PHYSICAL REVIEW B 93, 054510 (2016)

Superconductivity and other collective phenomena in a hybrid Bose-Fermi mixture formed by a polariton condensate and an electron system in two dimensions

Ovidiu Cotlet¸,1,* Sina Zeytinoglu,ˇ 1,2 Manfred Sigrist,2 Eugene Demler,3 and Atac¸ Imamogluˇ 1 1Institute of Quantum Electronics, ETH Zurich,¨ CH-8093 Zurich,¨ Switzerland 2Institute for Theoretical Physics, ETH Zurich,¨ CH-8093 Zurich,¨ Switzerland 3Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA (Received 8 October 2015; revised manuscript received 15 January 2016; published 9 February 2016)

Interacting Bose-Fermi systems play a central role in condensed matter physics. Here, we analyze a novel Bose-Fermi mixture formed by a cavity exciton-polariton condensate interacting with a two-dimensional electron system. We show that that previous predictions of superconductivity [F. P. Laussy, Phys.Rev.Lett.104, 106402 (2010)] and excitonic supersolid formation [I. A. Shelykh, Phys.Rev.Lett.105, 140402 (2010)] in this system are closely intertwined, resembling the predictions for strongly correlated electron systems such as high-temperature superconductors. In stark contrast to a large majority of Bose-Fermi systems analyzed in solids and ultracold atomic gases, the renormalized interaction between the polaritons and electrons in our system is long-ranged and strongly peaked at a tunable wave vector, which can be rendered incommensurate with the Fermi momentum. We analyze the prospects for experimental observation of superconductivity and find that critical temperatures on the order of a few kelvins can be achieved in heterostructures consisting of transition metal dichalcogenide monolayers that are embedded in an open cavity structure. All-optical control of superconductivity in semiconductor heterostructures could enable the realization of new device concepts compatible with semiconductor nanotechnology. In addition the possibility to interface quantum Hall physics, superconductivity, and nonequilibrium polariton condensates is likely to provide fertile ground for investigation of completely new physical phenomena.

DOI: 10.1103/PhysRevB.93.054510

I. INTRODUCTION extends the above-mentioned prior work and shows that the predicted phase transitions are closely intertwined. Interacting Bose-Fermi systems are regarded as a promising The central finding of our work is that when screening platform for investigating novel many-body physics. Recent effects are taken into account, the long-range polariton- advances demonstrating mixtures of ultracold bosonic and electron interaction is peaked at a wave vector q that is fermionic atomic gases have intensified research efforts in this 0 determined by the distance between the 2DES and the quantum class of systems. Feshbach resonances in two-body atomic well (QW) hosting the polaritons. Remarkably, increasing the collisions can be used to tune the strength of interactions polariton condensate occupancy by increasing the resonant into the strong-coupling regime, allowing for the investigation laser intensity leads to a substantial softening of the polariton of competition between various phase transitions such as dispersion at q (near q ) which in turn enhances the strength supersolid formation and superconductivity. Motivated by two r 0 of the polariton-electron interaction, making it even more recent proposals, we analyze a solid-state Bose-Fermi mixture strongly peaked. Leaving a detailed analysis of competition formed by an exciton-polariton Bose-Einstein condensate between superconductivity and potential charge density wave (BEC) interacting with a two-dimensional electron system (CDW) state associated with polariton mode softening as an (2DES). Unlike most solid-state systems, the interaction open problem, we focus primarily on the superconducting strength between the polaritons and electrons can be controlled phase transition. by adjusting the intensity of the laser that drives the polariton After introducing the system composed of a bosonic system. We find that this system can be used to reach the polariton condensate interacting with a 2DES in Sec. II A, strong-coupling regime, evidenced by dramatic softening of we investigate its strong-coupling limit in Sec. II B. Here, the polariton dispersion at a tunable wave vector. we summarize the effects of many-body interactions, leaving Before proceeding, we remark that the interaction between the more detailed calculations to Appendix A. In Sec. III we a 2DES and an indirect-exciton BEC has been theoretically use the theoretical framework developed earlier and analyze shown to lead to the formation of an excitonic supersolid [1,2]. the interactions between a polariton condensate and a 2DES This prior work however did not take into account the effect self-consistently. We notice that the strong interactions can of the exciton BEC on the 2DES. Concurrently, the effect of a lead to instabilities both in the condensate and in the 2DES. We polariton condensate on a 2DES has been investigated without investigate quantitatively the instability of the 2DES towards considering the back-action of electrons on the polaritons superconductivity [3–5] while also taking into account the and it has been predicted that the 2DES can undergo a effect of the 2DES on the BEC. We also comment briefly superconducting phase transition [3–5]. Our work unifies and on the instability of the 2DES towards the formation of an unconventional CDW ordered state as a consequence of the renormalized electron-polariton interaction becoming strongly * [email protected] peaked at wave vector qr . In Sec. IV we investigate how to

