Pairing by a dynamical interaction in a metal

It is our great pleasure to present this paper for the special issue of JETP devoted to 90th birthday of Igor Ekhielevich Dzyaloshinskii. His works on correlated electron systems are of highest scientific quality. He made seminal contributions to quantum magnetism, superconductivity, Fermi-liquid theory, and to other subfields of modern condensed matter physics. In a number of works, including the famous ones on the interplay between d-wave superconductivity and antiferromagnetism at the beginning of “high -$T_{c}$ era” (Refs.  Zheleznyak et al. (1997); Dzyaloshinskii (1987); Dzyaloshinskii and Yakovenko (1988)), Igor Ekhielevich used the renormalization group technique to obtain the flow of the couplings upon integrating out fermions with higher energies. In this study, we apply the same RG technique to analyze the pairing instability in systems with critical dynamical pairing interaction. We consider this as a natural extension of his works. Happy birthday, Igor Ekhielevich, and the very best wishes.

The pairing near a quantum-critical point (QCP) in a metal and its interplay with non-Fermi-liquid behavior in the normal state, is a fascinating subject, which attracted substantial attention in the correlated electron community after the discovery of superconductivity (SC) in the cuprates, Fe-based systems, heavy-fermion materials, organic materials, and, most recently, twisted bilayer graphene Monthoux et al. (2007); Scalapino (2012); Norman (2014); Maiti and Chubukov (2014); Keimer et al. (2015); Shibauchi et al. (2014); Fernandes and Chubukov (2016); Fradkin et al. (2010); Yang et al. (2006); Fratino et al. (2016); Sachdev (2018); Coleman (2015); Cao et al. (2018). Itinerant QC models, analyzed in recent years, include models of fermions in spatial dimensions $D\leq 3$, various two-dimensional models near zero-momentum spin and charge nematic instabilities, and instabilities towards spin and charge density-wave order with either real or imaginary (current) order parameter, 2D fermions at a half-filled Landau level, Sachdev-Ye-Kitaev (SYK) and SYK-Yukawa models, strong coupling limit of electron-phonon superconductivity, and even color superconductivity of quarks, mediated by gluon exchange. These problems have also been studied analytically and using various numerical techniques. We refer a reader to Ref. Abanov and Chubukov (2020), where the extensive list of references to these works has been presented.

From theory perspective, pairing near a QCP is a fundamentally novel phenomenon because an effective dynamic electron-electron interaction, $V(q,\Omega)$, mediated by a critical collective boson, which condenses at a QCP, provides a strong attraction in one or more pairing channels and, at the same time, gives rise to a non-Fermi liquid (NFL) behavior in the normal state. The two tendencies compete with each other: fermionic incoherence, associated with the NFL behavior, destroys the Cooper logarithm and reduces the tendency to pairing, while an opening of a SC gap eliminates the scattering at low energies and reduces the tendency to a NFL. To find the winner of this competition (SC or NFL), one needs to analyze the set of integral equations for the fermionic self-energy, $\bar{\Sigma}({\bf k},\omega)$, and the gap function, $\Delta({\bf k},\omega)$, for fermions with momentum/frequency $({\bf k},\omega)$ and $(-{\bf k},-\omega)$.

We consider the subset of models, in which collective bosons are slow modes compared to dressed fermions, for one reason or the other. In this situation, which bears parallels with Eliashberg theory for electron-phonon interaction Eliashberg (1960), the self-energy and the pairing vertex can be approximated by their values on the Fermi surface (FS) and computed within the one-loop approximation. The self-energy on the FS, $\bar{\Sigma}({\bf k},\omega)$, is invariant under rotations from the point group of the underlying lattice. The rotational symmetry of the gap function $\Delta({\bf k}_{F},\omega)$ and the relation between the phases of $\Delta({\bf k}_{F},\omega)$ on different FS’s in multi-band systems are model specific. E.g., near an antiferromagnetic QCP in a system with a single FS, the strongest attraction is in the $d-$wave channel. In each particular case, one has to project the pairing interaction into the irreducible channels, find the strongest one, and solve for the pairing vertex for a given pairing symmetry.

