## Introduction

Superconductivity exemplifies the dramatic effects of interactions in many-body quantum systems1. In conventional superconductors, electrons exploit the electron-phonon interaction to overcome the Coulomb repulsion2,3 by producing a highly retarded attraction that pairs electrons. This process requires a large ratio between the Fermi and Debye energies EF/ωD > > 14. A challenge to this mechanism is posed by superconductivity in low carrier density metals near polar quantum critical points (QCPs). Such materials, typified by doped SrTiO3 (STO)5, exhibit bulk superconductivity down to carrier densities of order 1019 cm−3, where the relevant phonon frequency exceeds the Fermi energy5 by orders of magnitude. Yet despite this inversion of energy scales, experiments6,7 indicate a conventional s-wave condensate, with a ratio of gap to transition temperature 2Δ/Tc ≈ 3.5 in agreement with BCS theory7.

Several theories have been advanced to explain superconductivity in polar metals using the conventional electron-phonon interaction8 and its extension to include plasmons9,10,11,12. Recently, it has been proposed that the underlying polar quantum criticality is a key driver in the pairing13,14,15,16. However, this appealing idea encounters a difficulty, for the critical modes of a polar QCP are transverse optical (TO) phonons for which the conventional electron-phonon coupling vanishes at low momenta17,18,19. Alternative phonon coupling mechanisms requiring spin-orbit coupling or multiband effects20,21,22 (such as Dirac points23,24) have also been examined and possible superconductivity23,25,26 has been discussed.

We reassess superconductivity in quantum critical polar metals guided by two key observations: first, that the strong ionic screening associated with the enhanced dielectric constant severely weakens the electronic Coulomb interaction (Fig. 1a); second, that in the absence of strong spin-orbit coupling the transverse optical phonon modes, decoupled from the electron charge, can be likened to dark matter, for like baryons in the cosmos, the electrons do not directly interact with the the intense background of zero-point dipole fluctuations. Furthermore like dark matter, the presence of the TO modes is only revealed to the electrons via their stress-energy tensor. In particular, the electrons interact with the energy density of the TO phonons. We model this coupling by the Hamiltonian27,28,29:

$${H}_{{{{{{{{\rm{En}}}}}}}}}=g\int {d}^{3}x{\rho }_{e}({{{{{{{\bf{x}}}}}}}}){\left({\bf P}({{{{{{{\bf{x}}}}}}}})\right)}^{2}$$
(1)

where ρe(x) = ψ(x)ψ(x) is the electron density, $$({\bf P}({{{{{{{\bf{x}}}}}}}}))^{2}$$ is proportional to the energy density of the local polarization P and g is a coupling constant with the dimensions of volume. Microscopically, this interaction can be thought of as arising from the short-range effects of the Coulomb force within a unit cell of the material. The presence of an additional charge at the conduction electron site modifies the potential profile for the ions. A local increase in the electron density attracts the surrounding positively charged ions, reducing the distance between them. This causes the local “effective spring constant” of the phonons to rise in regions of high electon density. The natural units for this interaction are therefore atomic ones, i.e. unit cell volume, and should not be extremely different between different materials.

This coupling suppresses the zero-point fluctuations of the polarization in the vicinity of electrons, which in turn lowers the chemical potential of nearby electrons (Fig. 1b), creating an attractive potential well. To lowest order, the resulting attractive potential is described by the virtual exchange of pairs of TO phonons (see Supplementary Information for details), allowing us to link these ideas to two recent observations: first, that two-phonon exchange appears to drive the anomalous “high-temperature” T2 resistivity of polar metals29,30, and second that two-phonon processes may drive superconductivity31, reviving an old idea27. At the same time, energy fluctuations are present in the vicinity of all continuous phase transitions, implying that their coupling to electrons must be relevant for a broad range of systems.

Here we study the consequences of the coupling to energy fluctuations of the order parameter (1) in quantum critical polar metals. We find that polar quantum criticality results in long-range “gravitational" interactions that mediate attraction between the electrons in an electromagnetically neutral background. This attraction overcomes the critically screened Coulomb repulsion and leads to superconductivity at extremely low densities. With increasing carrier concentration, the long-range character of the attraction is concealed due to shorter distances between electrons, leading to a dome-like superconducting region in the phase diagram. The application of our theory to SrTiO3 shows good agreement with the observed magnitude and doping dependence of Tc. We predict that the impact of this novel mechanism will be considerably enhanced in two-dimensional quantum critical polar systems.

