4. Introduction to the theory of superconductivity
4.2 Green’s functions for superconductors: the Nambu-Gor’kov formalism113
Fig. 4.6:The typical energy (q0) dependence of the dressed effective interaction for a fixed momentumq. In the low energy sector, up to the acoustic phonon frequencyωq, the interaction is attractive, leading to the condensation in-stability.
4.2 Green’s functions for superconductors: the Nambu-Gor’kov
Nambu realized that, by introducing the two components field operators Ψ(r) =
ψ↑(r) ψ↓†(r)
, Ψ†(r) =
ψ↑†(r) , ψ↓(r)
(4.76) the mean-field BdG Hamiltonian (4.42) can be written in the compact matrix form
HMF= Z
drΨ†(r)H(r) Ψ(r) (4.77) with
H(r) =
He+U(r) ∆(r)
∆∗(r) −[He∗+U(r)]
(4.78) the Hamiltonian matrix.
On the basis of the Pauli matrices 1=
1 0 0 1
τ1=
0 1 1 0
τ2=
0 −i i 0
τ3=
1 0 0 −1
(4.79) the Hamiltonian matrix has the form
H(r) =1[iImHe] +τ1[Re∆(r)] +τ2[iIm∆(r)] +τ3[ReHe+U(r)]. (4.80) In the absence of magnetic fields we can choose the phases of the fields such that the function∆is real, thereby killing the contribution of theτ2matrix.
Having introduced a matrix form for the Hamiltonian of our system, it is then nat-ural to introduce the matrix (Gor’kov) Green’s functionGout of the new field op-erators [101].
In analogy with the usual definition, the elements ofGatT= 0are Gab(r, t) =−i
~hT h
Ψa(r, t)Ψ†b(0,0)i
i (4.81)
whereT is the time-ordering operator anda, b∈ {1,2}. The Gor’kov Green’s func-tion is thus
G(r, t) =
−~ihT h
Ψ↑(r, t)Ψ†↑(0,0)i
i −~ihT h
Ψ↑(r, t)Ψ†↓(0,0)i i
−~ihT h
Ψ†↓(r, t)Ψ†↑(0,0)i
i −~ihT h
Ψ†↓(r, t)Ψ↓(0,0)i i
. (4.82) It is possible to introduce a diagrammatic representation of the matrixG(depicted as a thick double line ) and of each of its normal and anomalous components shown in (4.82). Along the convention of letting the arrows come out of the creation fields ψ†the diagrams associated toGand toGab(r, t)are depicted in Fig (4.2).
Notice that thefreeGreen’s function matrix is diagonal, since the anomalous terms come out of selective contractions of the interaction part.
In the momentum representation the Nambu field operators are Ψ(k) =
c↑(k) c†↓(−k)
(4.83)
=
Fig. 4.7:The diagrammatic representation for the exact Gor’kov Green’s function matrix, see (4.82). The thick double line represents the matrixGand the matrix elementsGabare represented by thick single lines with arrows com-ing out of fermionic creation operators.
and they satisfy the fermionic commutation relations
{Ψ(k),Ψ†(k0)}=1δkk0 , {Ψ(k),Ψ(k0)}= 0. (4.84) In the absence of magnetic fields the free fermionic Gor’kov Green’s function in the momentum-energy representation is
G(0)(k, E) =
1 E−εk 0
0 E+ε1
k
!
. (4.85)
The exact Green’s function can be obtained via the Dyson equation as [G(k, E)]−1=h
G(0)(k, E)i−1
−Σ(k, E) (4.86) whereΣ is the matrix self-energy. Equation (4.86) is diagrammatically shown in components in Fig (4.2).
= + +
= +
= +
= + +
Fig. 4.8:The diagrammatic representation of the Dyson equation for the exact Gor’kov Green’s function componentsGab, see (4.86). Normal as well as anomalous selfenergy insertions are evident. Notice that, since the vertices always involve a fermionic part of the typeψ†ψ, the arrows ”continue” in each of them and nopaircreation or annihilation takes place.
