Renormalization Group Study of a Fragile Fermi liquid in dimensions
We present a calculation of the low energy Greens function of interacting fermions in dimensions using the method of extended poor man’s scaling, developed here. We compute the wave function renormalization and also the decay rate near the Fermi energy. Despite the lack of damping characteristic of 3-dimensional Fermi liquids, we show that quasiparticles do exist in dimensions, in the sense that the quasiparticle weight is finite and that the damping rate is smaller than the energy. We explicitly compute the crossover from this behavior to a 1-dimensional type Tomonaga-Luttinger liquid behavior at higher energies.
Recent experimental work E0 (); E1 (); E2 (); E3 (); E4 (); E5 () on the angle resolved photoemission (ARPES) have investigated weakly two dimensional systems. These are equivalently viewed as weakly coupled 1-dimensional chains, and exhibit the characteristics of 1-dimensional Tomonaga-Luttinger type systems with anomalous dimensions, exhibiting a crossover at lowest energies to a Fermi liquid type behavior, with a finite but very small value of the quasiparticle weight . The small scale of here is related to the almost 1-d nature of the systems. Fermi liquids with a small but non zero also arise in other important condensed matter systems in 0, 2 , 3 and dimensions. A small in the latter arise due to strong correlations, rather than reduced dimensionality. Historically the Gutzwiller wave function Gutzwiller () provided a first example of such a behavior, suggesting a strong correlation induced vanishing near the Mott insulating state. This was made especially explicit in the work of Brinkman and Rice Brinkman-Rice (). In 0 dimensions, the asymmetric single Anderson impurity model Hewson-rg (); RG1 (); RG2 (); RG3 (); RG4 () (AIM) provides a well studied and classic example. Here one finds an exponentially small from the Bethe Ansatz solution Hewson-Rasul (), in the limit where the occupancy of the impurity level . In the Hubbard model, which is solvable numerically by the dynamical mean field theory dmft () (DMFT), one finds a vanishing as the electron density tends to the Mott insulating value , with possibly small correctionsecfl-dmft () to the exponent for very small . In other dimensions various approximations- such as the slave particle field theories- suggest a similar small value of in the metallic state found near the Mott insulating limit. We may provisionally call this group of metallic systems with a small , whatever the origin of its small scale, as “Fragile Fermi Liquids” (FFL).
We next consider the important issue of the damping rate in order to refine this notion. Recent work on the large U Hubbard or the - model using extremely correlated Fermi liquid (ECFL) theory ecfl-dmft (); Gweon-Shastry (); Shastry-Asymmetry (); ecfl (); ecfl-aim () gives an interesting insight into the nature of the quasiparticle damping near the insulating limit, which agrees in remarkable detail with the results of DMFTecfl-dmft (). In the large d model at low energies, one finds that the quasiparticle Greens function at the Fermi momentum, including the damping, can be expressed as
where and are energies on the scale of the bandwidth. Since the imaginary part gives us the damping of the quasiparticles here, this expression goes beyond the domain of the Landau Fermi liquid theory. The Landau theory merely says that the damping is of without giving the scale of the damping, nor does it specify the terms beyond the leading order. Thus the Greens function including damping exhibits an scaling, with an unexpected and prominent odd in corrections to damping as in Eq. (1). This cubic term helps in understanding the ARPES line shapes in very strongly correlated metals as shown in Ref. (Gweon-Shastry, ; Shastry-Asymmetry, ), and also in the thermopower of correlated matter Ref. (Palsson-Kotliar, ). It may be viewed as one of the signatures of extreme correlations, in addition to their role in diminishing . For the AIM, a similar expression for the low energy Greens function to quadratic order results in the extension of the Fermi liquid theory in the interesting work of HewsonHewson-rg (). In the following we focus on effects of small brought about by dimensionality rather than strong correlations. Therefore we shall be content to ignore the cubic term and discuss the leading quadratic term alone. Taking the above examples as benchmarks, we refine the notion of the Fragile Fermi Liquids. These may be characterized as having quasiparticles endowed with a small , with a damping (smaller than the energy) on an energy scale that itself shrinks with .