Away from a QCP, the effective $V(\Omega)$ tends to a finite value at $\Omega=0$. In this situation, the fermionic self-energy has a FL form at the smallest frequencies, and the equation for $\Delta(\omega)$ is similar to that in a conventional Eliashberg theory for phonon-mediated superconductivity; the only qualitative distinction for electronically-mediated pairing is that $V(\Omega)$ by itself changes below $T_{c}$ due to feedback from fermionic pairing on collective modes. At a QCP, the situation is qualitatively different because the effective interaction $V(\Omega)$, mediated by a critical massless boson, is a singular function of frequency. Quite generally, such interaction behaves at the smallest $\Omega_{m}$ on the Matsubara axis as $V(\Omega_{m})\propto 1/|\Omega_{m}|^{\gamma}$, where $\gamma>0$ is some exponent. (Fig. 1). This holds at frequencies below some upper cutoff $\Lambda$. At larger $\Omega_{m}>\Lambda$, the interaction drops even further, and can be safely neglected.

In these notations, BCS pairing corresponds to $\gamma=0$. The superconducting transition temperature $T_{c}$ inthe BCS case can be most straightforwardly obtained by computing the pairing susceptibility $\chi_{pp}$ in the order-by-order expansion in the coupling and identifying the temperature at which it diverges. The series contain the powers of the product of the Cooper logarithm $\log{\Lambda/T_{c}}$ and the dimensionless coupling $\lambda$, defined such that $1+\lambda$ is the ratio of the renormalized and the bare electron masses. The series are geometrical, i.e., $\chi_{pp}=\chi_{0}(1+\lambda\log{\Lambda/T_{c}}+(\lambda\log{\Lambda/T_{c}})^{2}+...)=\chi_{0}/(1-\lambda\log{\Lambda/T_{c}})$. We will see that for $\gamma>0$, the pairing kernel still has a marginal $1/|\omega|$ form, and, as a result, the series for $\chi_{pp}$ still contain logarithms. However, contrary to the BCS case, where $1/|\omega|$ comes from the fermionic propagator, the marginal exponent $-1$ for $\gamma>0$ is the sum of the exponent $\gamma$ from the interaction and $1-\gamma$ from the fermionic self-energy (see below). As a result, the logarithmic series are not geometric, and we will see that $\chi_{pp}$ does not diverge down to $T=0$. We show that the pairing instability still exists, however one needs to go beyond the logarithmic approximation to obtain it.

We show that there exists a special case, which falls in between $\gamma=0$ and $\gamma>0$. This is the limit $\gamma\to 0+$, when $V(\Omega_{m})\propto\log{\Lambda/|\Omega_{m}|}$. In this limit, the pairing instability still can be detected by summing up the series of logarithms for $\chi_{pp}$. However, the series are not geometric, and some extra efforts are needed to sum them up to find the form of $\chi_{pp}$ and detect the pairing instability at $T\sim\Lambda e^{-\pi/(2\sqrt{\lambda})}$.

The model with the interaction $V(\Omega_{m})\propto 1/|\Omega_{m}|^{\gamma}$ displays a very rich physics, and our group has studied it over the last few years. This physics is particularly interesting for $\gamma\leq 2$, where a wide range of the pseudogap (preformed pair) behavior emerges, and for $\gamma>2$, when the new, non-superconducting ground state emerges Wu et al. (2020a). In this communication, we will not discuss these values of $\gamma$ but instead focus on small $\gamma$ and discuss in some detail the difference between BCS pairing at $\gamma=0$ and the pairing at $\gamma=0+$ and at $\gamma>0$. We derive the Eliashberg equation for the pairing vertex and analyze it within logarithmic perturbation theory and beyond it. We then convert the integral Eliashberg equation into the approximate differential equation and obtain and solve the corresponding renormalization group (RG) equation.

