Strong coupling behavior of the neutron resonance mode in unconventional superconductors

Strong coupling behavior of the neutron resonance mode in unconventional superconductors

Patrik Hlobil Karlsruher Institut für Technologie, Institut für Theorie der Kondensierten Materie, 76128 Karlsruhe, Germany    Boris Narozhny Karlsruher Institut für Technologie, Institut für Theorie der Kondensierten Materie, 76128 Karlsruhe, Germany    Jörg Schmalian Karlsruher Institut für Technologie, Institut für Theorie der Kondensierten Materie, 76128 Karlsruhe, Germany Karlsruher Institut für Technologie, Institut für Festkörperphysik, 76128 Karlsruhe, Germany
September 10, 2019
Abstract

We analyze whether and how the neutron resonance mode in unconventional superconductors is affected by higher order corrections in the coupling between spin excitations and fermionic quasiparticles and find that in general such corrections cannot be ignored. In particular, we show that in two spatial dimensions () the corrections are of same order as the leading, one-loop contributions demonstrating that the neutron resonance mode in unconventional superconductors is a strong coupling phenomenon. The origin of this behavior lies in the quantum-critical nature of the low energy spin dynamics in the superconducting state and the feedback of the resonance mode onto the fermionic excitations. While quantum critical fluctuations occur in any dimensionality , they can be analyzed in a controlled fashion by means of the -expansion (), such that the leading corrections to the resonance mode position are small. Regardless of the strong coupling nature of the resonance mode we show that it emerges only if the phase of the superconducting gap function varies on the Fermi surface, making it a powerful tool to investigate the microscopic structure of the pair condensate.

The emergence of a resonance mode in the inelastic spin excitation spectrum below the superconducting transition temperature has become an important indicator for unconventional superconductivity in a range of correlated materials. First observed RossatMignot91 ; Mook1993 ; Fong1995 ; Fong1996 ; Bourges1996 in YBaCuO, the phenomenon occurs in other cuprate superconductors Fong1999 ; He2001 ; He2002 , in heavy-electron superconductors Petrovic2001 ; Stock2008 ; Stockert2008 , and in iron-based materials Christianson2008 ; Inosov2010 . Below , one observes essentially two effects in the inelastic neutron spectrum: (i) the low-energy spectral weight is suppressed for energies , where is the magnitude of the superconducting gap; and (ii) a sharp peak occurs at that is centered around a finite momentum . Usually, coincides with the ordering vector of a nearby antiferromagnetic state. For , the imaginary part of the dynamic spin susceptibility at the momentum can be described as

(1)

where is the spectral weight of the resonance mode while the imaginary part of the incoherent part vanishes for .

Figure 1: Spin spectrum for different gap symmetries in the superconducting state. For sign-changing gap symmetry a resonance occurs at in the spin gap and the continuum is governed by particle-hole damping of superconducting excitations.

A promising explanation for the resonance mode that permits detailed comparison with experiment was obtained within an one-loop approach Abanov2000 ; Eschrig2000 ; Abanov2001 ; Eschrig2002 ; Abanov2002 ; Eschrigreview ; Korshunov2008 ; Maier2009 ; Eremin2005 ; Manske2001 . Within this approach, collective excitations of the superconductor are sensitive to the coherence factors of the BCS-like wave function. The coherence factors determine scattering-matrix elements for (i) interactions between Bogoliubov quasiparticles and (ii) interactions between quasiparticles and the pair condensate. In the case of spin-spin coupling (where the scattering matrix is odd under time reversal), the latter processes leads to the emergence of the resonance mode if the phase of the superconducting gap function takes distinct phases at momenta and (assuming that both belong to the Fermi surface). This effect makes neutron scattering sensitive to the internal structure of condensed pairs and allows one to identify unconventional pairing.

