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Electron correlation is a measure of the errors that are inherent in the Hartree-Fock theory or orbital models. When the electron density is high, correlation is weak and the traditional electronic theory works well. However, at a low density of electrons correlation effects become strong and the traditional theory fails to describe the electron system correctly. Therefore, the electron correlation plays a radical role in such materials as high-temperature superconductors and heavy fermions, etc. To date, there is no agreement on how to deal with higher-order terms (correlation energy) in the series of electron’s ground state energy although a method that is termed diagrammatic iteration approach (DIA) was developed more than one decade ago by the authors of this article. That is why no consensus on the origin and mechanism of superconductivity has been engaged in superconductivity community. From the viewpoint of methodology, the DIA is indeed an approach to higher-order terms from the lower-order ones, i.e. it is a new method to show how to go beyond the random phase approximation (RPA) step by step by iteration. Here, we are logically presenting it to the community of modern physics with more analyses and hope to attract more attention to it and promote its applications.

Nowadays, looking around theoretical literatures of condensed matter physics, it is obvious that methodology of dealing with an electron system remains to be based on the Hartree-Fock (HF) theory plus a variety of approximations that are used to estimate the correction of electron correlation effects to the total energy of the electron system and other relevant properties caused by the correlation. In the case of a high-density electron system like a metal or alloy the conventional electronic theories work well, but it fails to provide correct information to interpret and understand phenomena observed in experiments with a low-density electron system, such as a hightemperature superconductor (HTS). The long-term arguments existing in the community of superconductivity on the mechanism of a high transition temperature since 1986 after the discovery of the first superconductor of the cuprate type are attributed to the lack of sufficient attention to the need of a developed many-particle theory that is capable of well treating the electron correlation effects. It seems that the great success of conventional electronic theories in terms of the mean field approach (MFA) in metals, semiconductors and many other fields of condensed matter physics leads one to attempt to polish the existing conventional theories for explaining and understanding the “anomalies” observed with cuprate superconductors, where the word of anomaly is relative to the observations from conventional superconductors. The anomalies are, as a matter of fact, the natural outcomes of a cuprate superconductor at specified conditions. They should be derived from an elaborate and sophisticated theory accounting for the electron correlation effects. Although we had worked out a theory in terms of the diagrammatic interaction approach (DIA) more than one decade ago to deal with superconductivity regardless of highor low-transition temperature [

For the routine discussions and derivations of second quantization for a particle system we use the results from Gerald D. Mahan [

The full electron gas Hamiltonian is often too complicated to use for the more elaborate many-body theory. Quite often this is approximated by a model Hamiltonian which has a simple form. Usually, these model Hamiltonians look very simple but it still impossible to solve them exactly. Often it is even difficult to solve them approximately!

The homogeneous electron gas is a model that is frequently used to learn about electron correlation effects. The homogeneous model is also called Jellium model. We can think of taking the positive charge of the ions and spreading it uniformly about the unit cell of the crystal. Of course, the homogeneous electron gas then has no crystalline structure. It has the Hamiltonian

where the first term is kinetic energy and the second represents pair interactions between two particles with spins of σ and σ'. Moreover, v_{q} in Equation (1) is the representative of the Coulomb interaction in k-space and usually thought of as the Fourier transformation, equal to 4πe^{2}/q^{2}, of the Coulomb interaction V(r) = e^{2}/r. It seems that there is nothing wrong here, but the problem is just from here as shown elsewhere [_{q} here is no longer that simple.

Since Equation (1) stays insolvable, we has to manage to simplify it gain. For example, if we adopt the socalled Cooper’s pairing condition of k + k' = 0 and σ + σ' = 0 as said in the BCS (Bardeen-Cooper-Schrieffer) theory [

However, even if the Hamiltonian has been further simplified with the Cooper’s pairing condition, it remains unresolvable because of the complicated and unknown interaction V(k, k'). The simplification of BCS [_{D} to be the cutoff frequency. This outcome of α = 0.5 fits to the experimental data observed then from some conventional superconductors. The assumption and simplification of V(k, k') to be the negative constant forces BCS to accept another rather strange concept— attractive interaction between two electrons due to the mediation of phonons! If V(k, k') is positive, there would be no solution at all to the BCS Hamiltonian. This paradoxical concept has confused us for more than half a century and was eventually proved to be improper [

There are two electronic properties that are important: the self-energy of an electron of momentum p and the total energy of the N-electron system.

