# Superconductivity near a ferroelectric quantum critical point in ultra low-density Dirac materials

###### Abstract

The experimental observation of superconductivity in doped semimetals and semiconductors, where the Fermi energy is comparable or smaller than the phonon frequencies, is not captured by conventional theory. In this paper, we propose a mechanism for superconductivity in ultra low-density Dirac materials based on the proximity to a ferroelectric quantum critical point. We consider both ionic and covalent crystals. We derive a low-energy theory that takes into account the strong Coulomb forces of the polarization modes and the direct coupling between the electrons and the soft phonon modes. We show that in the case of ionic crystals, Coulomb repulsion is strongly screened by the lattice polarization near the critical point. Using a renormalization group analysis in the low-density limit, we demonstrate that the effective electron-electron interaction is dominantly mediated by the transverse phonon mode. We find that the system generically flows towards strong electron-phonon coupling. Hence, we propose a new mechanism to simultaneously produce an attractive interaction and suppress strong Coulomb repulsion, which does not require retardation. We obtain qualitatively similar results for covalent crystals, though the Coulomb screening in this case is much weaker. We then apply our results to study superconductivity in the low-density limit. We find strong enhancement of the transition temperature upon approaching the quantum critical point. Finally, we also discuss scenarios to realize a topological -wave superconducting state in covalent crystals close to the critical point.

###### Contents

- I Introduction
- II Model
- III Renormalization group analysis near the critical point
- IV Superconductivity
- V Conclusions
- VI Acknowledgments
- A One-loop diagrams
- B RG equations for ionic crystals
- C RG equations for covalent crystals
- D Comments about crystal anisotropy
- E Decomposition of the effective interaction into pairing channels

## I Introduction

A key ingredient in superconductivity is the pairing between electrons. In spite of the strong Coulomb repulsion in free space, at low energy electrons experience an effective attraction and form bound states – the Cooper pairs. Thus, superconductivity essentially relies on a mechanism that concomitantly reduces the Coulomb repulsion and generates an attractive interaction. In metals, the attractive interaction results from the interchange of longitudinal phonons that couple to the electronic density. This mechanism requires retardation, namely, the crystal vibration must be much slower than the electronic motion. In terms of energy scales, this implies that the Fermi energy is much larger than the Debye frequency. In the intermediate frequency regime, between these two scales, the Coulomb repulsion is logarithmically suppressed, while the phonon interaction is unaffected Tolmachev and Tiablikov (1958); Morel and Anderson (1962). As a result, the net interaction between electrons may become attractive below the Debye energy.

From this perspective, systems of low carrier concentration, such as doped semimetals and semiconductors, are not expected to exhibit superconductivity. This is because they have a low Fermi energy, which is comparable to, or even smaller than, the typical Debye frequency, and thus does not allow for the dynamical screening of the repulsion. Moreover, the superconducting transition temperature is exponentially sensitive to the density of states, which is typically two orders of magnitude smaller in semimetals and semiconductors when comparing with a standard metal.

Surprisingly however, superconductivity in doped semimetals and semiconductors is ubiquitous. It was first discovered in SrTiO Schooley et al. (1964) and later in many other materials Bustarret (2008). To the best of our knowledge ^{1}^{1}1Superconductivty has been measured in Zr doped SrTiO where the density was argued to be even lower Eagles (2016), the lowest density superconductors discovered to date (in reverse order) are Tl doped PbTe Matsushita et al. (2006), Sr doped BiSe Liu et al. (2015), YPtBi Butch et al. (2011), SrTiO Lin et al. (2014) and elemental Bismuth Prakash et al. (2017). It is noteworthy that, except for SrTiO, all of these materials are either narrow band topological insulators or topological semimetals. The common feature is a nearly touching conduction and valence bands with Dirac-like dispersion.

It is also interesting to notice that SrTIO, PbTe, and SnTe naturally reside close to a paraelectric-ferroelectric phase transition Rowley et al. (2014); Jantsch et al. (2001), which can be tuned in various manners. Recent experiments show that the superconducting transition temperature in SrTiO depends on the distance from the ferroelectric quantum critical point Stucky et al. (2016); Rischau et al. (2017); Rowley et al. (2018). Thus, it is natural to study the relation between the ferroelectric quantum critical point and superconductivity.