## Results

Our theory is built on an isotropic model for the polar metal, with an action S[ψ, ψ, P] = Se + SC + SEn. Here SEn = ∫dτHEn is the energy fluctuation term (1), $${S}_{e}={\sum }_{k}{\psi }_{k}^{{{{\dagger}}} }({\epsilon }_{\bf k}-i{\varepsilon }_{n}){\psi }_{k}$$, is the electronic action, in terms of the Fourier transformed electron field ψk, where k ≡ (iεn, k) is a four-vector containing the Matsubara frequency εn = (2n + 1)πT and wavevector k. The action of electrostatic interactions and the polar phonons is given by

$${S}_{C}=\mathop{\sum}\limits_{q}\left[\frac{{\left|e{\rho }_{q}-{(\nabla \cdot {{{{{{{\bf{P}}}}}}}})}_{q}\right|}^{2}}{2{\varepsilon }_{0}{\varepsilon }_{1}{{{{{{{{\bf{q}}}}}}}}}^{2}}+\frac{{\omega }_{n}^{2}+{\omega }_{T0}^{2}+{c}_{s}^{2}{{{{{{{{\bf{q}}}}}}}}}^{2}}{2{\varepsilon }_{0}{{{\Omega }}}_{0}^{2}}{\left|{{{{{{{\bf{P}}}}}}}}(q)\right|}^{2}\right]$$
(2)

where the total charge densities eρe − P and q ≡ (iωn, q), where ωn = 2nπT. ε1 is the bare dielectric constant, Ω0 is the ionic plasma frequency, ωT0 is the transverse optical mode frequency and cs is the speed of the transverse optical mode. $${\omega }_{T0}^{2}$$ vanishes at the QCP. The Gaussian coefficients of the polarization, $${\delta }^{2}{S}_{C}/\delta {P}_{a}(-q)\delta {P}_{b}(q)={D}_{ab}^{-1}(q)$$ in (2), separate into transverse and longitudinal components

$${D}_{\alpha \beta }^{-1}(q)={D}_{L}^{-1}(q){\hat{q}}_{\alpha }{\hat{q}}_{\beta }+{D}_{T}^{-1}(q)({\delta }_{\alpha \beta }-{\hat{q}}_{\alpha }{\hat{q}}_{\beta })$$
(3)

where $${D}_{L,T}^{-1}(q)=({\omega }_{n}^{2}+{\omega }_{T0,L0}^{2}+{c}_{s}^{2}{q}^{2})/{\varepsilon }_{0}{{{\Omega }}}_{0}^{2}$$ are the inverse longitudinal and transverse phonon propagators. The longitudinal optical mode frequency $${\omega }_{L0}^{2}={\omega }_{T0}^{2}+{{{\Omega }}}_{0}^{2}/{\varepsilon }_{1}$$ is shifted upwards by the Coulomb interaction.

We first consider the case where g = 0. Integrating over the longitudinal modes, we find that the Coulomb interaction becomes

$${\tilde{S}}_{C}=\mathop{\sum}\limits_{q}\left[{\left|{\rho }_{q}\right|}^{2}\frac{{e}^{2}}{2{\varepsilon }_{0}\varepsilon (q){{{{{{{{\bf{q}}}}}}}}}^{2}}+\frac{{\left|{{{{{{{{\bf{P}}}}}}}}}^{T}(q)\right|}^{2}}{2{D}_{T}(q)}\right],$$
(4)

where

$$\varepsilon ({{{{{{{\bf{q}}}}}}}},i{\omega }_{n})={\varepsilon }_{1}+\frac{{{{\Omega }}}_{0}^{2}}{{\omega }_{n}^{2}+{c}_{s}^{2}{{{{{{{{\bf{q}}}}}}}}}^{2}+{\omega }_{T0}^{2}}$$
(5)

is the renormalized dielectric constant and $${P}_{a}^{T}(q)=({\delta }_{ab}-{\hat{q}}_{a}{\hat{q}}_{b}){P}_{b}(q)$$ are the transverse components of the polarization. Most importantly, in action (4) the quantum critical transverse polar modes are entirely decoupled from the electronic degrees of freedom, motivating the dark matter analogy.