The most general form forΣcan be written, on the basis of the Pauli matrices (4.79), as
Σ(k, E) =1[E−EZ(k, E)] +τ1[Z(k, E)∆(k, E)] +τ3[δεk], (4.87) again having chosen to kill the contribution of theτ2matrix. In (4.87) the so-called
”mass-renormalization” functionZ(k, E)has been introduced, together with the pairing function∆(k, E)and the single particle normal correctionδεk. We will see shortly that indeed the parameter ∆(k, E)coincides with the gap we are used to from the BCS theory, and the reason of the name for theZ function will become clear. In the same spirit of the BCS treatment we will consider thenormalselfenergy corrections as already incorporated in the single particle dispersionεk, so thatδεk= 0and the selfenergy matrix does not contain anyτ3component (this is however not a crucial point, but it helps in simplifying the aspect of our expressions).
From (4.85,4.86,4.87) we extract the form for the matrixG
G(k, E) =1[E Z(k, E)] +τ3[εk] +τ1[Z(k, E)∆(k, E)]
E2Z2(k, E)−ε2k−Z2(k, E)∆2(k, E) . (4.88) By studying the poles of this single particle Green’s function we obtain informa-tions about the spectrum of the excitainforma-tions of the system. From the vanishing con-dition of the denominator in (4.88) we get the excitation energy
E2= ε2k
Z2(k, E)+ ∆2(k, E) (4.89) closely resembling the Bogolons dispersion (4.61). Indeed it can be shown that in the BCS caseZ(k, E) = 1and the reality of∆comes from our phase choice (i.e. in the absence of magnetic fields∆can always be chosen to be real).
Where the normal dispersionεkvanishes (i.e. at the Fermi level) we obtain the ex-citation gap∆, justifying the choice of its name. Analogously, if the normal disper-sionεkhas the formεk ∼~2k2/2mthe factorZ clearly renormalizes the electronic effective mass (again justifying the name chosen for it).
Up to now we did not make any assumption or choice about the form of the self-energy with respect to the interaction and the fermionic Green’s functions. To do so we can start discussing the perturbative treatment of thematrixGreen’s functions in line with the standard perturbation schemes for normal metals.
We have seen that the anomalous averages for the original fermionic operators come out naturally fromnormalaverages of the Nambu field operators (see 4.82).
Therefore we will have to consider a perturbation treatment for the matrix Green’s functionwhich is completely analogous to the one we are used to innormalmetals.
Every Green’s function will be replaced by the corresponding matrixG.
In order to incorporate the interactions in our treatment we notice the identity Ψ†(k+q)τ3Ψ(k) =c†↑(k+q)c↑(k) +c†↓(−k)c↓(−k−q)−δq,0. (4.90) A four fermion interaction term as in (4.26) is therefore written as
Hint= 1 2
X
k,k0,q
V(k,k0,q)
Ψ†(k+q)τ3Ψ(k) Ψ†(k0−q)τ3Ψ(k0)
. (4.91)
In real cases this effective interaction is mediated by additional bosonic fields which must be coupled to the fermionic ones (we will see an explicit example in the next chapter).
In conventional superconductivity the bosonic fields are the dressed phonons me-diating the residual attractive interaction close to the Fermi level. The dynamical bosonic field propagator entering the effective interaction will in general have a momentum-energy dependence, meaning a delay in the real time e-e coupling. The many body treatment we set up here is naturally suitable to treat these dynamical effects as well.
At this point the apparatus for the perturbative treatment is ready. We have to write selfenergy contributions in terms of the matrix Green’s function and of the media-tors of the interactions.
Everyselfconsistentselfenergy diagram will in general be function of itself, through the form of the exact Green’s functions. This implies we will find a system of non-linear integral equations linking the two functionsZand∆.
Let us take, for example, the selfconsistent Fock term due to the interaction (4.91), depicted in Fig(4.2).
k−k0, E−E0
k0, E0
Fig. 4.9:The Fock type selfenergy correction to the single particle fermionic Gor’kov Green’s functionG(0)(k, E). Integration over internal momenta and frequency is implied.