A variety of theoretical frameworks have been proposed to discuss superconductivity in the limit of low density, including polar phonons Gurevich et al. (1962); Takada (1980); Savary et al. (2017); Rowley et al. (2018); Gastiasoro et al. (2019), plasmons Takada (1978, 1980); Ruhman and Lee (2016, 2017), multi-band effects Koonce et al. (1967); Binnig et al. (1980), soft optical phonons Appel (1969), the charge Kondo effect Matsushita et al. (2005), and instantaneous attraction Eagles (1969); Gor’kov (2017). It is particularly important to single out the seminal contribution of the authors of Ref. Gurevich et al. (1962), who pointed out an essential ingredient in any theory of low-density superconductivity: a long-ranged attractive interactions. Only an interaction with a range that is comparable to the interparticle distance can lead to a sufficiently large coupling constant and drive an instability to superconductivity. This is similar to the phenomenon of Wigner crystallization, where long-ranged Coulomb interaction dominates the kinetic energy in the dilute limit rather than the high-density of states limit. Thus, the constraint of long-ranged attraction narrows down the range of viable pairing mechanisms in the low-density extreme limit. Such interaction may result from a dynamically screened Coulomb repulsion Gurevich et al. (1962); Takada (1978), fluctuations of an order parameter close to a quantum critical point Chubukov and Schmalian (2005); Lederer et al. (2015); Metlitski et al. (2015); Wang et al. (2016) or Goldestone mode fluctuations in certain types of spontaneously broken continuous symmetries Watanabe and Vishwanath (2014).

It has been theoretically proposed that pairing in SrTiO is mediated by the fluctuations near a ferroelectric (FE) quantum critical point (QCP) Edge et al. (2015); Wölfle and Balatsky (2018); Kedem (2018). However, these studies leave a few important questions open. To understand them let us first quickly review some basic facts regarding the FE QCP. This transition is essentially a structural transition where the order parameter is a vector (a lattice distortion) that spontaneously breaks inversion and rotational symmetry in the ordered state. For example, consider the diatomic ionic crystal in Fig. 1. In the ferroelectric phase, the two ions in the unit cell are distorted from their cubic Bravais lattice points. Because they have a charge imbalance (ionic), they induce a uniform electric polarization density. Thus, dynamically, the transition is described by a soft optical phonon mode associated with the relative displacement of the two charged ions. Such a phonon mode has three polarizations: one longitudinal-optical (LO) and two transverse-optical (TO) modes. However, in contrast to naive expectation, the dipolar interactions between these distortions prevent the LO mode from becoming soft at the trasition point Roussev and Millis (2001). Consequently, the soft bosonic modes associated with the dynamiccs of the FE QCP are purely transverse, which are typically only weakly coupled to the Fermi surface. Thus, the main question that remains unanswered by Refs. Edge et al. (2015); Wölfle and Balatsky (2018); Kedem (2018) is whether these transverse modes can couple strongly to gapless electrons. For more details we refer to a recent comment written by one of us Ruhman and Lee (2019).

In this paper, we answer this question. We show, for the particular case of a Dirac material, how the transverse modes couples to gapless electrons in the long-wavelength limit (see Sec. II). We then use the renormalziation group (RG) approach to study the tendency of this theory close to the critical point in Section III. We find that the combination of the strong polar dynamics and a nearly touching conduction and valence bands of the Dirac theory leads to strong screening of the Coulomb repulsion alongside with an enhancement of the electronic coupling to the soft transverse phonon mode. Thus, the phonon-mediated attraction by the transverse modes generically overcomes the Coulomb repulsion close to a FE QCP. Interestingly, this flow takes place even at zero electronic density. Therefore, it is distinct from the standard Tolmachev-Anderson-Morel mechanism to obtain attraction Tolmachev and Tiablikov (1958); Morel and Anderson (1962); it does not require the phonon frequency to be smaller than the Fermi energy. Finally, in Sec. IV, we analyze the possible superconducting instabilities from the interaction mediated by the critical phonon (ferroelectic) mode. We find strong enhancement of the transition temperature due to the enhancement of the electron-phonon coupling close to the critical point.