In order to explore further this notion of Fragile Fermi liquids, it would be of value to have solvable models that give detailed results for the damping, along with the required small Z. In this work we study weakly coupled dimensional systems resulting in a Fermi liquid where is very small, as described in the first paragraph. In view of the physics described by Eq. (1), our goal is to compute not only , but also the damping of the quasiparticles, through a controlled calculation within a dimensional model system, with . We expect that the quadratic behavior of the damping in Eq. (1) would be lost in the case of dimensions, but nevertheless the damping would be small relative to the energy of the quasiparticle. It is of interest then to check if the scaling survives, to the extent possible with the proximity of the Tomonaga-Luttinger behavior at exactly 1-d. For this purpose we study a sufficiently simple model that allows an asymptotically exact calculation, using the renormalization group, of the low energy Greens function, including the damping. This would also enable us to study the crossover from a Fragile Fermi liquid at the lowest energies, to a Tomonaga-Luttinger type behavior at higher energies, and thereby make contact with the experiments E0 (); E1 (); E2 (); E3 (); E4 (); E5 (). The model considered is the simplest one in dimensions, and is essentially the same as the one studied in the early work of Ueda and Rice (UR) in Ref. (T1, ). For our purposes, it turns out to be necessary to compute the scale (or frequency) dependent , and not just the static limit of this object. We will denote the static limit as . Furthermore, we are able to calculate the crossover from high to low energy behavior on a crossover scale that depends on . At low energies we obtain asymptotically a Fragile Fermi Liquid behavior: the leading damping term of Eq. (1) becomes with an dependent energy scale . The result is summarized as:
where is the anomalous dimension in 1-d and is a constant around , with a singular low energy behavior of
In view of the singularity of the final behavior of the damping term at small is , which is smaller than the energy of the particle . Putting these together we find
exhibiting an scaling, apart from the weak logarithmic correction and setting .
It is amusing to note that although our calculation Eq. (4) is designed for , if pushed somewhat bravely to , suggests that the singularity of in 2 dimensions would be weak, and give rise to a quadratic damping with possibly corrections. This is indeed correct as we know from other works. Using the full solution of the crossover problem, we compute the spectral function of an electron at the Fermi point from high to lowest energies, for a few typical values of the initial coupling constants.
We next summarize the literature and discuss what is the new result in this paper. UR performed a renormalization group (RG) analysis for small and showed that for a Fermi liquid (FL) fixed point emerges, while has a line of fixed points which maps to the Tomonaga-Luttinger model Giamarchi () with anomalous dimension (defined below in greater detail). This line of fixed points arises from the competition between the Peierls and Cooper channels. Further interesting theoretical work on this model has been undertaken in Ref. (T2, ; T3, ; T4, ; T5, ; T6, ; T7, ). For instance at small , Castellani, Di Castro and Metzner Ref. (T4, ) computed the value of the quasiparticle weight , their result is again a non analytic dependence on , with a slightly different set of exponents , and should be compared with our result for reported in Eq. (2). For a fixed our expression Eq. (2) would give a somewhat bigger magnitude of , but it is qualitatively similar. As far as we are aware Eq. (3), Eq. (4) and Eq. (5) are new.
Also new in our work is the method we use for our calculations. Our results stated above require the calculation of the full dependent self energy for the model of Ueda and Rice. For this purpose we have developed a renormalization group (RG) procedure that is a modification of the Wilsonian RG approach for fermions, presented pedagogically in the excellent review by Shankar Shankar (). The modification becomes necessary because the approach outlined in Ref. Shankar, leads to difficulties when one tries to use it for calculating the frequency dependent self energy. The difficulties as well as the main features of the new method, which we refer to as the Extended Poor Man’s Scaling (EPMS) prescription because it is very much in the spirit of Anderson’s celebrated “poor man’s scaling” approach to the Kondo problemRG1 (), are discussed briefly in the next section, and in greater detail in Section III. Our calculations also employ a different simplification of the momentum integrations that arise in dimensions as compared to the discussion in Ref. T1, . The new simplification is also summarized in the next section, and presented in detail in in Section III. We emphasize that this simplification is merely for ease of calculation, and we expect that the physics of significance that we discuss will be valid beyond the simplification.