The pairing problem at small $\gamma$ attracted a lot of attention in the last few years from various physics sub-communities of physicists Mross et al. (2010); Metlitski et al. (2015); Raghu et al. (2015); Mahajan et al. (2013); *raghu2; *raghu3; *raghu4; *raghu5; Wang et al. (2017, 2018); Fitzpatrick et al. (2015b); Wang and Chubukov (2013); Wu et al. (2019a); Son (1999); Chubukov and Schmalian (2005); Schäfer and Wilczek (1999); Pisarski and Rischke (2000); Wang and Rischke (2002); Damia et al. (2020) . The pairing interaction $V(\Omega)\sim 1/|\Omega|^{\gamma}$ with $\gamma\ll 1$ emerges for fermions near a generic particle-hole instability in a weakly anisotropic 3D system (more explicitly, in dimension $D=3-\epsilon$, where $\epsilon\sim\gamma$, Refs.  Mross et al. (2010); Metlitski et al. (2015); Raghu et al. (2015) ). A similar gap equation with small $\gamma$ holds for the pairing in graphene Khveshchenko (2009). The model with $\gamma=0+$ describes the pairing in 3D systems and color superconductivity of quarks due to gluon exchange Son (1999); Schäfer and Wilczek (1999); Pisarski and Rischke (2000); Wang and Rischke (2002). The $\gamma=0+$ model yields a marginal Fermi liquid form of the fermionic self-energy in the normal state and was argued Varma et al. (1989); *Littlewood_92; Varma (2020) to be relevant to pairing in the cuprates and, possibly, in Fe-based superconductors.

The structure of the paper is at follows. In the next Section we present the set of coupled Eliashberg equations for the pairing vertex $\Phi(\omega_{m})$ and the fermionic self-energy $\bar{\Sigma}(\omega_{m})$ and combine them into the equation for the gap function $\Delta(\omega_{m})$. In Sec. IV we analyze the structure of the logarithmic perturbation theory for $\gamma=0+$ and $\gamma>0$. In Sec. V we go beyond perturbation theory and re-express the integral Eliashberg equation as an approximate differential equation for the pairing vertex and solve it. We show that for $\gamma=0+$, the solution coincides with the result of summation of logarithmic series. For $\gamma>0$, we show that the system does become unstable towards pairing if the interaction exceeds a certain threshold. In Sec. VI we analyze the pairing at small $\gamma$ from RG perspective and reproduce the results of the previous Section. We present our conclusions in Sec. VII.

We consider itinerant fermions at the onset of a long-range particle-hole order in either spin or charge channel. At a critical point, the propagator of a soft boson becomes massless and mediates singular interaction between fermions. A series of earlier works on spin-density-wave order, charge-density-wave order, Ising-nematic order, etc (see Ref.Abanov and Chubukov (2020) for references) have found that this interaction is attractive in at least one pairing channel. We take this as in input and project boson-mediated interaction into the channel with the strongest attraction. As we said, at small frequencies, the interaction scales as $V(\Omega_{m})\propto 1/|\Omega_{m}|^{\gamma}$. We incorporate dimension-full factors, like the density of states, into $V(\Omega_{m})$ and treat it as dimensionless. We assume that the power-law form holds up to the scale $\Lambda$, and set $V(\Omega_{m})=0$ above this scale. To keep $V(\Omega_{m})$ continuous at $\Lambda$ and also to transform gradually between $\gamma=0+$ and $\gamma>0$, we use the following form for $V(\Omega_{m})$ (Fig.1):

Here $\lambda$ is a dimensionless coupling, and ${\bar{g}}$ has units of energy. We assume that $\lambda\ll 1$, but the ratio $\lambda/\gamma$ can be arbitrary. At $\gamma=0+$, the last term in (1) can be expanded in $\gamma$ and yields the logarithmic interaction

The conventional BCS/Eliashberg case in this notations is obtained by extending (1) to the case when a pairing boson has a finite mass. In this situation, $\Omega_{m}$ is replaced by a constant below a certain scale. In BCS theory, this scale is assumed to be comparable to $\Lambda$, such that $V(\Omega_{m})\approx\lambda_{BCS}$ up to $|\Omega_{m}|\sim\Lambda$, and zero at larger $|\Omega_{m}|$.