Normally, low carrier density metals are considered strongly interacting, for the ratio of Coulomb to kinetic energy, determined by rs = 1/(kFaB), where $${k}_{F} \sim {n}_{e}^{1/3}$$ is the Fermi momentum and $${a}_{B}=\frac{4\pi \varepsilon {\hslash }^{2}}{{m}^{*}{e}^{2}}$$ the Bohr radius, is very large at low densities. However, in a quantum critical polar metal, the large upward renormalization of the dielectric constant, Eq. (5), severely suppresses the interaction between the electrons. Indeed, the dielectric constant at the relevant electronic scales at low densities is $$\varepsilon \sim \varepsilon ({{{{{{{\bf{q}}}}}}}},\,{\omega }_{n}){|}_{q=2{k}_{F},{\omega }_{n}={E}_{F}}\approx \frac{{{{\Omega }}}_{0}^{2}}{{(2{c}_{s}{k}_{F})}^{2}}\gg 1$$ at the polar QCP, leading to rs 1. Furthermore, the electronic corrections to the dielectric constant, given in random phase approximation (RPA) by $$\delta {\varepsilon }_{RPA}=\frac{{e}^{2}}{{q}^{2}{\varepsilon }_{0}}{{{\Pi }}}_{e}({{{{{{{\bf{q}}}}}}}},\,{\omega }_{n})$$, where Πe(q, ωn) is the dynamical susceptibility (Lindhardt function) of the electron gas, can be neglected. Indeed, $$\frac{\delta {\varepsilon }_{RPA}}{\varepsilon }{|}_{q=2{k}_{F},{\omega }_{n}={E}_{F}} \sim {r}_{s}\,\ll\, 1$$.

The regime considered here lies in stark contrast to the conventional case, where the relevant longitudinal phonon frequency is much smaller than EF. In this regime, the electronic plasmon appears at low energies vFqωn. However, its contribution to pairing is suppressed by the factor ε−110. Thus, in what follows we neglect this possibility, approximating the electron dynamical susceptibility by its long wavelength, low-frequency limit as in (5).

We next consider the effect of turning on the coupling to energy fluctuations in (1). The presence of a finite electron density ne = 〈ρe(x)〉 leads to a shift in the phonon frequency:

$${\omega }_{T}^{2}({n}_{e})={\omega }_{T0}^{2}+2g{n}_{e}{\varepsilon }_{0}{{{\Omega }}}_{0}^{2},$$
(6)

which naturally explains the suppression of the polar state by charge doping, universally observed in polar metals15,32. Though this effect only shifts the position of the polar QCP, it must be included when considering properties as a function of electronic density. For example a system that is initially polar with $${\omega }_{T0}^{2} \, < \, 0$$ and g > 0 will have a polar QCP at a finite density, as is observed in Ca-Sr substituted SrTiO315.

The coupling of the energy fluctuations to the electron density fluctuations δρe(x) = ρe(x) − ne, cannot be integrated out exactly. Interactions with critical fluctuations near QCPs can be relevant perturbations in a scaling sense33, destabilizing the Fermi liquid ground state already at weak coupling33. In our case, however, the interaction Eq. (1) preserves the Fermi liquid. Assuming the dynamical critical exponent z = 1 and taking the scaling dimension of momentum [q] = 1, one obtains [g] = 2 − d, irrelevant in 3D (see Supplementary Information for details), implying that the system remains a Fermi liquid even at the QCP (whereas in the known cases of non-Fermi liquid behavior the interaction remains at least marginal33). Thus, we can consider its effects perturbatively for weak coupling. Integrating out the field P(x) to lowest order in g, we obtain an effective interaction between electrons:

$${{\Delta }}S=\frac{1}{2}\int {d}^{4}x{d}^{4}{x}^{\prime}\delta {\rho }_{e}(x){V}_{En}(x-{x}^{\prime})\delta {\rho }_{e}({x}^{\prime})+O({g}^{3})$$
(7)