Ii The Model
The partition function Schulz (); Solyom () for the model of interacting fermions in one dimension (1-d) without umklapp processes (and assuming zero temperature) that we study in this paper can be written as the Fermionic functional integral
where and are Grassman numbers; and for the right branch; and for the left branch; , and are dimensionless quantities defined respectively as , , in the left branch and in the right branch with , , being the physical Matsubara frequency, momentum and momentum cutoff respectively. Furthermore, we have used the abbreviated notations: , , and . The dimensionless coupling constants , where have their usual meanings as the coupling constants used in the literature, some times referred to as “g-ology” Solyom (), with or corresponding to the backward (Fig.1(a), i.e., , and or to the forward (Fig. 1(b), i.e., inter-branch scattering terms, and or corresponding to the intra-branch forward scattering term (Fig. 1(c)).
As is well knownSolyom (), standard diagrammatic perturbation theory in powers of the coupling constants and for the self energy and other properties of the model in Eq. (7) in 1-d lead to (logarithmic) divergences. Such divergences are best handled by scaling or RG approachesSolyom (); Shankar (); T1 () either of which leads to the same scaling equations for the effective coupling constants as a function of a ”running” momentum cutoff as high momentum fermion degrees of freedom are recursively eliminated and the cutoff is continuously reduced. As mentioned in the introduction, in this work we are interested in performing a detailed RG calculation of the frequency dependent self energy and Greens function of the model in Eq. (7), which is clearly more demanding than finding the scaling equations for the coupling constants. Furthermore, we would like to extend such a calculation to dimensions where we should be able to see the Fermi liquid emerging from a non Fermi liquid state.
In extending the existing Wilsonian RG prescription, which is well explained in Shankar’s review article, to the calculation of self energy in dimensions, we encounter two difficulties. The first difficulty is that the Wilsonian RG, becomes cumbersome for performing calculations of self energies and in particular the quasiparticle weights Shankar (); Metzner (). The reason is that the rules of Wilsonian RG require that all the momentum labels in the internal propagators in diagrammatic perturbation theory, equivalent to intermediate excited states in traditional perturbation theory, must correspond to the fast or high momentum degrees of freedom that are being eliminated. As discussed in detail in Section III, momentum and energy conservation then leave the self energy unchanged until the running momentum cutoff is half of the original momentum cutoff, and this renders the method difficult to implement. To solve this problem, we propose a modification of Wilson’s scheme, whose mode elimination process is in the spirit of Anderson’s poor man’s scaling approach RG1 () to the Kondo problem. We will refer to as the Extended Poor Man’s Scaling (EPMS). It differs from the Wilson scheme in that it only requires the intermediate states that are eliminated to involve at least one high energy or fast mode, while in the Wilson scheme all the eliminated states are required to involve only fast modes. This procedure leads to contributions to the self energy arising continuously from the very beginning of the reduction of the momentum cutoff, and makes it easier to track its evolution from high to low frequency scales. The procedure is argued to be self-consistent for the current problem, and as a check we verify that the various exponents and other properties calculated using the new procedure agree with available results from the literature.
The second difficulty has to do with the angular integrals that arise in extending the calculations to non-integer dimensions. To deal with this, we propose a simpler prescription for dealing with dimensions than used earlierT1 (), which we argue is valid when and are small (see Section III.D for the details). Using such a prescription and the second order EPMS method, we obtain the flow equations for the coupling constants and for the factor, and numerical as well as exact limiting results for the factor and . They all show crossover behaviors, with the emergent crossover scale being given by or , where ( is the running cutoff). When or , the system shows 1-d-Tomonaga-Luttinger type behavior, while it approaches the higher dimensional limit and shows Fermi liquid behavior if or . Also, we show that when , where is the anomalous dimension from the 1-d limit, one obtains a “Fragile Fermi Liquid” low energy behavior, with extremely small .
The rest of this paper is organized as follows. In Section III, we discuss the difficulties in calculating the self energy and the Z factor using Wilsonian RG in slightly greater detail, and then outline the EPMS method and our prescription for calculations in dimensions. In Section IV, we discuss the second order flow equations for the coupling constants and their solutions obtained using our prescriptions and show that they are in agreement with those in the literature. Sections V and VI contain our central, new results for the and for the leading behavior of the self energy in dimensions. Section VII has a brief discussion about the breaking down in dimensions of the ‘Lorentz-invariance’ that is a characteristic feature of the asymptotic (low ) behavior of correlation functions of interacting fermions in 1-d. In Section VIII we summarize the main points of the paper. Since we make repeated comparisons of the EPMS method to the Wilsonian RG, for convenience we have summarized the salient aspects of the latter in Appendix A. The full details of the EPMS prescription are presented in Appendix B. Readers who are unfamiliar with RG calculations in the context of 1-d fermionic systems are likely to find the rest of the paper more accessible if they go through the Appndix A first.