Below we measure all quantities with the dimension of energy, i.e., $T,\Omega_{m},\Lambda$, in units of ${\bar{g}}$, i.e., introduce ${\bar{T}}=T/{\bar{g}},{\bar{\Omega}}_{m}={\Omega_{m}}/{\bar{g}}$, and ${\bar{\Lambda}}=\Lambda/{\bar{g}}$. Throughout this paper we assume that $\bar{\Lambda}\gg 1$.

Earlier works on the pairing mediated by a soft collective boson found that a boson is overdamped due to Landau damping into a particle-hole pair and can be treated as slow mode compared to a fermion, i.e., at the same momentum $q$, a typical fermionic frequency is much larger than a typical bosonic frequency. This is the same property that justified Eliashberg theory of phonon-mediated superconductivity. By analogy, the theory of electronic superconductivity, mediated by soft collective bosonic excitations in spin or charge channel, is often referred to as Eliashberg theory, and we will be using this convention.

In Eliashberg theory, one can explicitly integrate first over the momentum component perpendicular to the Fermi surface and then over the component(s) along the Fermi surface, and reduce the pairing problem to a set of coupled integral equations for frequency dependent self-energy $\bar{\Sigma}(\omega_{m})$ and the pairing vertex $\Phi(\omega_{m})$.

At $T=0$, the coupled Eliashberg equations are

where ${\tilde{\bar{\Sigma}}}(\bar{\omega}_{m})=\bar{\omega}_{m}+\bar{\Sigma}(\bar{\omega}_{m})$. In these equations, both $\bar{\Sigma}(\omega_{m})$ and $\bar{\Phi}(\omega_{m})$ are real functions. Observe that we define $\bar{\Sigma}(\omega_{m})$ with the overall plus sign and without the overall factor of $i$. In the normal state ($\bar{\Phi}\equiv 0$), we have at $\gamma>0$, and $|\bar{\omega}_{m}|\ll 1$

At $\gamma=0+$,

The superconducting gap function $\bar{\Delta}(\bar{\omega}_{m})$ is defined as the ratio

The equation for $\bar{\Delta}(\omega_{m})$ is readily obtained from (III):

This equation contains a single function $\bar{\Delta}(\bar{\omega}_{m})$, but for the cost that $\bar{\Delta}(\bar{\omega}_{m})$ appears also in the r.h.s., which makes Eq. (7) less convenient for the analysis than Eqs. (III).

For a generic $\gamma>0$, the coupled equations (III) for $\bar{\Phi}$ and ${\tilde{\bar{\Sigma}}}$ describe the interplay between the two competing tendencies – one towards superconductivity, specified by $\bar{\Phi}$, and the other towards incoherent NFL normal-state behavior, specified by ${\tilde{\bar{\Sigma}}}$. The competition between the two tendencies is encoded in the fact that ${\tilde{\bar{\Sigma}}}$ appears in the denominator of the equation for $\bar{\Phi}$ and $\bar{\Phi}$ appears in the denominator of the equation for ${\tilde{\bar{\Sigma}}}$. In more physical terms, a self-energy ${\tilde{\bar{\Sigma}}}$ is an obstacle to Cooper pairing, while when $\bar{\Phi}$ is non-zero, it reduces the strength of the self-energy and renders fermionic coherence.