For completeness, in every step of the way, we compare our analysis with the case of covalent crystals (non-ionic) where both LO and TO modes are soft at the critical point. In this case, the screening of the Coulomb repulsion is much weaker (only logarithmic). Interestingly, however, the interplay between phonon-mediated attraction and Coulomb repulsion opens a possibility of topological -wave superconductivity in a certain range of parameters.

Before proceeding to the analysis itself, we would like to note that the case of a Dirac theory is somewhat simpler than the case of a finite Femri surface. This allows us to form a clearer picture of the fate of the transition and puts the calcualtions under control. However, we also draw motivation from a relastic system: the ionic alloy PbSnTe, which undergoes a FE phase transition at Jantsch et al. (2001). The transition can be described by the spontaneous formation of a relative displacement between the Pb and Te lattices, therefore, it leads to a gapless optical phonon mode (see Fig. 1). When is further increased above , the alloy undergoes a second, topological phase transition, between a trivial insulator and a topological crystalline insulator Hsieh et al. (2012). The topological transition entails gapless Weyl points close to the -points of the Brillouin zone Liang et al. (2017). When doped with Tl or In atoms, this alloy becomes metallic and superconducting, with the transition temperature exhibiting a peak at some intermediate value of Parfen et al. (2001).

Thus, it seems doped PbSnTe is perfectly suitable for our theory as it includes a FE QCP, a small Fermi surface with Dirac dispersion and superconductivity. However, it is important to note that the situation is more complex. In pure PbTe, for example, superconductivity appears only when doped with Tl. Additionally, it has been found that the superconducting state emerges only above a critical density where additional electron pockets become populated Giraldo-Gallo et al. (2018). Nonetheless, we find the question of the ferroelectric quantum critical flcutations coupled to gapless fermions an interesting problem, which is definitely relevant to this alloy and possibly relevant to other system.

## Ii Model

We now consider the low energy effective field theory of a Dirac semimetal near the ferroelectric transition. The Euclidean (imaginary time) action is given by the sum

where the first three terms describe the dynamics of the fermions , the optical phonon field and the Coulomb field , while the latter three describe their interactions. The Coulomb field should be considered as a Hubbard-Stratonovich decomposition of the instantaneous Coulomb interaction. Now we specify these terms in detail.

### ii.1 Quadratic terms

The electron term – The electron quadratic term (motived by the model of the PbTe crystal Hsieh et al. (2012)) reads

(1) |

where is a four-component Dirac spinor, denotes different fermionic flavors, and . Parameters , and stand for electron velocity, Dirac mass, and Fermi energy, respectively. We use Hermitian gamma matrices and , where are usual Pauli matrices. Notice that here we have assumed an isotropic dispersion by taking the same velocity in all directions. The anisotropic case does not modify the main qualitative results of this paper, and therefore we comment on it in Appendix D. For generality, we have assumed a non-zero mass term and a finite Fermi energy . However, we will neglect them in our RG analysis, assuming that they are much smaller than other relevant energy scales.

We note two important discrete symmetries of Eq. (1): inversion symmetry and time reversal symmetry . The action of these symmetries in terms of Dirac matrices is given by and , respectively, where is complex conjugation.

The phonon term – Next, we consider the dynamics of the phonon modes, which become soft at the FE phase transition. To have an intuitive picture in mind, we consider the scenario in which the FE order is dominantly generated by a lattice distortion. For simplicity we consider a cubic ionic crystal with two atoms in the unit cell (the rocksalt structure of the IV-VI semiconductors, see Fig. 1). We label the two sublattices by and corresponding to the “blue” and “red” ions, which have equal in magnitude and opposite sign charges. Each sublattice has a corresponding phonon displacement field and . As usual, there are two modes: a gapless acoustic mode given by the sum and a gapped optical branch given by the difference . Near the FE transition, the optical branch becomes nearly gapless and is described by the effective action

(2) | ||||

Here and are the longitudinal and transverse phonon velocities, respectively, is the phonon mass, which is the tuning parameter to the transition, and is the lowest order symmetry allowed anharmonic correction to the phonon energy (where we have neglected additional anisotropic terms allowed by the cubic symmetry Roussev and Millis (2001)).

The Coulomb term – The third quadratic term is that of the Coulomb potential:

(3) |

where is the bare dielectric constant, which accounts for the contribution of core electrons. This contribution is due to the transitions between the high-energy atomic configurations, and does not include the contributions from the lattice dynamics or electronic interband transitions close to the Dirac point.