Unconventional superconductivity often occurs in close proximity of competing states with long-range order. Consequently, the concomitant quantum criticality requires an investigation of the microscopic structure of the superconducting state (and in particular, of the resonance mode) that goes beyond the usual one-loop approach. It is well known, that itinerant systems in the vicinity of a spin-density-wave quantum-critical point are characterized by the energy scale of the normal state spin excitation spectrum Hertz1976 ; Moriya1985 ; Millis1993 that vanishes at the quantum critical point, where the magnetic correlation length diverges. As a result, precisely at those points on the Fermi surface that are connected by the magnetic ordering vector (i.e. and ) the quasiparticle lifetime for energies above deviates from the standard Fermi-liquid result Abanov2003 ; Metlitski2010 . In two- or three-dimensional systems, these sets of points are referred to as hot spots or hot lines of the Fermi surface, respectively. So far, it is unclear whether or not quantum-critical fluctuations that are relevant at higher energies and contribute to the incoherent contribution in Eq. (1) lead to any feedback on the spectral features of the resonance mode. For example, if higher order vertex corrections to the dynamic spin susceptibility are governed by excitations with energies smaller than (and thus behave similar to Fermi-liquid quasiparticles), then the weak coupling picture is expected to be robust. On the other hand, if such virtual excitations are quantum critical, i.e. have typical energies larger than , the analysis becomes more subtle me ; Onufrieva .

Another open issue is related to the sensitivity of the resonance mode with respect to the variation of the phase of the superconducting order parameter on the Fermi surface. Whether or not this is the case if one takes into account higher orders in perturbation theory needs to be explored. The relevance of strong coupling behavior for the resonance mode is also suggested by the observation of a nearly universal ratio of and in a wide range of systems Greven2009 , which one would not expect from weak coupling theory.

In this paper we evaluate self-energy and vertex corrections to the dynamic spin susceptibility in the superconducting state and determine higher order corrections to the neutron-resonance mode. For , we find that both, self-energy and vertex corrections, cause significant changes in the resonance and cannot be ignored, except for very weak coupling strength. Near a magnetic quantum-critical point these corrections are of same order as the leading one-loop result, revealing that the resonance mode is a strong coupling phenomenon. Self-energy corrections are primarily caused by singularities in the fermionic spectrum that were caused by the resonance mode in the first place. In contrast, vertex corrections are dominated by quantum-critical fluctuations contributing to , due to the fact that virtual processes lead to the emergence of the resonance mode. In order to develop a controlled theory of the resonance mode, we perform an -expansion around the upper critical dimension , that reveals how quantum-critical fluctuations affect the dynamic spin susceptibility as function of the dimensionality of the system. These results demonstrate that the theory of Refs. Abanov2000, ; Eschrig2000, ; Abanov2001, ; Eschrig2002, ; Abanov2002, ; Eschrigreview, ; Korshunov2008, ; Maier2009, ; Eremin2005, ; Manske2001, is applicable for three-dimensional superconductors including moderately anisotropic materials. On the other hand, for the neutron resonance mode is a strong coupling phenomenon and our results show that no controlled theory for the effect exists so far. Finally, we demonstrate that higher-order vertex corrections only lead to a resonance mode if the phases of the gap at and are distinct.

I The spin fermion model

Consider an unconventional superconductor in the vicinity of a spin density wave instability. Low-energy spin excitations of the system (i.e. paramagnons) can be described Abanov2003 ; Metlitski2010 in terms of a spin-1 boson that is characterized by the dynamic spin susceptibility

(2)

where and are the wave-vector and frequency, is the antiferromagnetic ordering vector and determines the distance to the instability. The spin dynamics is described by the self-energy . Hereafter, refers to the retarded susceptibility, while is used for the corresponding Matsubara function. Similar notations are used below for fermionic Green’s functions and self-energies. The spin dynamics, encoded in , is a consequence of coupling of the collective spin degrees of freedom to low-energy particle-hole excitations. At low energies and in the normal state, the dominant contribution to the imaginary part of for comes from the fermionic quasiparticles in the vicinity of the hot spots (or hot lines) on the Fermi surface (defined by the relation , where is the bare fermionic single-particle dispersion measured relative to the Fermi energy). In this paper, we consider commensurate magnetic order, where is equal to a reciprocal lattice vector, i.e. .

Let be the creation operator of a fermion with the momentum and spin index . Coupling between spin fluctuations and fermionic quasiparticles is described by the following term in the Hamiltonian Abanov2003 ; Metlitski2010

(3)

where operators are given in real space and is the vector of the Pauli matrices. A microscopic derivation of this model is possible in the limit of weak electron-electron interactions and may be based upon a partial resummation of diagrams in the particle-hole spin-triplet channel. In this case, it is usually not permissible to approach the regime in the close proximity of the magnetic critical point, which for generic Fermi surface shape requires a threshold strength of the interaction. However, we may consider this spin-fermion model as a phenomenological theory of low-energy quasiparticles coupled to spin fluctuations that is valid only at energies small compared to the initial electron bandwidth. At low energies, fermions near the hot spots determine the spin dynamics and are crucial for the spin-fluctuation-induced pairing state. In this case the electronic spectrum near the hot spots may be linearized, (where is the quasiparticle velocity at momentum ). In what follows, we assume that is small compared to the corresponding fermionic scales, which implies smallness of the dimensionless parameter

(4)

Here and are the projections of at the hot spot onto directions that are parallel and perpendicular to , respectively, and is the number of hot spots (lines), or generally the number of fermion flavors that couple to the spin excitations. In what follows, we assume and use , see Fig. 2.