The full set of the equations for electron-mediated pairing generally must contain the third equation, describing the feedback from the pairing on the bosonic propagator. This feedback is small in the case of electron-phonon interaction, but is generally not small when the pairing is mediated by a collective mode because the dispersion of a collective mode changes qualitatively below $T_{c}$ (Refs. Abanov et al. (2003, 2001a)). In this work, we only consider the computation of $T_{c}$ and will not discuss system behavior below $T_{c}$. For this computation, it is sufficient to restrict with the two equations (III). Moreover, we can (i) treat $\bar{\Phi}(\bar{\omega}_{m})$ as infinitesimally small and neglect it in the denominator of Eq. (III), and (ii) use Eqs. (4) or (5) for ${\tilde{\bar{\Sigma}}}$. For a small, but finite $\gamma>0$, the linearized equation for the pairing vertex is

Observe that the overall coupling is just a number, equal to one, because the factor $\lambda/\gamma$ in $V(\bar{\Omega}_{m})$ cancels out with the same factor in the self-energy. This factor is still present in the denominator, but in the term, which contains $|\bar{\omega}^{\prime}_{m}|^{\gamma}$ and becomes irrelevant at the smallest frequencies

At $\gamma=0+$, the linearized equation for $\bar{\Phi}(\bar{\omega}_{m})$ is

As a first try, we analyze the linearized equation for the pairing vertex perturbatively, by adding up a trial, infinitesimally small $\bar{\Phi}_{0}$ to the r.h.s of Eqs. (8) and (9) and computing the pairing susceptibility perturbatively, order-by-order. Such approach has been commonly used for BCS pairing. The perturbation series there contain singular Cooper logarithms. The logarithmic singularity can be cut either by a finite $T$ or at $T=0$, by a finite total incoming bosonic frequency, $\bar{\Omega}_{tot}$. For consistency with other cases, we set $T=0$ and keep $\bar{\Omega}_{tot}$ finite. The result of order-by-order analysis for a BCS pairing is well known:

The ratio $\bar{\Phi}(\bar{\Omega}_{tot})/\bar{\Phi}_{0}$ (the pairing susceptibility) diverges at $|\bar{\Omega}_{tot}|=\bar{\Lambda}e^{-1/\lambda_{BCS}}$ and becomes negative at smaller $|\bar{\Omega}_{tot}|$, indicating that the normal state is unstable towards pairing. (In a more accurate description, the pole in $\bar{\Phi}(\bar{\Omega}_{tot})$ moves from the lower to the upper half-plane of complex frequency Abrikosov et al. (1965)).

We now do the same calculation for $\gamma>0$. The kernel in Eq. (8) is still marginal at $\bar{\omega}^{\prime}_{m},\bar{\omega}_{m}\ll 1$, but, as we said, now the scaling dimension $-1$ is the sum of $-\gamma$, coming from the interaction, and $-1+\gamma$, coming from the self-energy. This implies that perturbation theory still contains logarithms, but in distinction to BCS, each logarithmical integral $\int d\bar{\omega}^{\prime}_{m}/|\bar{\omega}^{\prime}_{m}|$ runs between the upper cutoff at $|\bar{\omega}^{\prime}_{m}|\sim\bar{\omega}_{max}=(\lambda/\gamma)^{1/\gamma}$ and the lower cutoff at $|\bar{\omega}^{\prime}_{m}|\sim|\bar{\omega}_{m}|$. Because the lower cutoff is finite, we can set $\bar{\Omega}_{tot}=0$. Summing up the logarithmical series, we then obtain Abanov et al. (2001b)