where

$${V}_{En}(x-{x}^{\prime})=-\!2{g}^{2}{{{{{{{\rm{Tr}}}}}}}}\left[D{(x-{x}^{\prime})}^{2}\right]$$
(8)

is recognized to be an attractive density-density interaction resulting from two-phonon exchange, Fig. 1b. At criticality, the contribution to Eq. (8) of the transverse modes stems from their propagator

$${D}_{ab}^{tr}({{{{{{{\bf{x}}}}}}}},\,\tau )=\frac{{\varepsilon }_{0}}{{c}_{s}}{\left(\frac{{{{\Omega }}}_{0}}{2\pi }\right)}^{2}\frac{1}{{{{{{{{{\bf{x}}}}}}}}}^{2}+{c}_{s}^{2}{\tau }^{2}}({\delta }_{ab}-{\hat{x}}_{a}{\hat{x}}_{b}),$$
(9)

leading to an attractive long-range interaction of the form

$${V}_{En}^{cr}({{{{{{{\bf{x}}}}}}}},\,\tau )=-\frac{\alpha }{{\left({{{{{{{{\bf{x}}}}}}}}}^{2}+{c}_{s}^{2}{\tau }^{2}\right)}^{2}},$$
(10)

where $$\alpha={\left(\frac{g{\varepsilon }_{0}{{{\Omega }}}_{0}^{2}}{2{\pi }^{2}{c}_{s}}\right)}^{2}$$, which plays the role of an emergent “gravitational” attraction between electrons at a polar QCP. These interactions also act on the TO phonons via the quartic term in their action33,34 (see Supplementary Information for details). In what follows we will not consider this effects, assuming the relevant momenta and energies to be always larger than the critical region in the energy-momentum space set by the phonon-phonon interactions.

Away from criticality, (10) is valid for space-time separations smaller than the correlation length ξ = cs/ωT. The qualitative form of the interaction at finite momentum and frequency transfer, relevant for pairing, is obtained by a Fourier transform of this expression with a long-distance cutoff ξ

$${V}_{En}(i{\omega }_{n},{{{{{{{\bf{q}}}}}}}})\, \approx \, -\frac{\alpha }{{c}_{s}^{2}}\int\nolimits_{{a}_{0}}^{\xi }\frac{{e}^{i[({{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{\bf{x}}}}}}}})+{\omega }_{n}\tau ]}}{{x}^{4}}{d}^{4}x\\ \sim -\left(\frac{2{\pi }^{2}\alpha }{{c}_{s}}\right)\ln \left[\frac{{a}_{0}^{-1}}{\max ({\xi }^{-1},{\omega }_{{{{{{{{\bf{q}}}}}}}}}/{c}_{s})}\right]$$
(11)

where $${\omega }_{q}=\sqrt{{\omega }_{n}^{2}+{c}_{s}^{2}{{{{{{{{\bf{q}}}}}}}}}^{2}}$$. The characteristic electronic momentum and energy transfer scales are q ~ kF ~ n1/3 and ωn ~ EF ~ n2/3, respectively, resulting in q−1 ~ n−1/3, cs/ωn ~ n−2/3. Furthermore, following (6), a finite electron density leads to a finite correlation length of the order ξ ~ n−1/2. Consequently the interaction character changes with density, and can be described by an effective interaction $${V}_{En}^{Pair}({{{{{{{\bf{x}}}}}}}},\tau )$$ obtained by inverse Fourier transforming the final result of Eq. (11) (Fig. 2). For a low density polar metal that is critical at zero doping (ωT0 = 0) one expects the interaction to be cut off at the Fermi momentum kF, while being essentially independent of frequency, since EF/cskF. This is identical to an instantaneous repulsion, nonlocal in real space (Fig. 2, middle):

$${V}_{En}^{Pair}({{{{{{{\bf{x}}}}}}}},\tau ){|}_{{c}_{s}{k}_{F}\gg {E}_{F},{\omega }_{T}({n}_{e})} \sim \frac{\delta (\tau )}{{\left|{{{{{{{\bf{x}}}}}}}}\right|}^{3}}$$
(12)