### ii.2 Coupling terms

Electron-Coulomb coupling – We start with the coupling between the Dirac electrons and the Coulomb potential

(4) |

where is the electronic density.

Phonon-Coulomb coupling – The coupling of the ferroelectric phonon modes to the Coulomb potential follows from Eq. (4) by noting that the deviations of the “red” and “blue” ionic density from equilibrium are given by and . Given that the ionic charges are of equal magnitude and opposite signs, the coupling of the lattice to the Coulomb field is given by

(5) |

where is the ionic charge on “blue” sites (charge on “red” sites equals ).

Notice that the form of the coupling (5) implies that only the longitudinal phonon mode couples to the Coulomb field. We also point out that after integrating Eq. (5) by parts one gets a dot product between the polarization density and the electric field . Therefore, this equation can also be viewed as the action of a dipole moment density in an electric field. Finally, in the case of a non-polar covalent crystal (e.g. elemental bismuth) all atoms in the unit cell are neutral, leading to a vanishing coupling .

Electron-Phonon coupling – We now consider the coupling between the Dirac electrons and the phonon modes. We write down this coupling from general symmetry arguments. The phonon mode is a time-reversal invariant vector. Inspecting all possible local Dirac bilinears specified in Table 1, we find that the only Dirac bilinear that forms a time-reversal symmetric vector and, thus, is allowed to couple to the phonon displacement field is . Therefore, the corresponding coupling is given by

(6) |

In Eq. (1), we have assumed the Dirac cones occur at the inversion symmetric points in the Brioullin zone. In the case they do not, the coupling can also include inter-flavor scattering.

We emphasize that, for simplicity, we consider a rotationally symmetric model in the main text. We discuss the possible effects of the cubic anisotropy in Appendix D.

Bilinear | ||||
---|---|---|---|---|

## Iii Renormalization group analysis near the critical point

We now use RG to analyze the theory introduced in the previous section. Summing up Eqs. (1)-(6) we have

(7) |

The coefficients and account for the renormalization of the dynamical terms.

We apply the standard momentum-shell RG scheme Fisher (1974) by separating fields into short- and long-scale parts according to (analogously with fields and ), followed by the integrating out the high-energy part within an infinitesimal cylindrical momentum-frequency shell . Here, is a momentum UV cutoff corresponding to the scale at which electron dispersion can be considered linear, and is “RG time”. As the second step, we further rescale momenta, frequencies, and the long-wavelength parts of the fields according to

(8) |

to restore the UV cutoff back to . Above, is the dynamical exponent, and , , are engineering field dimensions. This rescaling leads to the tree-level RG flows of the couplings after coarse-graining by the factor (the argument is suppressed for brevity):

(9) |

It should be mentioned that the choice of dynamical and field exponents is somewhat arbitrary here since it does not affect the flow of dimensionless coupling constants Radzihovsky (2011); Kozii et al. (2017). The special choice , , , , and makes the theory scale invariant. This is a non-interacting fixed point. At this fixed point, and are relevant perturbations, while and are marginal at the tree level. Since is the tuning parameter for the FE transition, we will assume it small close to the critical point. In what follows, we focus on two distinct cases: The case of ionic crystals with and the case of covalent crystals with .

### iii.1 Ionic crystals ()

#### iii.1.1 Fixed point theory

Near the non-interacting fixed point introduced above, is relevant and, at the tree level, obeys the following RG equation:

(10) |

Thus, in the case of ionic crystals, grows rapidly to strong coupling. Therefore, we should first derive the effective low-energy theory with large coupling (of the order of UV cutoff) and then proceed to the RG analysis of the resulting theory. We can integrate out the Coulomb field , which generates the following terms

(11) | |||

The first term is the standard Coulomb interaction. The second term can be viewed as a phonon mass generated in the longitudinal sector (note that it is independent of the magnitude of the momentum). This mass generation is the well-known LO-TO splitting in ionic crystals Mahan (2013). Finally, the last term is the Fröhlich coupling between the longitudinal mode and electronic density.