The spin-fermion model can be described by an effective action that in the superconducting state takes the form

(5)

where

is the extended Gor’kov-Nambu-spinor. Here, we use the following notations

The matrices are the usual Pauli matrices and in Eq. (5) we combine the Matsubara frequencies and momenta into and use the short-hand notation

In the basis of the extended spinor , the bare fermion propagator is given by

(6)

The corresponding self-energy matrix in the superconducting state can be written as

(7)

where we defined the matrices

(8)

Using this definitions the dressed Green’s function can be expressed as

(9)

Explicitly, we obtain for the matrix Green’s function:

(10)

The resulting gap function will, as usual, be determined from the solution of the corresponding self-consistency equations.

i.1 Normal-state behavior

In the normal state spin fluctuations can decay into gapless electron-hole excitations which leads to overdamped spin dynamics in agreement with observations obtained in various neutron scattering experiments RossatMignon1991 ; Inosov2010 . The corresponding dynamic susceptibility

(11)

where is given by Eq. (4), can be obtained by evaluating the bosonic self-energy Abanov2000

(12)
Figure 2: Parametrization of the linearization of a hole-like Fermi surface with corresponding AF vector . For the visualization of the Fermi surface experimental fits to a tight binding model of Bi-2212 were used Kordyuk2003 .

In order to calculate the one-loop diagrams it is convenient to linearize the spectrum around the hot spots with , which dominate the integrals. As can be seen in Figure 2 we can linearize , where is the Fermi velocity at the corresponding hot spot. Each of the hot spots contribute equally, which allows us to focus on one of them. Along the same lines we can also linearize the connected hot spot at via

(13)

The velocities and are the perpendicular and parallel projections of on . Introducing new integration variables it is now possible to approximate

(14)

The signs refer to different gap symmetries; we consider to be constant around the hot spots. In order to simplify our calculations we will set in future calculation, which is a suitable approximation for many known unconventional superconductors like Bi-2212 (compare with Fig. 2) .

Under the assumption that we can neglect the momentum dependence of the self-energy near the hot spots, the self-energy (12) yields

The static contribution renormalizes the bare “mass” and determines the correlation length via .

In two dimensions (), coupling of normal-state fermionic quasiparticles with overdamped spin fluctuations leads to renormalization of the fermionic spectrum. Already at one-loop level, one finds non-trivial behavior of the fermionic self-energy at the hot-spots Abanov2003 :

(15)

Here the frequency plays the role of the crossover scale. Indeed, for energies below the self-energy (15) may be approximated by the Fermi-liquid-like expression with the dimensionless coupling constant . However, at higher energies the fermionic spectrum exhibits non-Fermi-liquid behavior as the self-energy (15) on the imaginary axis becomes proportional to the square root of the frequency, .

For our subsequent analysis, it will be important to determine the fermionic self-energy for arbitrary dimensions using the -expansion with the small parameter

(16)

Similarly to Eq. (15), we find the non-Fermi-liquid behavior at high energies

(17)

that is characterized by the coupling constant

(18)

and the energy scale

(19)

where contains the information about the surface of a unit sphere in dimensions. On the real axis this yields in the non-Fermi liquid regime

(20)

For , we find with the characteristic frequency , where is the bosonic momentum cutoff. On the real axis this becomes

(21)

Note, that Eq.(21) holds only for momenta on the hot lines, in contrast to Ref. MFL, , where within the marginal Fermi-liquid phenomenology the same frequency dependence is assumed everywhere on the Fermi surface.