First of all, the inter-electronic coupling constant is defined as

where e—electron charge,

and

where n and n_{s} are the density of electrons in 3D and 2D, respectively. It can be seen that the higher the density of electrons, the smaller the coupling constant. The condition of

Another important dimensionless parameter r_{s}, characteristic length of the system of interest, is adopted to be a criterion of the electron density, and defined as below

where _{s}, the smaller the parameter r_{s}. In terms of this definition, kinetic energy E_{k} is calculated to be _{g} is given by [

for 3D and

for 2D, where the first term is kinetic energy, the second is the exchange energy, and all of the rest terms together construct the correlation energy E_{c}. In Equation (6.b) ε_{r} is the so-called ring energy, which is usually evaluated approximately in consideration of the ring-diagrams and the second-order exchange diagrams. Many authors have also obtained the results shown in Equations (6.a) and (6.b) [

It is seen that when the density of electrons is high (r_{s} = 1), kinetic energy prevails the potential energy (exchange and correlation energies) and the latter is hence negligible. Thus, the free-particle model works well and the corresponding results from the free-particle model are accurate. The conventional perturbation theory was developed based on the free-electron model. It is obvious that the free-electron model is good for such a high electron density as it is in a conventional metal, but not accurate for a low density of electrons in which r_{s} is close to or greater than 1 and the coupling constant α is also greater than 1. It is easy to see that when r_{s} > 2, |E_{x}| > E_{k}. Therefore, as the density of electrons keeps decreasing, E_{c} is no longer negligible, nor is the free particle model valid.

The different types of the elastic or inelastic, magnetic (spin-dependent) or non-magnetic processes of scattering may cause electrons to become localized or nearly localized. In particular, if the material under investigation has a low density of electrons and/or a high density of defects (or impurities), inter-electron coupling constant α is close to or greater than 1. Meanwhile, the parameter characterizing the scattering of electrons k_{E} l is also close to 1, where k_{F} is the Fermi momentum and l = v_{F} τ is the mean free path.

Therefore, strictly speaking, the conventional treatment of electrons in bulk electronic materials does not apply to a dilute electron gas because the methods, based on the use of the lowest order of perturbations, i.e., the RPA in interaction and the Born approximation in scattering, are no longer valid for this case where correlation effects play a major role since these methods treat electrons as if they were free. In other words, the so-called vertex correction—higher-order correction in the self-energy part or two-electron interaction kernel, while using diagrams, must be taken into account. The incorporation of correlation in the interaction (scattering) allows us to expand the existing methods and to develop a new method in electronic theory. The question is how we can go beyond the RPA approximation with the diagrammatic method.

It is also well known [

In general, there are two methods to deal with an electronic system: Hamiltonian and quantum field theory. Conventionally, we in general seek a proper Hamiltonian for a system of particles and then find out its approximate solution. The BCS theory is a good example of the attempt of this kind for conventional superconductors. The Hubbard model with a large positive constant U which has been believed to be the Hamiltonian appropriate for HTS looks simple enough, but to date a solution to it was obtained only in one dimension. Therefore, even if the assumption of a constant interaction potential is valid in some special cases, we are still in the difficulty of obtaining its solution. The major issue here is how much we can do with a Hamiltonian and how much information we are able to extract from it.

Nevertheless, the quantum field theory appears to be better in dealing with a many-particle system in the case of strong coupling. By means of the diagrammatic method, if we manage to construct an equivalent diagram for the interaction potential energy and then converts it to an integral equation, we can get the transition condition by the pole condition of the equation, which corresponds to a phase transition.

The difficulty we meet in trying to go beyond RPA, however, rests in the lack of an acceptable approach to take into account higher-order corrections to the self-energy of a single electron due to correlations. The partial summation that has been used in this approach often give rise to diverse and sometimes controversial outcomes and conclusions because some of the higher-order terms may be canceled out and a partial summation may happen to pick up only the terms that lead to divergence. A survey of literature on superconductivity theory during the past two decades clearly shows a severe shortage of the works based on quantum field theory. Its importance in condensed matter physics is unquestionable since it is the only way to approach the detailed information of interactions of a many-particle system.