The generated mass term for the longitudinal phonon mode is of the form . Consider the limit of large , such that , where is the UV cutoff. We can further integrate out the massive longitudinal phonon mode and only keep the leading order terms in the expansion. This procedure generates the standard dynamically screened Coulomb interaction

(12) |

where

(13) |

is the dynamical dielectric constant, which manifestly satisfies the Lyddane-Sachs-Teller relation Mahan (2013).

Close to the critical point , which implies that the dielectric constant scales as and diverges at low energies and momenta. Thus, the effective fine-structure constant , which signifies the strength of the Coulomb interaction, becomes highly irrelevant and flows quickly to zero. It means that the Coulomb interaction is effectively screened by the longitudinal phonon mode. In the end, the FE critical point is controlled by the following effective field theory

(14) |

where

(15) |

is the projector to the plane transverse to . The couplings and here are weakly renormalized after integrating out the longitudinal mode.

Before continuing we would like to make an important remark. We have arrived at a low-energy effective theory describing soft fluctuations of the phonon modes, where the only coupling to the charged fermions is the direct coupling Eq. (6). This is a somewhat counter intuitive result, as the phonon distortions generate huge dipolar moment with strong Coulombic forces. One might expect that these Coulomb forces are still dominant over the direct coupling we consider. We note, however, that the same Coulomb forces that cause the phonon to couple strongly to electrons also cause strong coupling of the phonons to themselves – the coupling that caused the generation of the mass term and the LO-TO splitting. Thus, the polarization of the soft modes is precisely the part of the phonon that does not generate a dipolar moment. Thus, the only relevant coupling to the dynamical soft modes is Eq. (6).

#### iii.1.2 One-loop RG analysis

Now we analyze the effective field theory for ionic crystals (14) within the one-loop RG approach. To get rid of the exponents , which in principle can be chosen arbitrary, we focus on the dimensionless quantities which are independent of these engineering dimensions Kozii et al. (2017); Radzihovsky (2011). First, we derive coupled RG equations for the ratio of the phonon to electron velocities and the dimensionless electron-phonon coupling constant (the details of the calculation can be found in Appendices A and B):

(16) |

The most important result that can be extracted from these equations is that the electron-phonon coupling flows to the strong-coupling regime, see Fig. 2. Consequently, we conclude that the 3+1D ferroelectric quantum critical point in a Dirac semimetal considered in this paper is generically a strongly-coupled problem, even if the original UV value of the coupling constant is small. This may be contrasted with standard QED in three dimensions, where the flow of the coupling is towards weak coupling and the low-energy effective theory is the ?ree Dirac dispersion with renormalized parameters Isobe and Nagaosa (2012). In the next section we discuss the possible superconducting instabilities resulting from this flow to strong coupling.

Our RG equations were derived under the assumptions of the zero Dirac fermion mass and Fermi energy, while the one-loop approximation is valid provided the coupling remains small. Given the flow to the strong coupling, it is important to understand what stops the RG flows. Here we estimate the scale at which becomes of order 1 and defer the discussion of a finite Dirac mass/Fermi energy to Sec. IV. In realistic materials, the Fermi velocity is much bigger than the phonon velocity, thus, one can set in Eq. (III.1.2). Then, the equation for the flow of can be readily integrated. Completely neglecting the mass of the soft mode, we find that grows to at the RG scale which corresponds to the momentum scale

(17) |

Here is the initial UV value of the coupling constant at the scale .

Another natural scale that serves as a cutoff for our RG equations is set by the flow of the (dimensionless) mass of the transverse phonon mode , which determines the critical region:

(18) |

where is the dimensionless phonon-phonon interaction. Assuming that and are small compared to the UV value , the solution of this equation with the exponential accuracy reads as . The critical regime is determined by the condition , which corresponds to the RG scale or, equivalently, momentum scale ^{2}^{2}2More accurately, in case of finite (but small) and , in Eq. (19) should be replaced with .

(19) |

If , the theory flows to the strong coupling regime before the phonon mode gets massive. Our RG equations are only applicable then down to . In the opposite case, , the RG flow should be stopped at , where the transverse phonon mode becomes massive and can be integrated out. At this scale, the system leaves the critical regime, while the coupling between phonons and fermions still remains weak. The corresponding phase diagram is shown in Fig. 3. We will consider the latter case in more detail in the next section in context of superconductivity.