in analogy with the instantaneous Newtonian gravity emerging at low energies from the “relativistic" interaction (10). This case is also realized for a system away from a QCP (ωT0 > 0) at intermediate densities (cskFωT(ne), EF, see Fig. 2). In the high density limit, the frequency-dependence of the interaction can be no longer ignored and it is suppressed for frequencies beyond cskF, qualitatively similar to a usual phonon-mediated attraction (Fig. 2, rightmost region). Finally, if ωT(ne) EF, cskF (leftmost region of Fig. 2), which can be realized away from the QCP at the lowest densities, the interaction $${V}_{En}^{Pair}({{{{{{{\bf{x}}}}}}}},\,\tau )$$ reduces to an instantaneous local attraction.

A more detailed calculation of the interaction potential (see Supplementary Information for details) yields

$${V}_{En}(i{\omega }_{n},\,{{{{{{{\bf{q}}}}}}}})= -\left(\frac{2{\pi }^{2}\alpha }{{c}_{s}}\right)\left(\log \left[\frac{{{{\Omega }}}_{T}}{{\omega }_{T}({n}_{e})}\right]-f\left(\frac{{\omega }_{q}}{{\omega }_{T}({n}_{e})}\right)\right),\\ f(x)= \frac{\sqrt{4+{x}^{2}}}{2x}\log \left[\frac{\sqrt{4+{x}^{2}}+x}{\sqrt{4+{x}^{2}}-x}\right]-\frac{1}{2},$$
(13)

where ΩT is the upper cutoff for the optical phonon frequency throughout the Brillouin zone. Here it is important to note, that since the integral is logarithmically divergent, large momenta of the order of the Brillouin zone size contribute significantly. At such momenta, the dispersion is expected to deviate from simply quadratic. Thus in practical calculations, an average value of cs should be used in the prefactor of (13), which we take in the spirit of Debye approximation to be equal to $${\overline{c}}_{s}={{{\Omega }}}_{T}/({(6{\pi }^{2})}^{1/3}/{a}_{0})$$. We remark that the contribution of the longitudinal modes to the induced interaction can be shown to be negligible in the critical low-density regime, where ΩTωT(ne), cskF and $$\frac{{{{\Omega }}}_{0}^{2}}{{c}_{s}^{2}{k}_{F}{a}_{B}^{-1}}\,\gg\, 1$$ (see Supplementary Information for details).

### Superconductivity at low densities

We now show that the attractive interaction mediated by the energy fluctuations always leads to superconductivity at low densities close to the polar QCP. In particular, the density is assumed to be in the middle regime of Fig. 2, i.e. EF, ωT(ne)  2cskF, allowing us to neglect ωn in the interaction. Averaging the repulsive Coulomb ((4), (5)) and the attractive (13), interactions over the Fermi surface, i.e. $$\langle V(k-k^{\prime} )\rangle={\langle V({k}_{F},{k}_{F},\theta )\rangle }_{\theta }$$, we obtain (see Supplementary Information for details):

$$\lambda=N(0)\left(\left(\frac{2{\pi }^{2}\overline{\alpha }}{{\overline{c}}_{s}}\right)\left\{\log \left[\frac{{{{\Omega }}}_{T}}{2{c}_{s}{k}_{F}}\right]+1\right\}-\frac{({e}^{2}/{\varepsilon }_{0}){c}_{s}^{2}}{{{{\Omega }}}_{0}^{2}}\right),$$
(14)

where $$N(0)=\frac{{k}_{F}{m}^{*}}{2{\pi }^{2}{\hslash }^{2}}$$ is the density of states and $$\overline{\alpha }$$ is equal to α with $${c}_{s}\to {\overline{c}}_{s}$$. In deriving this we used that ωT ~ (gn)1/2, EF ~ n2/3cskF ~ n1/3. Most importantly, we find that at low enough doping the two-phonon attraction inevitably overcomes the Coulomb repulsion due to the logarithmic enhancement of the former. In particular, the total interaction is attractive for densities below

$${n}_{cr}=\frac{1}{4{a}_{0}^{3}}{e}^{-3\gamma },\;\gamma=\frac{128{\pi }^{5}{c}_{s}^{2}{\overline{c}}_{s}^{3}{e}^{2}}{{{{\Omega }}}_{0}^{6}{g}^{2}}$$
(15)