In weak coupling, it is well known that the Hamiltonian method is in good agreement with the diagrammatic approach. For example, in the study of superconductivity the BCS theory [^{}give rise to consistent results. In strong coupling, however, the Eliashberg theory fails as the BCS theory does. Therefore, in order to obtain a result beyond RPA a technique to be used to go beyond RPA must be developed. The diagrammatic iteration approach was hence born in our previous work [^{}

Now, let us brief the methodology of DIA. The well-known Migdal’s expression [

where p and k are the four-dimensional vectors, G(k) and D(k) are the dressed electron and boson Green’s functions, respectively. The boson Green’s function satisfies the so-called Dyson equation _{0}(k) being the zeroth-order boson Green’s function, and

where V(k) is the matrix element of the bare Coulomb interaction. In the case of the electron-phonon interaction

where

where Γ(p, k) is the three-point vertex function and, in the ladder approximation, is defined as [

where K(p, p') denotes the kernel of the integral equation for the vertex function and describes the many-particle correlations. The interaction kernel K(p, p') in Equation (9) satisfies Ward’s identity [

The DIA’s core is iteration. Namely, starting from the known information—the single-particle self-energy in RPA, wherein Γ is taken to be 1, we obtain the two-particle interaction kernel K in terms of Ward’s identity,

where G is Green’s function of electron and p four-dimensional momentum and σ spin. Then, with the newly obtained K and the RPA’s Γ = 1 we can calculate the vertex part Γ according to Equation (9), in the first step beyond RPA. The vertex part obtained here can be substituted into Equation (7) for Γ to re-calculate the selfenergy and then the interaction kernel K based on Ward’s identity at the first step beyond RPA. By now, the first loop of the iteration has been completed and all of the quantities of Σ, K and Γ in the first step beyond RPA have been calculated. The second loop is that in terms of the self-energy Σ and the vertex Γ obtained in the first step beyond RPA we are able to recalculate K and Γ in the second step beyond RPA, and then Σ again. Thus, the second loop is completed. Of course, from Equation (8) irreducible polarizability can also be calculated in each step in terms of the corresponding vertex part Γ. By doing so step by step we can sequentially receive all the possible information in principle to any step beyond RPA. However, mathematical complexity makes it unrealistic to do so analytically to higher orders. Of more difficulty is that we are unable to get the summation of the complicated kennels of all the orders even if we could get several analytic expressions. Thus, no success has been made analytically. Instead, a diagrammatic technique is recommended.

Based on Feynman’s diagrammatic symbols used in quantum filed theory: a solid line with an arrow directed on one side represents Green’s function of an electron and a dashed one stands for Green’s function of a boson, we may construct the diagram of the spin-dependent single particle self-energy

In

However, when coupling is strong, from the viewpoint of the diagrammatic method, the high-order corrections are no longer negligible and the problem now becomes how to find out diagrammatic structure of the vertex function. In principle, there are infinite number of Feynman diagrams that represent all of the possible interaction patterns. Of more importance is now to have a rule or regulation of constructing them. An intuitive way is to follow the order of the boson line. A preliminary evaluation shows that the convoluted relationship among the self-energy Σ interaction kernel K and the vertex function Γ leads to a variety of developments of the diagrams as shown below, and the weight of each of the diagrams, even if they are in the same order, is different and that the contribution from a diagram in a higher order may be larger than that from a low-order diagram. A variety of partial summations may lead to completely different or conflict results. Therefore, only a complete summation all over the possible diagrams is reliable and acceptable. To this end we have developed a method to reach this goal going out of RPA step by step. This is the so-called diagrammatic iteration approach briefly described as below.

In order to do so he first thing needed is to find the diagrammatic functional derivative operation. Clearly, from the mathematical definition of a derivative the functional derivative with respect to the electron Green’s function

It is worthwhile to indicate that we may have two different models in dealing with the interaction potential: a given or renormalized interaction. The first model means a simple boson propagator represented by a dashed line as shown on the right hand side of

For the first model—a given interaction, by means of the diagrammatic functional derivative technique a cut on the electron line of the RPA self-energy diagram on

However, for the second model—a renormalized interaction, a cut of any electron line on the RPA diagram of

For a given interaction, first of all, we try to go beyond RPA analytically.