Finally, we discuss the flow of the dimensionless phonon-phonon interaction , which corresponds to the anharmonicity of the lattice oscillations:

(20) |

This equation, again, can be easily analyzed in the physical case . Then, since is a marginally relevant parameter, eventually also flows to strong coupling. It is straightforward to show, however, that this flow does not introduce any new cutoff, as can reach order 1 no sooner than at given by Eq. (17), which is realized in the large- limit (i.e., when the term proportional to on the right-hand side of Eq. (20) can be neglected). It is also interesting to note that sufficiently large in Eq. (20) can, in principle, drive negative, thus indicating a first-order transition into the ferroelectric state. Since we consider hardly realizable in real physical systems, we do not study this possibility in detail here.

Another interesting result that can be inferred from the RG equations is the flow of the electron and phonon velocities (here we fix the dynamical critical exponent ):

(21) |

We see that one of the physical properties of the ferroelectric critical point in Dirac materials is the reduction of the velocities under RG for both the transverse phonon modes and the Dirac fermions. Furthermore, as is shown in Fig. 2, for , the velocity ratio flows to one of two possible values or , depending on whether the initial value of is smaller or larger than , respectively, with . If , the flow is always towards .

So far we only considered a rotationally symmetric model with isotropic electron and phonon velocities. For , however, there is no symmetry that forbids anisotropic terms that manifest the symmetry of the underlying lattice. Nevertheless, the accounting for these terms does not modify main qualitative results described above. Hence, we focus on the isotropic case for the rest of the paper for simplicity, and defer the discussion of possible anisotropies to Appendix D.

### iii.2 Covalent crystals ()

Now we perform similar RG analysis for covalent crystals exemplified by elemental bismuth. While the main qualitative results, such as flow to strong coupling, in this case are the same as for ionic crystals, certain important differences should be discussed. In particular, as mentioned above, the optical phonon distortion generates a negligible amount of polarization in covalent crystals. Consequently, the effective theory for these materials is described by Eq. (7) with . As a result of this important difference, the argumentation of Section III.1 about the screening of Coulomb interaction by longitudinal phonons no longer holds. Instead, one should keep track of the flows of the parameters and , in addition to those considered in Eq. (III.1.2). Focusing again on dimensionless parameters that do not depend on engineering dimensions and , we find the following system of coupled one-loop RG equations:

(22) |

where we defined , , , and .

Since both the longitudinal and the transverse phonon modes become massless at the transition in covalent crystals, they should be treated on equal footing. Consequently, one could in principle consider two (not independent) dimensionless couplings and , which quantify the electron-electron interaction strength mediated by the transverse phonons and the longitudinal phonons, respectively. It is straightforward to show, however, that, in the physical limit , is marginally irrelevant, while flows to strong coupling. Indeed, in this limit, first two equations of (22) take form

(23) |

while analogous equation for would read as

(24) |

Similarly to the case of ionic crystals, electron-phonon coupling flows to the strong-coupling regime, while the Coulomb interaction becomes suppressed under RG. The important difference, however, is that now is only marginally irrelevant and flows to zero much slower. The reason for this difference is that the Coulomb screening in covalent crystals is due to interband (between particle and hole bands) transitions, which is much weaker than the screening due to lattice polarization in ionic crystals.

The flow of the phonon velocities can also be easily studied in the limit . Analogously to ionic crystals, flows to zero in this regime. The flow of , on the other hand, is sensitive to the number of flavors and to the initial conditions, as well as to the scale that stops RG. For instance, at sufficiently large , is increased under RG.

Finally, the flows of the phonon-phonon coupling and the phonon mass are qualitatively similar to the case of ionic crystals, so we do not consider them here in detail.

## Iv Superconductivity

In the previous section we have analyzed the RG flow of the electron-phonon coupling near a ferroelectric quantum critical point. We found that, generically, the critical point is unstable and flows to strong electron-phonon coupling, while the Coulomb interaction flows to weak coupling. As a result, we anticipate that the effective electron-electron attraction mediated by the ferroelectric phonon modes will become dominant over the Coulomb repulsion. Hence, the natural next step in our study is to apply this result to superconductivity.

