveals, however,that the theorem states only the irrelevance of the firstorder vertex corrections, but not for the first-order selfenergy corrections. If the irrelevance of the former corrections is true, the same must be true for the latter, giventhat the WI always holds. In this regard, the Migdal’s theorem actually proves the relevance of the G0 W0 approximation, which is indeed confirmed by the results in Figure 3. (ii) The preference of the G0 W0 approximation to
the Eliashberg theory applies only to the calculation of
Tc ; the normal-state property in a metal as representedby Z(iωp ) is better described by the latter, as can be seenin Figure 4(a). (iii) Since Tc in all the three schemes varies
very much with the change of μ∗ , those schemes are notgood enough for the first-principles calculation of Tc , aslong as μ∗ is given phenomenologically. Thus, we needto develop a scheme to determine μ∗ from first principles. The present author pursued such a scheme in theframework of the G0 W0 approximation [57], according


p

G 0W0

Figure . (a) Renormalisation factor Z(iωp ) and (b) the gap
function (iωp ) normalised by (iπT) calculated for fcc Al at T =. K in the three diﬀerent schemes.

Downloaded by [University of Tokyo] at 21:33 08 January 2016

to which the gap function is not considered as a function of the frequency variable but the momentum one andthe gap equation to give Tc is derived for the momentumdependent gap function (p) as


ε p
(p )
tanh
K p,p ,
2ε p
2Tc

(39)

where the pairing interaction K p,p is defined by
K p,p =∞0|ε p | + |ε p |2d 2
W (p− p , i),π
 + (|ε p| + |ε p |)2(40)

with W(q) given by Equation (26) in which (q) is
replaced by 0 (q). This scheme was successfully applied
to real materials such as the n-type SrTiO3 [58] andthe graphite intercalation compounds [59,60]. (iv) Firstprinciples determination of μ∗ is also made possible by
SCDFT [38–41], according to which the fundamental gap
equation is just the same as that in Equation (39), but
the pairing interaction is different from the one in Equation (40). Because the form in Equation (40) includes thecontribution from the plasmons [57] in a very naturalway [61,62], it is strongly recommended to employ Equation (40) for future implementation of SCDFT [63].

4. Conclusion
In summary, we have discussed the relevance of the G0 W0approximation in obtaining some of physical quantities inboth normal and superconducting states. Due to the existence of the WI, this approximation remains to be usefulin a much wider interaction range than one might naively
imagine, probably up to the medium coupling range inwhich most of the real materials are involved.

Acknowledgements
Y. Takada thanks H. Maebashi and S. Ishii for valuable discussions for the result in the normal state.

Disclosure statement
No potential conflict of interest was reported by the author.

Funding
Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan: Innovative Area ‘Materials Design throughComputics: Complex Correlation and Non-Equilibrium
Dynamics [grant number 22104011]

MOLECULAR PHYSICS

Downloaded by [University of Tokyo] at 21:33 08 January 2016

References[1] L. Hedin, Phys. Rev. 139, A796 (1965).
[2] M.S. Hybertsen and S.G. Louie, Phys. Rev. Lett. 55, 1418
(1985).
[3] M.S. Hybertsen and S.G. Louie, Phys. Rev. B 34, 5390
(1986).
[4] F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61,
237 (1998).
[5] W. Aulbur, L. Jönsson, and J. Wilkins, in Solid State
Physics, edited by H. Ehrenreich and S. Spaepen (Academic, New York, 2000), Vol. 54, p. 1.
[6] S. Ishii, K. Ohno, Y. Kawazoe, and S.G. Louie, Phys. Rev.
B 63, 155104 (2001).
[7] E. Kikuchi, S. Ishii, and K. Ohno, Phys. Rev. B 76 (2007)
075325.
[8] G. Baym and L.P. Kadanoff, Phys. Rev. 124, 287 (1961).
[9] G. Baym, Phys. Rev. 127, 1391 (1962).
[10] W.-D. Schöne and A.G. Eguiluz, Phys. Rev. Lett. 81, 1662
(1998).
[11] K. Delaney, P. García-Gpmzález, A. Rubio, P. Rinke, andR.W. Godby, Phys. Rev. Lett. 93, 249701 (2004).
[12] A. Stan, N.E. Dahlen, and R. van Leeuwen, J. Chem. Phys.
130, 114105 (2009).
[13] C. Rostgaard, K.W. Jacobsen, and K.S. Thygesen, Phys.
Rev. B 81, 085103 (2010).
[14] F. Caruso, P. Rinke, X. Ren, A. Rubio, and M. Scheffler,Phys. Rev. B 88, 075105 (2013).
[15] P. Koval, D. Foerster, and D. Sánchez-Portal, Phys. Rev. B
89, 155417 (2014).
[16] A. Kutepov, S.Y. Savrasov, and G. Kotliar, Phys. Rev. B 80,
041103(R) (2009).
[17] F. Bruneval, F. Sottile, V. Olevano, R.D. Sole, and L.
Reining, Phys. Rev. Lett. 94, 186402 (2005).
[18] M. Shishkin, M. Marsman, and G. Kresse, Phys. Rev. Lett.
99, 246403 (2007).
[19] Y. Takahashi, Nuovo Cimento Ser. 10 6, 370 (1957).
[20] Y. Takada, Phys. Rev. B 52, 12708 (1995).
[21] Y. Takada, Phys. Rev. Lett. 87, 226402 (2001).
[22] C.F. Richardson and N.W. Ashcroft, Phys. Rev. B 50, 8170
(1994).
[23] H. Maebashi and Y. Takada, Phys. Rev. B 84, 245134(2011).
[24] Y. Takada, J. Superconductivity 18, 785 (2005).
[25] H. Maebashi and Y. Takada, J. Phys. Soc. Jpn. 78, 053706(2009).
[26] G. Stefanucci, Y. Pavlyukh, A.-M. Uimonen, and R. van
Leeuwen, Phys. Rev. B 90, 115134 (2014).
[27] D.F. DuBois, Ann. Phys. (N.Y.) 7, 174 (1959).
[28] D.F. DuBois, Ann. Phys. (N.Y.) 8, 24 (1959).
[29] Y. Takada and H. Yasuhara, Phys. Rev. B 44, 7879 (1991).
[30] G.M. Eliashberg, Soviet Phys. JETP 11, 696 (1960).
[31] W.L. McMillan, Phys. Rev. 167, 331 (1968).