Iii The Extended Poor Man’s Scaling method and The expansion prescription
iii.1 Difficulties in calculating the Self Energy and the Z factor using Wilsonian RG
In this subsection, we discuss the difficulties in calculating the self energy and the Z factor using Wilsonian RG in slightly greater detail. In particular, the quasiparticle weight comes from the frequency derivative of the self energy, with (the external) . In Wilsonian RG, the first dependent contribution to the self-energy comes from the two-loop “sunrise” diagrams like the one shown in Fig. (2) (a). In a one-dimensional system, the contribution from this diagram to the self-energy at a certain step of the RG is proportional to the integral (using rescaled internal momenta and frequencies as in Eq. (77) of the Appendix A)
where the delta functions arising from frequency and momentum conservation have been used to carry out the integral over and . The subscripts on the (momentum) integral signs are used to denote the constraints that the integrated momenta belong to the eliminated shell: , and the function , with defined to be zero unless , keeps track of the same constraint on . For , . By carrying out the frequency integrals using contour integration, it is straightforward to verify that non-vanishing contributions to the integral for can come only from the regions , , or , , . Either of these is incompatible with the momentum shell constraints on and , unlessKopietz2 () . So the above contribution to the self energy vanishes for , and there would certainly be no contribution from infinitesimal mode elimination, with . Thus, nonvanishing frequency dependent contributions to the self energy can arise only from the one loop (or Hartree) diagrams involving the frequency dependent two body and three body vertexes like the ones in Fig. (14), which are “irrelevant” in the RG sense. But even this contribution will not appear in the first few steps of the RG, i.e., not until the running cutoff is reduced to . Thus it becomes cumbersome to calculate self-energy contributions beyond one-loop using the Wilsonian RG Metzner (). Therefore we propose the EPMS scheme, which makes the calculation of two loop contributions to the self energy relatively easier.
iii.2 The EPMS method
Now we introduce the procedure of EPMS by elucidating the similarities and the differences between EPMS and Wilsonian RG. EPMS is different from Wilsonian RG Shankar () in its way of mode elimination. In Wilsonian RG, we calculate diagrams with the constraint that all the internal modes are only fast modes. In EPMS, we calculate the diagrams with the modified constraint that at least one of the internal modes is a fast mode. In this sense, EPMS can be regarded as a field theory version of Anderson’s poor man’s scaling method RG1 (). The sunrise diagrams in Fig. (2) attempt to depict the difference by way of an example. We note that EPMS still retains the spirit of RG in that we integrate out high energy degrees of freedom and study the low energy effective theory Shankar (); RG2 (). As discussed above, Wilsonian RG is not very convenient for calculating the two loop contributions to the self energy because the non-vanishing contributions come from formally irrelevant two or three body vertexes produced in previous steps, and furthermore do not appear until the running cutoff reduces to . EPMS proposes to overcome this difficulty by taking into account all the contributions (that would have arisen in the subsequent steps of Wilsonian RG) from some of the formally irrelevant two and three body vertexes at the same time when those vertexes are produced. With this idea, non-vanishing contributions to the frequency dependent self energy, for example, appear at the very first step of EPMS, and are accumulated continually from the EPMS mode elimination process. The same form of effective action is obtained after mode elimination in EPMS as in Eq. (77) (see Appendix A) from Wilsonian RG, but with different multiplicative renormalization factors , and . Apart from this, the steps involving the rescaling of frequencies, momenta, and fields in EPMS are the same as in Wilsonian RG.