We emphasize that in our scenario, both for ionic and covalent crystals, the enhancement of the attractive interaction over the Coulomb repulsion does not require finite electron density, in contrast to the Anderson-Morel theory. Nonetheless, this does not imply that the superconducting transition temperature does not depend on density. At least at weak coupling, , a finite density of states is essential to obtain a finite . Therefore, we will now relax our previous assumption about the Fermi energy exactly at the Dirac point and assume a finite Fermi momentum . As before, we separately consider the cases of ionic and covalent crystals. We also focus on the paraelectric side of the transition, i.e., consider systems possessing both time-reversal and inversion symmetries in the normal state.

### iv.1 Ionic crystals

As we have shown in Sec. III, one can define two scales and given by Eqs. (17) and (19), which denote the divergence scale of the electron-phonon coupling and the phonon mass , respectively. When , diverges first and thus the flow is terminated before reaches strong coupling (this regime is denoted by the shaded regions in Fig. 3). In what follows we consider this weak coupling limit, where the BCS approach is applicable, and leave the strong coupling regime for a future work.

The additional scale we have introduced, , can, in principle, also put the flow to a halt when the running scale becomes of order . Thus, depending on the ratio between and , one may again consider two cases. The first case, , is close to the standard Anderson-Morel scenario with the phonon-associated scale being smaller than and we do not consider it here in detail. Since we are interested in understanding superconductivity at very low density, we focus on the opposite case . In this limit, the screening of the Coulomb repulsion by longitudinal phonons occurs well above the Fermi scale, as disscused below Eq. (13), and we obtain a Fermi liquid with static phonon-mediated attraction.

To obtain an effective low-energy interaction, we allow the system to flow according to the RG equations derived in Sec. III.1 until it reaches the scale . We then use Eq. (14) to integrate out the transverse phonon mode, which is massive at this scale, with the effective propagator that can be considered frequency- and momentum-independent. This procedure results in the attractive interaction Hamiltonian

(25) |

where the effective interaction constant

(26) |

is obtained from Eq. (III.1.2) in the limit of . To make the analysis similar to the conventional BCS at this point, we write Eq. (25) in the Hamiltonian formalism (and use instead of ). This became possible since at the scale the phonon-mediated interaction can be considered static, , analogously to BCS theory.

#### iv.1.1 Projection onto the conduction band

Now we analyze the superconducting instabilities due to interaction (25). We assume that the Fermi energy is much larger than the superconducting gap, , hence, the conventional weak-coupling BCS-like treatment is applicable. In this case, it is convenient to project all operators onto the conduction band, thus significantly simplifying the model by reducing it from the original four-orbital to effective two-orbital. In the paraelectric phase, the only case we consider in this Section, both time-reversal and inversion symmetry are present in the normal state, hence, all energy bands remain double degenerate even in presence of strong spin-orbit coupling. The electron states are characterized by two-component spinor . In the presence of spin-orbit coupling, however, components are not spin eigenstates anymore, but rather eigenstates in some band basis. The choice of this basis, however, is not unique. For concreteness, we choose the so-called manifestly covariant Bloch basis (MCBB), in which transforms as an ordinary spin-1/2 Fu (2015). To find this basis, we diagonalize Hamiltonian which corresponds to Eq. (1), and choose the band eigenstates to be fully spin-polarized along the z-axis at the origin of the point group symmetry operations (see also Refs. Kozii and Fu (2015) and Venderbos et al. (2016) for more details). The eigenvectors and in the MCBB that correspond to the states in the conduction band are given by

(27) |

where corresponds to the electron/hole band, respectivly, and we defined . The mapping onto the MCBB then simply implies the transformation , and can schematically be written as , where is a projector onto MCBB. It is straightforward to show then that the Dirac bilinear , which couples to a soft ferroelectric mode, projects onto

(28) |

where is the Levi-Civita tensor, and here are Pauli matrices acting in MCBB.