where $$\gamma \sim 40\frac{{a}_{B}}{{a}_{0}}\frac{{E}_{h}{{{\Omega }}}_{T}^{5}}{{{{\Omega }}}_{0}^{6}}$$ for $$g \sim {a}_{0}^{3}$$. For strongly polar materials, Ω0 ΩT and hence γ 1, so this restriction is unimportant. Moreover, from (14) the attractive part of the interaction behaves as $$\lambda \propto {k}_{F}\left\{\log ({{{\Omega }}}_{T}/(2{c}_{s}{k}_{F}))+1\right\}$$, describing a dome-shaped behavior of the attractive coupling constant peaked at kF = ΩT/2cs corresponding to a density:

$${n}_{max}=\frac{1}{3{\pi }^{2}}{\left(\frac{{{{\Omega }}}_{T}}{2{c}_{s}}\right)}^{3}\approx \frac{{({\overline{c}}_{s}/{c}_{s})}^{3}}{4{a}_{0}^{3}}.$$
(16)

As the phonon dispersion flattens near the Brillouin zone edges, the average $${\overline{c}}_{s} \, < \, {c}_{s}$$ so that $${n}_{max}{a}_{0}^{3}\,\ll\, 1$$, corresponding to a dilute charge concentration below half filling. Finally, away from the QCP (i.e. ωT0 ≠ 0), the Coulomb screening is reduced, resulting in an additional repulsive term $$\sim 2\pi {e}^{2}{\omega }_{T}^{2}({n}_{e})/({{{\Omega }}}_{0}^{2}{k}_{F}^{2})$$ (c.f. (5)). Due to its singular nature at kF → 0 this sets a lower bound on the density $${n}_{min} \sim {\xi }^{-3}{[\gamma /\log ({{{\Omega }}}_{T}/{\omega }_{T}({n}_{e}))]}^{3/2}/3{\pi }^{2}$$ where the interaction is attractive. Therefore, both proximity to the QCP and sufficiently low densities are required to generate an attractive interaction.

Let us now discuss the critical temperatures of the resulting superconductor. At low densities, the interaction is essentially instantaneous (see Fig. 2). The critical temperature can then be found in the non-adiabatic weak-coupling limit to be Tc ≈ 0.28EFe−1/λ (which includes vertex corrections to the interaction)35,36. Due to the exponential dependence on the coupling constant, one expects Tc to have a dome-like shape with a maximum at nmax as in (16). The theory developed here also has important consequences for the dependence of Tc on external tuning parameters (e.g. pressure) in the vicinity of a polar QCP. Neglecting the residual Coulomb term in (14) and thus assuming the dominance of the energy-fluctuation attraction, one obtains

$$\frac{d\log {T}_{c}}{d\log {\omega }_{T}({n}_{e})}\propto \left\{\begin{array}{ll}-\frac{1}{{\log }^{2}\frac{{{{\Omega }}}_{T}}{{\omega }_{T}({n}_{e})}},&{\omega }_{T}({n}_{e})\,\gg\, {c}_{s}{k}_{F}\\ -\frac{{\omega }_{T}^{2}({n}_{e})\log \left(\frac{{c}_{s}{k}_{F}}{{\omega }_{T}({n}_{e})}\right)}{{({c}_{s}{k}_{F})}^{2}},&{\omega }_{T}({n}_{e})\,\ll\, {c}_{s}{k}_{F}.\end{array}\right.$$
(17)

For ωT(ne) cskF, a singular dependence of Tc on tuning parameter near the QCP is observed. However, at intermediate densities (ωT(ne) cskF, Fig. 2, middle panel), the coupling constant is almost independent of the TO phonon frequency, so the tuning sensitivity will be much weaker.