The above results for the normal-state fermionic dynamics demonstrate that the upper critical dimension for non-Fermi liquid behavior of the fermionic spectrum at the hot-spots is . Near three dimensions we can develop an -expansion which is controlled for arbitrary . As we show below, in the limit (i.e. for ) the -expansion is not reliable anymore. One might hope that an expansion with respect to can be developed. As shown in Refs. SSLee2009, ; Metlitski2010, for and gapless fermions in the normal state, the usual loop expansion does not correspond to an expansion in , making a controlled expansion in a complicated task, amounting to the summation of all planar diagrams. An important question is whether the dynamics in the superconducting state, where fermions are gapped, is still plagued by similar problems.

i.2 Pairing instability

In order to investigate the emergence of the resonance mode, we will consider the spin-fermion model deep in the superconducting state. For this we need an estimate of the superconducting gap amplitude at low temperatures. Here, we obtain this quantity by combining the numerical solution of Ref. AbanovEPL01, with the linearized gap equations near (for varying dimension ). Since these gap equations were solved elsewhere Bonesteel1996 ; Son1999 ; AbanovEPL01 ; AbanovEPL01b ; Chubukov2005 ; Moon2010 , we merely summarize the key results to make the article self-contained and in order to introduce the notation used throughout this paper.

In the superconducting state we express anomalous averages through the self-energy and determine this quantity, along with the associated gap function self-consistently. Since the dominant contribution to the bosonic self-energies comes from the hot spots, one obtains for these momenta . In the case of cuprate superconductors, the minus sign corresponds to -wave pairing. In the case of the iron-based superconductors, the minus sign corresponds to the state or a d-wave state, depending on the typical spin-momentum vector .

For , the gap equation determining was solved in Refs. AbanovEPL01, ; AbanovEPL01b, . It was found that the amplitude of the gap function is proportional to the instability temperature , with . Thus, in what follows we will merely determine and use it as an estimate for the gap amplitude in the superconducting state. Related pairing problems with singular pairing interactions were discussed in the context of gauge-field induced pairing in quantum-Hall double layers Bonesteel1996 , color superconductivity Son1999 and the strong coupling behavior in problems with massless boson exchange in three dimensions Chubukov2005 . Quantum-critical pairing with power-law dependence of the pairing interaction were studied in Ref. Moon2010, . In what follows we summarize the key results for quantum-critical pairing as a function of .

The one-loop fermionic self-energy matrix in Nambu-space follows from Eq. (5):

(22)

Using Eq. (7) and this self-energy we obtain the functions:

(23)

The normal and anomalous Green’s function in the superconducting state are thus given by Eq. (10). The self-energies near the hot spots are weakly momentum-dependent and therefore we assume the dispersion correction for the determination of the superconducting transition temperature, because the frequency dependence is dominant in the term. Integrating over fermionic energies then yields the linearized Eliashberg equations eli ; car [noting that ]

(24)

that determine . The self-energies are evaluated at the momenta and , which is suppressed in the notation. The effective coupling function in Eq. (24) is given by the integral

(25)

Here integration over momenta is performed over the components of the bosonic momentum that are parallel to the Fermi surfaceChubukov2005 . The result of the integration is given by

(26)

with the energy scale defined in Eq. (19).

The Matsubara gap function obeys the linearized equation

(27)

It is convenient to bring this equation to the form

(28)
Figure 3: dependence of the universal function .

At the quantum critical point, where , it holds that the transition temperature must be determined by a critical value of the coefficient in front of the Matsubara sum. Then the ratio should take a universal value yielding . Away from the critical point, the transition temperature may be written in the form

(29)

with universal function of the dimensionless coupling constant defined in Eq. (18). For , we recover the BCS behavior . However, in this regime magnetic correlations are so short-ranged that our continuum theory is no longer the appropriate starting point. On the other hand, if the coupling constant is larger than unity, the pairing is quantum-critical and . In Fig. 3 we show the numerical dependence of the strong coupling limit as a function of the dimensional expansion parameter . From the numerical solution of the gap equation we find for the case of two dimensions (the full numerical dependence on is shown in Fig. 4). Although these results are obtained in the limit of large , the calculation is well controlled in the limit of small . In our subsequent analysis we therefore use in the regime of strong magnetic correlations (i.e. for ).

Figure 4: dependence of the universal function for . The dashed curve is an exponential fit to the expected weak-coupling behavior and the dots represent the numerical values of the strong coupling calculation.