Taking into account the spin index in Equation (7), assuming D(k) = D_{0}(k) and Γ(p, k) = 1 in it, and using the

identity

Substituting Equation (9) with the spin indices and the kernel of Equation (10) into Equation (7), we obtain the self-energy incorporating the vertex correction in the first step (order) beyond RPA. Using Ward’s identity one more time for the single-particle self-energy that incorporates the vertex correction with the kernel of Equation (10), we express the interaction kernel up to the second step (order) as

where

and

Instead of doing so further, we use diagrams to represent the above equations. The diagrams for these interactions are given in

_{2}(p, p'), corresponds to the first order exchange interaction incorporating the vertex correction of the first order. The last diagram (d), (

interaction D_{0}(k) < 0. The diagrams of the vertex part & the self-energy with the interaction kernel, described by Equation (11) and Equation (12), are given in

As an example, we can evaluate kernels K_{2} (p, p') and K_{3} (p, p') in the case of the non-retarded interaction, i.e., when D_{0}(k) = D_{0}(k), where D_{0}(k) is a given frequency-independent interaction. If we assume that the electron Green’s function is given by

where

and

where

where

According to the above descriptions and skills introduced on

Let us replace the kernel by the effective two particle interaction,

[

To evaluate

where

We consider the case when the given interaction is similar to the electron-phonon interaction, i.e.

We first consider the case of

where

and the density of states per spin NF (for example, in an isotropic 3D or 2D electron spectrum

^{*} is the effective mass and p_{F} is the Fermi momentum).

where

Correspondingly, substituting

where the analytic continuation _{Q} in Equation (17) is real and positive, we have [

It can be seen that the quantity _{c}, the temperature at which a pole first appears in

which is exactly the same transition temperature obtained in the BCS theory [

As seen from the above discussions, we have adopted the DIA in a 3D electron gas of a given Interaction. Since it pertains to a weak coupling system, the electron correlation effects should be so weak that MFA is good enough as done by BCS [

What we have done step by step to go beyond RPA seems to be unnecessary in the case of a weak coupling system because the higher-order interaction kernel like K_{2} is eventually proved to be negligible as shown above and thus we obtain exactly the same results as BCS. However, as mentioned above, we intend to present in this article how DIA works and how to proceed because DIA procedures for a given interaction are simpler than a renormalized interaction. Of course, for a weakly coupled system we do not need adopting DIA. However, for a system like a cuprate superconductor the higher transition temperature T_{c} can be reached only by means of the model of a renormalized interaction in terms of DIA [

In summary, we have indicated several issues existing in the electronic theory. This conceived a strong demand of a development of the existing theory that was established based on the concept of MFA in the case of weak coupling. Also, we have briefly introduced a new approach—diagrammatic iteration approach. DIA indeed made a development of the methodology in the diagrammatic approach to the region beyond RPA in order to fully account for electron correlation effects. The DIA exhibits its sophisticated logic and seasonality and is one more step close to the truth. We believe that DIA is a better approach in describing a strongly coupled electron gas. One of objectives of this article is to attract more attentions to and discussions of it. Although DIA was originally devoted to a unified description of highand low-temperature superconductivity yet it can be widely used in condensed matter physics and other fields where strong coupling plays an important role.

The sincere hope of the authors is to see more and more scientists working on field theory which can apply this approach to their studies and further develop it. We are confident that the final consensus on the mechanism of superconductivity will be engaged in the Coulomb interaction with phonons promoting the isotope effect and that quantum field theory must replace Hamiltonian to obtain better solutions in the subjects of condensed matter physics where electron correlation effects are not negligible. Moreover, it may also be useful in atomic or nuclear physics.

The authors wish to show our appreciation of the support from Chongqing Academy of Science and Technology, China, with an internal fund for ensuring the completion of the article. Also, our thanks go to Liu Yi and Zhang Peng for their assistance in drawing the plots.