### 2D polar metals

Similar calculations can be performed in two dimensions. While at tree level, Eq. (1) is marginal, the corrections due to quartic interactions between phonons introduce an anomalous dimension to the energy fluctuations, reducing their momentum-space singularities37, which allows to preserve the Fermi liquid in 2D (see Supplementary Information for details). We note that the anomalous dimension of the energy fluctuations makes them distinct from the perturbative two-phonon processes in the 2D case at low momenta/frequencies. Indeed, the presence of an anomalous dimension indicates that the phonons are no longer well-defined quasiparticles33 so that the energy fluctuations can no longer be uniquely defined as a two-phonon exchange. However, for momenta larger than a critical one (set by the phonon-phonon interactions), (11) is expected to be valid. The 2D Fourier transformation of expression (11) yields

$${V}_{En}^{2D}(i{\omega }_{n},{{{{{{{\bf{q}}}}}}}}) \sim \frac{{g}_{2D}^{2}{\varepsilon }_{0}^{2}{{{\Omega }}}_{0}^{4}}{4\pi {c}_{s}^{2}}\frac{1}{\max ({\omega }_{T}({n}_{e}),{\omega }_{{{{{{{{\bf{q}}}}}}}}})}$$
(18)

- a stronger singularity than in 3D. In the limit cskFωT(ne) the terms due to LO phonon energy fluctuation also have to be included, being of the same order, but this does not change the qualitative form of the interaction (see Supplementary Information for details). Finally, the bare Coulomb repulsion in 2D is given by $$\frac{{e}^{2}}{2{\varepsilon }_{0}q}$$. Screening with the polar mode and conduction electrons, however, reduces it to

$${V}_{C}^{2D}(i{\omega }_{n},\;{{{{{{{\bf{q}}}}}}}})=\frac{{\varepsilon }_{0}^{-1}}{\frac{{{{\Omega }}}_{0}^{2}{l}_{0}{{{{{{{{\bf{q}}}}}}}}}^{2}}{{\omega }_{n}^{2}+{\omega }_{T}^{2}({n}_{e})+{c}^{2}{{{{{{{{\bf{q}}}}}}}}}^{2}}+4\frac{{m}^{*}{e}^{2}}{{\hslash }^{2}}},$$
(19)

where l0 is the 2D layer thickness.

### Superconductivity in Doped SrTiO3

We now apply these results to doped SrTiO3. Figure 3a displays the doping dependence of Tc calculated using the parameters from the literature (see Supplementary Information for details) and taking the coupling g as a fitting parameter. The resulting value $$g/{a}_{0}^{3}\,\approx\, 0.68$$ is comparable to ones obtained from fits to the T2 resistivity ($$g/{a}_{0}^{3}\,\approx\, 0.5$$, note the factor of two in the definition of the coupling29) and doping dependence of the TO phonon frequency ($$g/{a}_{0}^{3}\,\approx\, 0.56$$ can be deduced from38). We assume an effective one-band model with mass m* = 4me. The Hall effect and quantum oscillation data39 indeed suggest that one of the three bands occupied in SrTiO3 contains the most carriers (by an order of magnitude) and has the largest effective mass, dominating the density of states. For the energy fluctuation interaction, Eq.(13), we assumed 2cskFEF, which holds in SrTiO3 for densities lower than 2.6  1019 cm−3. As discussed before (see Fig. 2), this allows us to approximate the interactions with an instantaneous one. In this approximation, 2Δ/Tc   takes the BCS value 3.5235,36, in accord with STM experiments7.

If the Coulomb repulsion (represented for intermediate densities by the second term in Eq. (14)) is neglected (gray dashed line in Fig. 3a) both the magnitude and the doping dependence of the critical temperature are in good agreement with experiment39,40 at all electron concentrations. The dome-like shape of Tc(ne) arises from the nonomonotonic dependence of the two-phonon attractive coupling constant on density, initially rising with the density of states, subsequently decreasing as the lower energy cutoff cskF grows. Importantly, the position of the Tc peak, nmax is approximately independent of the only free parameter of our theory, the coupling constant g - indeed, Eq. (16) yields nmax ≈ 5.5  1020 cm−3, close to the actual value. Including the Coulomb repulsion (black line in Fig. 3a, see Supplementary Information for details), the results based on the instantaneous approximation do not agree as well with the experimental data for densities higher than 2.6  1019 cm−3. Indeed, since in this regime EF 2cskF, frequency dependence of the interaction can be no longer ignored. Whilst a detailed analysis of the frequency-dependent equations is beyond the scope of the current work, we observe that while the Coulomb repulsion strengthens as a function of frequency (see Eq. (5)), the attraction from energy fluctuations, Eq. (11), becomes weaker. Thus we expect that the interaction will change its sign as a function of frequency, allowing for an enhancement of the pairing via the effects of retardation2,3. We note that a second superconducting dome has been observed at yet lower densities ne ~ 1018 cm−341,42; however, the Meissner effect is absent for this second dome5,42. This suggests a highly inhomogeneous state likely affected by the nature of the dopants43, effects that lie beyond the current model.