Finally, for the power-law dependence of the transition temperature (29) becomes

(30)

which is fully consistent with earlier results Son1999 ; Chubukov2005 . In a recent publication Chubukov2013 , it was shown that momentum dependent self-energies correct the numerical values of Eq. (29), yet do not modify the dependence. Here, we ignore these effects in the determination of the pairing amplitude. This is justified since we are only interested in order of magnitude of the pairing gap.

Ii Analysis of the resonance mode

Now we discuss the implications of the above picture for the behavior of the resonance mode in the vicinity of a magnetic quantum-critical point.

The analysis of the resonance mode as a spin-exciton in the superconducting state, caused by scattering between quasiparticles and the condensate, was investigated in Refs. Abanov2000, ; Eschrig2000, ; Abanov2001, ; Eschrig2002, ; Abanov2002, ; Eschrigreview, and based on the determination of the leading contribution to the bosonic spin self-energy. Key concepts for the emergence of the resonance mode can be carried over from the analysis of the leading order terms. To this end, we follow Abanov and Chubukov Abanov2000 and discuss the emergence of a resonance mode in the superconducting state. Generally, the imaginary part of bosonic self energy vanishes at for frequencies , where is the amplitude of the superconducting gap at the hot spot.  Within weak coupling theory holds that grows continuously at according to if and have the same phase. However, as soon as the phases of and differ, becomes discontinuous at . A key quantity for our analysis is therefore the height of this discontinuity:

(31)

Once the discontinuity in the imaginary part of translates into a logarithmic divergence of its real part at :

(32)
Figure 5: Schematic plot of the real and imaginary part for the bosonic self-energy with discontinuity at for a superconductor with varying phase . On the upper right side we plot a numerical one-loop analysis of the imaginary part for a realistic spectrum of Bi-2212, where we used experimental parameters of Ref. Kordyuk2003 .

Within one-loop approximation the susceptibility (2) with self-energy (32) yields Eq. (1). The resulting energy of the resonance mode is

(33)

with spectral weight

(34)

The resonance energy is bound to occur below the particle-hole continuum that sets in at , while the imaginary part of the incoherent contributions vanishes for , see Fig. 5. Below we will see that at one-loop order, the discontinuity is given by such that . The above results are correct as long as is of order . However, in the limit it was shown that Abanov2000 is determined by the leading low-frequency dependence of . Here, we focus on the former regime.

Our analysis of corrections to the spin-susceptibility that go beyond the leading order still yields that . The emerging discontinuity is then solely responsible for all of the qualitative features of the model, including Eqs. (1) and (33). In order to determine the self-energy of the collective spin excitations, we start from the action Eq. (5) and integrate out the gapped fermions, leading to a theory of the collective spin modes:

(35)

The usual skeleton expansion follows from expanding the logarithm. The overall factor in front of the second term is a consequence of the fact that and are not independent Grassmann fields since we had to extend the Nambu spinor due to the spin-changing interaction and one has to be careful in integrating out the fermionic degrees of freedom. Here, we use the identity Greiner

(36)

where is a Grassmann vector and a quadratic matrix. It is possible to write our path integrals in this form by using the symmetry

(37)

We then obtain

(38)

where we define and use that the determinant of is 1. The expansion of the logarithm leads to the known perturbation series and it is easy to see that we are allowed to replace with the initial matrix . In summary, the only difference to the usual integration over two independent Grassmann fields is the factor in front of the term of the effective action.

In the superconducting state the propagator matrix should be replaced by to make the theory self-consistent.

ii.1 Resonance mode at one-loop

Within the one-loop approximation the bosonic self-energy will be of order . The corresponding contribution to the action is given by

(39)

Here, the one-loop boson self-energy is

(40)

The self-energy is times the spin-susceptibility of fermions in the BCS theory. Using the standard mean-field approach (which here amounts to setting and constant in frequency) we find that on the real axis

(41)

The fermionic propagators in the superconducting state can be written as

with the coherence factors and superconducting dispersion at the hot spots. For zero temperature and positive Eq. (41) yields

(42)

Since the self-energy for negative frequencies can be easily obtained from we will restrict further calculations to . To analyze the resonance mode near the antiferromagnetic ordering vector, we evaluate at . The integral in Eq. (42) is dominated by fermions near the hot spots on the Fermi surface. Consequently, leading to a spin gap in the spectrum of the resonance mode.