Indeed, this was the first non trivial contribution considered by Eliashberg in his original treatment of the effects of the dynamical dressed phonons in the quasipar-ticle excitation spectrum of superconductors [102]. Thus
Σ(k, E) =i
Z dk0dE0
(2π)4 V(k,k0,k0−k;E−E0)τ3G(k0, E0)τ3. (4.92) Substituting the form (4.87) for the selfenergy and the exact Green’s function (4.88) we obtain
1[E−EZ(k, E)] +τ1[Z(k, E)∆(k, E)] =i
Z dk0dE0
(2π)4 V(k,k0,k0−k;E−E0)×
×τ3
1[E0Z(k0, E0)] +τ3[εk0] +τ1[Z(k0, E0)∆(k0, E0)]
E02Z2(k0, E0)−ε2k0−Z2(k0, E0)∆2(k0, E0) τ3. (4.93) The formal inclusion of a frequency dependence in the effective interaction is auto-matic.
By writing the matrix equation in components we come out with a system of cou-pled integral equations forZand∆, called the ”Eliashberg’s equations”.
It is in principle impossible to decouple exactly the mass renormalization equation from the gap one and even if this was done they would still be extremely difficult to solve separately. However, in some particular regimes, we can get a lot of im-portant informations about the spectrum.
We will not dwell here on the original system considered by Eliashberg for his ef-fective interaction. The interested reader can find more on that in [102, 103, 104].
We point out that the selfenergy diagram we just mentioned is only the simplest but non-trivial one (the Hartree contribution, being a constant energy shift, does not give rise to any new interesting effect). Further terms can be obtained by dress-ing the vertices, to reach the exact fermionic selfenergy.
In ordinary phonon-mediated superconductivity, however, it was shown by Migdal [105] that any additional bosonic insertion in the vertex part lead to corrections of orderp
m/M wheremis the electron mass andM the typical ionic mass. It was then natural to neglect these corrections and stick to the Fock term considered by Eliashberg.
In the next Chapter we will face the Eliashberg equations again and discuss their solutions in the case where dynamical bosonic gauge fields are the mediators of the residual attractive quasiparticle interaction.
In the previous chapters we have seen that several interesting phenomena take place close to the degeneracy of two LL of opposite spin at the Fermi energy.
This happens essentially because, at the crossing, the single particle fermionic en-ergies of the two species coincide and the residual interactions between them is the leading energy scale to modify the GS properties.
In particular, the GS can be restructured and its spin-related properties can show anomalous behaviours. For example, as we saw in chapter 3, without quasiparticle interactions we cannot get the shoulder in the polarization observed aroundevery CFLL crossing at the Fermi energy.
In this chapter we will consider the role of the residual interactions close to the de-generacy between two LL of fermions at the chemical potential.
Our approach will be essentially the following: we know from experiments that there is a point where the densities of the fermions belonging to the crossing modes coincide. This means that the two levels have an average filling1/2each. We also know that a half filled LL of fermions can be mapped into a Fermi liquid of CF.
Therefore we take, as a starting point, two Fermi liquids of CF (of two different species, which could be spin, or, more generally, pseudospins) and turn the inter-actions on.
The system we will consider is very close to what has been investigated in the field of bilayer Quantum Hall samples. A bilayer system can be realized in experiments by directly constructing two 2DES separated by a finite distancedor by produc-ing a wide 2DES where two eigenstates of the confinproduc-ing potential in thezdirection are occupied at the typical experimental densities. Clearly, in a bilayer system the pseudospin index coincides with the layer index.
In the following we will be mainly concerned in a single 2DEG where different fermion species coexist, being true spin eigenstates.
The main task will be to write the effective interaction coupling particles with the same as well as with opposite spins. The equal-spin interaction will essentially modify the single particle properties of CF, while the opposite-spin term can lead to more interesting effects. We will see, for instance, that the leading small en-ergy interaction between particles with opposite spins and momenta can be attrac-tive, leading to a condensation mechanism similar to what we just discussed in the framework of superconductivity [106]. The pairing instability of the CF liquid will be the main topic of the present chapter, and its outcomes will be discussed at the end of this section.
Along the chapter we will consider again the Chern-Simons Lagrangian, this time
for two species of fermions, and the related gauge field propagators mediating the residual CF interaction. To evaluate the superconductive properties of the system we will consider the Nambu-Gor’kov formalism and compute the energy gap in the spectrum by means of the Eliashberg equations described in the previous chapter.