As discussed above, proximity to a polar QCP should enhance Tc, particularly at low densities. Such a correlation has been observed for the cases of oxygen isotope substitution14 as well as Ca-Sr substitution15, pressure11 or strain44,45. In particular, the enhancement is observed to be more pronounced at low dopings15, in qualitative agreement with the arguments given above, see also Fig. 3a, green dashed line. Note also, that in the polar phase close to the QCP the interaction (1) would still lead to pairing due to fluctuations of the order parameter amplitude around the equilibrium value.

Above we have mentioned, that critical effects for pairing should be emphasized in 2D systems. We propose that thin films of SrTiO3 may provide a platform to explore the predictions of the energy fluctuation theory. Figure 3b displays the predicted Tc, using Tc = 0.15EFe−1/λ for 2D instantaneous pairing (cskFωT(ne))36), using coupling constants appropriate for a two-layer thick slab of SrTiO3. Assuming that the electrons and phonons are in the lowest lateral quantization state, we obtain g2D = g/(2a0) (see Supplementary Information for details). The results show that an appreciable Tc is obtained at low densities. Furthermore, Tc is highly sensitive to the approach to criticality: a slight decrease of the TO phonon frequency leads to an enormous increase of Tc.

## Discussion

New polar metals are being actively discovered46; our theory suggests that suppression of the polar structural transition, for example by pressure11, promises to reveal a number of new low-density superconductors. In particular, perovskites BaTiO3 and KTaO3 can be tuned to ferroelectric QCPs by pressure47 and strain48, respectively, and become metallic under doping32,49,50. Additionally, two-dimensional systems created from these materials, such as LaAlO3/SrTiO3 interfaces51 and surfaces of KTaO352 crystals have been shown to exhibit superconductivity. In the former case, the doping dependence and maximal value of Tc suggest that the pairing is driven by the same mechanism as in bulk SrTiO353. Also, a recently discovered intrinsic polar metal LiOsO354 has been predicted to host nodal points close to the Fermi level (which suggests that small Fermi surfaces with dilute concentrations can be present or created by doping)55 and to have a strain-tuned polar QCP56. In all of these cases, the role of energy fluctuation mechanism in pairing can be verified by the characteristic dependence of the phonon energy on electron concentration (Eq. (6)), scaling of Tc with phonon energy (Eq. (17)), and the presence of a related T2 component in resistivity at low densities28,29. At the same time, unlike typical scenarios of quantum critical pairing57, the normal state is expected to be a Fermi liquid.

Intriguingly, the “dark matter” scenario identified here in polar metals indeed has a counterpart in the fermionic superfluid scenarios of cosmological dark matter58,59, suggesting that quantum critical polar metals may provide a solid-state platform to study these proposals experimentally. In a broader context, energy fluctuations are ubiquitous at quantum critical points, and may thus play a role in electron pairing in many settings. One instance is the case of superconductivity occurring close to a nonpolar structural transition, driven by a phonon at a high-symmetry point, exemplified by tungsten bronze60,61,62 and the A15 compounds63. As in a polar metal, the critical phonon decouples from electrons at low momenta, raising the intriguing possibility of energy-fluctuation pairing. Energy fluctuations have been also proposed as alternative drivers of pairing near antiferromagnetic quantum critical points in strongly correlated metals64,65. Finally, the phenomenon of superconductivity mediated by energy-density fluctuations can also serve as a tool to probe “dark matter" aspects of the solid state, involving excitations that do not interact electromagnetically, such as those related to complex hidden orders66.

Note Added: While this work has been under review, Ref. 67 appeared, where a similar mechanism has been applied to SrTiO3 at ne 1018 cm−3.