9[32] P.B. Allen and R.C. Dynes, Phys. Rev. B 12, 905 (1975).
[33] P.B. Allen and B. Mitrović, in Solid State Physics, edited by
H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New
York, 1982), Vol. 37, p. 1.
[34] J.P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990).
[35] Y. Takada, J. Phys. Chem. Solids 54, 1779 (1993).
[36] Y. Takada, in Condensed Matter Theories, edited by M.
Casas, M. de Llano, J. Navarro, and A. Polls (Nova, New
York, 1995), Vol. 10, pp. 255–268.
[37] P. Morel and P.W. Anderson, Phys. Rev. 125, 1263
(1962).
[38] L.N. Oliveira, E.K.U. Gross, and W. Kohn, Phys. Rev. Lett.
60, 2430 (1988).
[39] S. Kurth, M. Marques, M. Lüders, and E.K.U. Gross, Phys.
Rev. Lett. 83, 2628 (1999).
[40] M. Lüders, M.A.L. Marques, N.N. Lathiotakis, A. Floris,
G. Profeta, L. Fast, A. Continenza, S. Massidda, and
E.K.U. Gross, Phys. Rev. B 72, 024545 (2005).
[41] M.A.L. Marques, M. Lüders, N.N. Lathiotakis, G. Profeta, A. Floris, L. Fast, A. Continenza, E.K.U. Gross, and
S. Massidda, Phys. Rev. B 72, 024546 (2005).
[42] We employ units in which  = kB = 1.
[43] I.E. Dzyaloshinskii and A.I. Larkin, Soviet Phys. JETP 38,
202 (1974).
[44] H. Maebashi and Y. Takada, Phys. Rev. B 89, 201109(R)(2014).
[45] W. Metzner, C. Castellani, and C.D. Castro, Adv. Phys. 47,
317 (1998).
[46] G. Niklasson, Phys. Rev. B 10, 3052 (1974).
[47] S.L. Adler, Phys. Rev. 126, 413 (1962).
[48] N. Wiser, Phys. Rev. 129, 62 (1963).
[49] Y. Nambu, Phys. Rev. 117, 648 (1960).
[50] S.Y. Savrasov and D.Y. Savrasov, Phys. Rev. B 54, 16487(1996).
[51] http://www.quantum-espresso.org/[52] P.B. Littlewood and C.M. Varma, Phys. Rev. B 26, 4883
(1982).
[53] D. Pekker and C.M. Varma, Annu. Rev. Condens. Matter
Phys. 6, 269 (2015).
[54] T. Gherghetta and Y. Nambu, Phys. Rev. B 49, 740 (1994).
[55] Y. Takada and T. Higuchi, Phys. Rev. B 52, 12720 (1995).
[56] A.B. Migdal, Soviet Phys. JETP 7, 996 (1958).
[57] Y. Takada, J. Phys. Soc. Jpn. 45, 786 (1978).
[58] Y. Takada, J. Phys. Soc. Jpn. 49, 1267 (1980).
[59] Y. Takada, J. Phys. Soc. Jpn. 51, 63 (1982).
[60] Y. Takada, J. Phys. Soc. Jpn. 78, 013703 (2009).
[61] R. Akashi and R. Arita, Phys. Rev. Lett. 111, 057006(2013).
[62] R. Akashi and R. Arita, J. Phys. Soc. Jpn. 83, 061016(2014).
[63] Y. Takada, in Carbon-Based Superconductors: Toward
High-Tc Superconductivity, edited by J. Haruyama (Pan
Stanford, Singapore, 2015), pp. 193–230.