Near the  threshold the imaginary part of the bosonic self-energy (42) exhibits a discontinuity. within the one-loop calculation, the height of the discontinuity is given by Abanov2000

(43)

with from Eq. (4). This result occurs for sign-changing gap . It is straightforward to analyze Eq. (42)  for the more general pairing-state with and . Now the discontinuity occurs at and is given by

(44)

The resonance occurs as long as the gap amplitude of both states connected by is finite and the phases of the pairing states are distinct. For the imaginary part of the boson self-energy will grow linearly with until it saturates when it reaches the band-width of the fermions. This general behavior can be seen in the numerical plot shown in Fig. 5.

ii.2 Higher order corrections to the resonance mode

In Ref. Abanov2003, it was shown that for , vertex corrections in the spin-fermion model lead to logarithmic divergences in the normal state. Evaluating the spin-fermion vertex corrections in the superconducting state one finds that this logarithmic divergency is cut off at the scale of the superconducting gap . On the other hand, an analysis of the gap equation for spin-fluctuation-induced pairing yields for arbitrary that quantum critical excitations with are important for the value of the transition temperature .  Therefore, we examine the higher orders in perturbation theory in more detail. Specifically, we are interested in corrections to the discontinuity of Eq. (31). Diagrammatically these corrections are given by

(45)

Here, the wavy lines correspond to Eq. (2) with the one-loop bosonic self-energy. The fermionic lines are the mean-field Green’s functions used in the previous section.

ii.2.1 Self-energy corrections

The first diagram in Eq. (45) takes into account the self-energy corrections to the fermionic Green’s functions. To the leading order these are calculated in Appendix A. The imaginary parts of the normal and anomalous self-energies

(46)

are gapped near the hot spots for frequencies , see also Ref. Abanov2001, . Excluding the strong coupling limit [see the discussion following Eq. (34)] we find that the excitations around are well separated from the continuum yielding sharp quasiparticle resonances. The minimal excitation energy is determined by the real part of the self-energy (46) and may be smaller than the mean-field gap at the hot spots. For zero temperature we can evaluate (41) for external momentum to

(47)

Obviously, there is still a spin gap of , which is as usual determined by twice the minimal excitation energy of the fermionic spectrum. The one-loop fermionic self-energies near the hot spot are functions that depend on the dispersion on the opposite side , see Appendix A. Note: Our approach takes into account leading momentum and frequency corrections which arise due to the interaction of the superconducting fermions with the collective boson mode, but not two-loop corrections in the fermionic self-energy. In the considered parameter regime these momentum and frequency dependencies are weak and in (47) we see that the contributions to the discontinuity come from fermions with which lie around the hot spot. Therefore we expand to leading order in momentum and frequency

(48)

where the coefficients etc. are computed numerically.

Figure 6: Discontinuity containing self-energy corrections relative to one-loop jump for parameter range .

With the help of the self-energy (48) we find for the fermionic Green’s functions in Eq. (47)

and correspondingly for the anomalous propagators. Here, one has to rescale the energy in the coherence factors as well. Since we are interested in the discontinuity of (47) at we can expand the arguments of the two delta functions around and evaluate the frequency integration. The integration is greatly simplified by the usual spectrum linearization around hot spots. Performing this analysis, we find that the minimal excitation energy of the particle-hole spectrum is still such that the spin gap of is unaffected by the self-energy corrections. As a result we find for the discontinuity

(49)

The ratio is shown in Fig. 6 as a function of for , see Appendix A for further details. Since only for the self-energy corrections are of order one for the physical regime of order unity. In the limit of large the parameters (for arbitrary ), see Appendix A. The result suggests that using an expansion, the self-energy corrections can be calculated controllably. However, previous resultsSSLee2009 show that for there are problems with the expansion in the normal state with a gapless fermionic spectrum. It is unclear whether these problems persist in the superconducting state discussed here.

ii.2.2 Vertex-Corrections

Now we turn in the examination of vertex corrections. Performing the perturbation theory in the extended spinor-space, indicated by the double-lined propagator matrices, we can express them as

(50)

where in the following fermionic Matsubara frequencies will be written with capital letters and bosonic ones with small letters. Due to spin rotation symmetry we can restrict ourselves to the zz-component of the bosonic self-energy. The sum has to be executed over all possible combinations of Gor’kov-Nambu Green’s functions with arrow conservation at each vertex

Figure 7: Possible diagrams for the next-order self-energy in the superconducting state. Note the arrow conservation at each vertex indicating the energy-momentum conservation of the theory.