The effective interaction (25) projected onto the conduction band has form

(29) |

#### iv.1.2 Pairing channels and transition temperature

To demonstrate the superconducting instabilities, we now decompose interaction (29) into pairing channels, analogously to how it has been done in Ref. Kozii and Fu (2015). The time-reversal invariant superconducting order parameter generally takes form

(30) |

where, again, is the Levi-Civita symbol. In systems with strong spin-orbit coupling, spin and angular momentum are not good quantum numbers. Instead, in systems with symmetry considered here, all possible orders are characterized by the total angular momentum . As was shown in Refs. Fu (2015) and Kozii and Fu (2015), the form-factors up to order have the form shown in Table 2. state corresponds to the conventional -wave pairing with , while sates are odd-parity p-wave, and transform as a pseudoscalar ( with ) and a vector ( with ) under the symmetry operations.

bilinear | Coefficient | Transverse | Longitudinal | |
---|---|---|---|---|

even | 1 | 2 | ||

odd | -1 | -2 | ||

odd | 1/2 | -3/2 |

Next, we restrict the effective interaction (29) to the Cooper channel with the zero total momentum by keeping terms with only. Focusing on the states near the Fermi surface, , it is straightforward to decompose Eq. (29) into the pairing channels Kozii and Fu (2015):

(31) |

where coefficients are listed in Table 2. The ellipsis on the right-hand side of Eq. (31) denote terms with ^{3}^{3}3
We note that for the odd-parity sectors where , the terms contain both and angular momenta. As a result, the decomposition is interaction dependent.. The contribution from these terms is numerically small, and we do not consider it in this paper.

Up to order only two channels are attractive and lead to a superconducting instability: the scalar with and the vector with ^{4}^{4}4We note that the values we have obtained here are not the same as in Ref. Kozii and Fu (2015), where . The reason for this difference is the presence of the projector onto the transverse mode, , which was not considered previously..
We thus, conclude that pairing in the s-wave channel is the most dominant superconducting instability. However, the projector significantly reduces in this channel without modifying it in the p-wave channel with . As we demonstrate in the next section when considering the covalent crystals, this difference may play significant role leading to an odd-parity superconductor if the Coulomb interaction is not vanishingly small.

The transition temperature can be estimated from Eq. (31) using the usual gap equation Mineev and Samokhin (1999):

(32) |

In case of the most attractive s-wave channel, it takes form

(33) |

where is the density of states at the Fermi energy per one spin projection per one Dirac node, and all quantities entering it are taken at the RG scale . We emphasize that the upper cutoff in Eq. (32) is not the phonon frequency, as in the standard BCS theory, but given by the Fermi energy. This situation is somewhat analogous to the superfluidity in a charge-neutral Fermi liquid, studied in Ref. Gor’kov and Melik-Barkhudarov (1961). We estimate transition temperature from Eq. (32) as

(34) |

The parameters , , and are the original (UV) values of the Fermi momentum, phonon velocity, and phonon mass, respectively, while is renormalized according to Eq. (26), and we used . Hence, the proximity to the ferroelectric critical point leads to a significant enhancement of . To emphasize this point, we rewrite Eq. (34) in the form

(35) |

where is the estimate for a transition temperature that we would obtain without taking into account the critical nature of the ferroelectric fluctuations. We see that, even within the weak-coupling approximation , we obtain huge enhancement of the transition temperature by a factor . This result is to some extent similar to the enhancement of by the critical nematic fluctuations obtained in Ref. Lederer et al. (2015).

Finally, we estimate the temperature that would correspond to a transition into the p-wave superconducting state :

(36) |

An additional factor in the exponent appears due to the averaging over the directions of vector in Eq. (32). is exponentially smaller than , and, consequently, -wave superconducting phase seems unreachable within the present scenario. However, we demonstrate in the next section that the presence of the repulsive Coulomb interaction can, under certain conditions, suppress -wave channel and drive a system into the odd-parity superconducting state.

### iv.2 Covalent crystals

Our analysis of superconductivity in covalent crystals is similar to the ionic case. There is, however, two important differences. First, the longitudinal phonon mode also becomes soft at a ferroelectric transition, consequently, there will be an additional contribution to the effective electron-electron interaction mediated by a longitudinal mode. Second, the Coulomb repulsion is not screened by the lattice polarization, but only by the interband transitions. As we showed in Sec. III.2, the correspondent coupling constants and are marginally irrelevant, see Eqs. (III.2) and (24). They flow to zero only logarithmically upon RG and, thus, should be taken into account in the weak-coupling regime we are considering here. As we show below in Sec. IV.2.1, the inclusion of the Coulomb interaction allows, upon proper tuning of the coupling constants, to realize a -wave superconductor.

The effective electron-electron interaction due to longitudinal phonons projected onto the conduction band has the same form as Eq. (25), but with the substitution and