In the following, we use second order renormalization of the one body vertex (or self energy) as an example. In Wilsonian RG, when we calculate the sunrise diagram in Fig. (2)(a) in the step (i.e., when the running cutoff is reduced from to ), all the (rescaled) internal momenta being integrated out are restricted to the shell as in Eq. (8). However, in EPMS, while one of the internal momenta being integrated out is still restricted to the shell, all the others could be either fast modes or slow modes, as depicted in Fig. (2)(b). The simplest way of doing this is as follows: First, prior to the step of EPMS, we calculate the net self energy to second order in the current values of the leading coupling constants:
Here, , denote the rescaled internal frequencies and momenta, and , the rescaled external frequency and momentum; like in Eq. (8), the momentum and frequency conserving delta functions have been used to calculate the integrals over and , with the remaining rescaled internal momenta being fully integrated, from to ; and is the square sum of the running coupling constants discussed and defined in Section V. Then we take the difference
as the incremental contribution to the self energy from the mode elimination step of the EPMS program. The factor is used in order to retain the same running coupling constants and relevance as in . From this we calculate the multiplicatively cumulative contributions to the renormalization coefficients and introduced in Appendix A as,
where the subscript ”” is used to denote that the contributions are from EPMS.
iii.3 Additional rules of EPMS
Although EPMS overcomes the difficulties of Wilsonian RG in calculating the self-energy contributions, it has some disadvantages. In Wilsonian RG, there is no divergence in any intermediate step because the upper and lower limits of integration are always finite numbers with the same sign. But there is no guarantee of this in EPMS; it is certain to work only when divergences that could in principle be present in get cancelled in calculating . The logarithmic divergence in one-dimension is an example.
Also, as discussed in the last subsection, the true difference between EPMS and Wilsonian RG is in the order in which diagrams are being summed. What is produced at a certain step in EPMS includes not only the contributions from that very step of Wilsonian RG, but also a set of terms from later steps and of higher order. This poses the problem of avoiding double counting in EPMS. In order to resolve the double counting issue, we have to add some additional rules into the EPMS procedure. First of all, given a specific order of calculations, only the highest order diagrams and tree diagrams are calculated with running coupling constants. For example, Fig. (2) and Fig. (14)(a) are the highest order loop diagrams in a second order calculation. On the contrary, the lower order diagram like Fig. (3) should be calculated with bare(original) coupling constant. Second, the contribution to lower order vertexes from formally irrelevant higher order vertexes is calculated using the original coupling constants. For example, in the Fermi gas model the original couplings of irrelevant vertexes are zero. So there will be no contribution from irrelevant vertexes to lower order vertexes in EPMS.
A more detailed discussion of the comparison between the EPMS calculations and the Wilsonian RG calculations is presented in Appendix B.
iii.4 The expansion prescription
The angular integrals that arise when one implements RG calculations in dimensions larger than 1-d are in general rather difficult to evaluate. In this paper, we are particularly interested in dimensions with . Drawing inspiration from Ref. (T1, ), we use the following prescription which should be valid for small values of and the external frequency . It relies on the fact T1 () that the Cooper (particle-particle) channels (see Fig. (4) b) do not depend sensitively on dimensionality while the Peierls (particle-hole) channels (see Fig. (4) a) do. This asymmetry can be understood as follows. For the marginal one-dimensional Cooper channels, the momentum transfer is zero, which means the two incoming (or outgoing) momenta are equal and opposite. In dimensions higher than 1, the outgoing momenta can be at an arbitrary angle relative to the incoming momenta. This property leads to the Cooper (BCS) instability in one, two and three dimensions Shankar (). On the other hand, in the marginal case for the one-dimensional Peierls channel, the momentum transfer is . In higher dimensions, the angle between incoming and outgoing momenta is strongly restricted if the momentum transfer is fixed and nonzero. Therefore the Peierls instability is suppressed by the angular integral in higher dimensions.
We propose the following simple prescription for expansion by considering this asymmetry as the leading effect arising from the extra dimensions that needs to be taken into account. Hence, when the Cooper channels are calculated in dimensions, we use the same formula as in 1-d. However, when calculating the Peierls channels in dimensions, we introduce an additional factor in an appropriate momentum integral. For example, in the case of sunrise diagram like the ones in Fig. (2), the momentum integral over (the one in the opposite direction relative to the other two) should include the factor . This introduction of is to be regarded as a purely mathematical device to approximately take into account the crucial effects of the extra dimensions. We note also that in the RG calculation, the in should always be in terms of the original scale. Otherwise, the rescaling of in would lead to the changes in the relevance of different terms and get in conflict with the fact that the relevancy of each term is the same in one, two and three dimensions Shankar (). We show in the next section that this prescription gives the same flow equations for the coupling constants as in Ref. (T1, ).