We see that the pairing susceptibility increases as $\bar{\omega}_{m}$ decreases, but remains finite and positive for all finite $\bar{\omega}_{m}$, even when $\bar{\Omega}_{tot}=0$. Re-doing calculations at a finite $\bar{\Omega}_{tot}$ we find the same result as in (11), with $|\bar{\omega}_{m}|$ replaced by max$(|\bar{\omega}_{m}|,{\bar{\bar{\Omega}}}_{tot})$. This implies that for $\gamma>0$, there is no signature of a pairing instability within the logarithmic approximation. The absence of a pairing instability within the logarithmic approximation generally implies that pairing does not develop at weak coupling and, if exists, is a threshold phenomenon. In our case, the dimensionless coupling in the series in Eq. (11) is a number equal to one, i.e., the problem we consider is not weak-coupling. A weak-coupling limit can be reached if we extend the model and make the pairing interaction parametrically smaller than the one in the particle-hole channel. In practice, this is done by multiplying the interaction in the pairing channel by $1/N$, where $N>1$ (Refs. Raghu et al. (2015); Wang et al. (2016); Abanov et al. (2019); *Wu_19_1; Abanov and Chubukov (2020); Damia et al. (2020)), while the interaction in the particle-hole channel is left intact  ¹¹1The extension to $N>1$ was originally rationalized by extending the original model to matrix $SU(N)$, with integer $N$, hence the notation. We treat $N$ as a continuous parameter. In the extended model, $\log{\bar{\omega}_{max}/|\bar{\omega}_{m}|}$ gets multiplied by $1/N$, and at large $N$ the problem becomes weak-coupling. We show below that indeed there is no pairing instability at large $N$.

The case $\gamma=0+$ falls in between BCS and $\gamma>0$ cases. Namely, we show below that the series of logarithms are not geometric, like the ones for $\gamma>0$, however by summing up the series one does find the scale of a pairing instability, like in BCS. We assume and than verify that for relevant $\bar{\omega}^{\prime}_{m}$, $\lambda\log{(\bar{\Lambda}/|\bar{\omega}^{\prime}_{m}|)}$ is small for $\lambda\ll 1$, and neglect this term in the denominator of (9). We keep $\bar{\Omega}_{tot}$ non-zero to avoid divergencies and for simplicity set $\bar{\omega}_{m}$ and $\bar{\Omega}_{tot}$ to be comparable. To logarithmic accuracy, we can then view $\bar{\Phi}$ as a function of a single parameter $\bar{\Omega}_{tot}$.

Because the interaction is logarithmic, perturbation series hold in $\lambda\log^{2}{(\bar{\Lambda}/|\bar{\omega}_{m}|)}$. Evaluating the pairing vertex in order-by-order calculations, we find (see Appendix for details)

At a first glance, the coefficients in (12) are just some uncorrelated numbers. On a more careful look, we realize that the series fall into

Accordingly, the pairing susceptibility diverges when the argument of $\cos$ becomes $\pi/2$, i.e., at $\bar{\Omega}_{tot}=\bar{\Lambda}e^{-\pi/2\sqrt{\lambda}}$. It is natural to associate this scale with $T_{c}$ (Ref. Son (1999)). The susceptibility also diverges at a set of smaller $\bar{\Omega}_{tot,n}=\bar{\Lambda}e^{-\pi(1+2n)/2\sqrt{\lambda}}$, but here we focus only on the highest onset temperature. We see that $T_{c}$ has exponential dependence on the coupling, like in BCS theory, however the argument of the exponent contains $\sqrt{\lambda}$ rather than $\lambda$. This in turn justifies the neglect of the self-energy, because for relevant frequencies, $\lambda\log{(\bar{\Lambda}/|\bar{\omega}^{\prime}_{m}|)}\sim\sqrt{\lambda}\ll 1$ (the corrections due to self-energy have been analyzed in Ref. Wang and Rischke (2002)).

To summarize, the case $\gamma=0+$ is similar to BCS in the sense that pairing occurs for arbitrary weak coupling ($T_{c}$ remains finite even if we replace $\lambda$ by $\lambda/N$ and set $N$ to be large). However, for and finite $\gamma>0$, there is no indication of the pairing instability within the logarithmic approximation.

We now analyze the linearized equation for the pairing vertex for $\gamma>0$ beyond the logarithmic approximation. To do this, we convert integral equation (8) into to a differential equation with certain boundary conditions and solve it non-perturbatively.