In principle, there could be other slightly different schemesMetzner () for dimensions. The reason for choosing our scheme is that it introduces the higher dimension effects without changing the interaction effects in 1-d qualitatively. In a 2-d system, there are three classes of interactionsMetzner (); Shankar (), i.e., back, forward and exchange scattering interactions. If we apply this 2-d classification of interactions directly in dimensions, both and in the 1-d model get regarded as back scattering terms. Then it would be hard to connect to 1-d case as well as look at the crossover behaviors. Instead, our prescription in dimensions could be imagined as saying that for both the and terms, the incoming as well as outgoing momenta are equal and opposite, but the outgoing momenta could be a bit off the incoming line. And we still take and as back and forward scattering interactions respectively. Such a generalization of the 1-d model does not change the nature of and in 1-d qualitatively and hence helps to understand the crossover behaviors between Tomonaga-Luttinger liquid in 1-d and Fermi liquid in higher dimensions.
Iv Second order flow equations for the coupling constants
In this section, we derive the second order flow equations for the coupling constants in 1+ dimensions. If we only look at the contribution to the marginal couplings, corresponding to all external momenta being at the Fermi surface (external ), the only non-vanishing diagrams are shown in Fig. (5) (the Peierls and Cooper channels involving the same branches are easily shown to give vanishing contributions) and the internal momenta would both have to be high momenta. Therefore, EPMS will give the same results as Wilsonian RG Shankar (). Since the two body vertexes are marginal, in calculating the diagrams we can use momenta and frequencies as per the original scale and change the limits on the momentum integrals to take into account the running cutoff without changing the result.
The Peierls channel contribution with and on different branches is:
where the last result is obtained for infinitesimal change in the running cutoff, as obtained by setting , and . The Cooper channel with and on different branches gives
Again, the last result above is for infinitesimal RG.
Then, for the EPMS step reducing the running cutoff from to , the incremental change in the coupling constants is given by , and . Hence we get the flow equations:
These equations are essentially the same as in Ref. (T1, ), except that appears here instead of in the reference. The equations show an emergent crossover scale . For , the dimensional equations behave like their 1-d versions Solyom (): , and . The solutions are
where and are the values of and respectively when entering the region, and . We can therefore write down the following approximate solutions setting , and choosing , using the small solutions:
We can see from these figures that and are fairly good approximations for the exact and in the two asymptotic regions and . In 1-d, if , then - this is referred to as the 1-d fixed point model; even when , and as . However these results no longer hold in the case of dimensions - for as well as for almost all other initial conditions, goes to zero asymptotically as for .
V Calculation of , the quasiparticle weight
To calculate , it is convenient to rewrite the interaction terms in action in Eq. (7) as
where is the opposite spin of . Since we are only interested in dimensions with , we use the same prescription as in Section III in our EPMS calculation of . Therefore the integrals including the functions look the same as in the 1-d case, except for the additional factor in the Peierls channels. In the second order sunrise diagram, it is easy to see that each of the two body interaction couplings above only couples to itself. The , and terms in the contributions to the self energy are all given by the same diagram, as shown in Fig. (2); so the net contribution is proportional to . Furthermore, there are contributions to the self energy proportional to coming from the diagrams in Fig. (7).
The calculation of the self energy using the EPMS prescription extended to dimensions using the sunrise diagrams like the ones shown in Fig. (2) and Fig. (7), labelled such that and are from the same branch (left branch for example), will hence involve the integrals
where in the last step we have set and . The last result is exactly what we would have obtained by including the dependent factor into the integral rather than into the integral at the outset; i.e., in the case with and from the same branch, including factor into integral is equivalent to including into integral. Henceforth, for convenience, we use in the following calculations of and in dimensions comparison with the formalism in the 1-d case. Also, since we are interested in the dependent part of the self energy, we set the external (at the Fermi surface) without loss of generality.
where, as before, . The integral will vanish after integrating over and unless the integrand has poles in different half planes; that is, either , , or , , ). Either set of conditions is impossible to satisfy, so the integral vanishes; hence the term does not contribute to .
Next, consider the contribution proportional to . The relevant integral is
Evaluating the frequency integrals by contour integration, we see that there are two regions of space which can lead to non-vanishing contributions: either , , or , , . In the former case, after simplifying we get the condition , , leading to the contribution
Likewise, in the latter case, we have the contribution
It is convenient at this stage to change into real frequencies by the analytic continuation () so that we are looking at the renormalization of the retarded Green function. Thus, from Eqs. (28) and (29), we get