2469-9950/2016/93(5)/054510(19) 054510-1 ©2016 American Physical Society OVIDIU COTLET¸ et al. PHYSICAL REVIEW B 93, 054510 (2016) reach the strong-coupling regime experimentally in order to observe these phase transitions. We find that superconductors with temperatures of a few kelvins can be obtained in tran- sition metal dichalcogenide (TMD) monolayers. We briefly summarize our results and provide a short description of new physics and applications enabled by our analysis in Sec. V.

II. THEORETICAL INVESTIGATION A. Description of the coupled electron-polariton system The system that we investigate is similar to the system in Ginzburg’s proposal for high-temperature exciton-mediated superconductors [6]. It consists of a 2DES in close proximity to a quantum well (QW) in which excitons can be created by shining a laser resonantly without influencing the 2DES. The whole system is embedded in a cavity formed by a pair of distributed Bragg reflectors (DBRs), which confine light and and allow for a strong interaction between excitons and photons. Due to the nonperturbative light-matter coupling the new eigenstates are composite particles called polaritons. The polaritons can form a BEC either under nonresonant or under direct resonant excitation by a laser [7]. The interaction between neutral polaritons and the electrons in the 2DES is due to the excitonic content of the polaritons and can be enhanced by enhancing the size of the dipole of the exciton using a dc electric field. In this scenario, an attractive interaction between electrons can be mediated by the polariton excitations of the BEC. As we will show below, the strength of the interaction is proportional to the number of polaritons in the condensate k which can be tuned experimentally. This attractive interaction kph/10 ph allows electrons to pair and form a superconducting state with q/kF a tunable critical temperature. The schematic design of the experimental setup is presented in the upper panel of Fig. 1. FIG. 1. Upper panel: The schematic of the semiconductor het- erostructure that is analyzed. The pumping laser that induces and sustains the BEC in the lower QW is not shown in the schematic 1. Polariton system in order not to complicate the figure. Lower panel: Bare polariton In order to briefly introduce polaritons we will follow (blue) and electron (red) dispersion in logarithmic scale. The vertical Refs. [7,8]. Since the whole system is translationally invariant, dashed lines are at the photon wave vector kph = 3.3Ec(0)/(c) the in-plane momentum k is a good quantum number. The (corresponding to the maximal momenta that we can investigate DBR mirrors assure the confinement of light along the vertical optically; the 3.3 factor comes from the GaAs index of refraction) direction leading to quantization. Furthermore, by tuning the and at kph/10 (roughly corresponding to the momentum where the distance between the DBR mirrors we can ensure that our polariton dispersion switches from photonic to excitonic). The param- = = cavity supports only one mode along the vertical direction, eters used are typical GaAs parameters: g0 2meV,me 0.063m0, = = × 11 −2 = = and we will work in this regime. We also neglect the coupling mh 0.046m0, ne 2 10 cm , Ec(0) Ex (0) 1.518 eV. between the exciton and the leaky and guided modes [9]. The interaction between the light field and the excitons in the lower and lower (LP) polaritons:   QW from Fig. 1 can be modeled by the Hamiltonian † † H = b b + b b , (2)   k,LP k,LP k,LP k,UP k,UP k,UP = † + † k k H Ex (k)ak,xak,x Ec(k)ak,cak,c k k where we introduced the upper and lower polariton dispersion:     + † + 1 2k2 g0(ak,cak,c H.c.), (1) (LP,UP ) = 2 + 2 + ∓ 2 + 2 k 4g0 δE (k) 4g0 , k 2 2mp (3) where ak,x and ak,c are the exciton and cavity-photon anni- hilation operators and g0 denotes the light-matter coupling where δE(k) = Ex (k) − Ec(k) is the energy difference be- strength. The above Hamiltonian can be diagonalized through tween the exciton and the cavity mode. Denoting the k = 0 a canonical transformation and the resulting particles are detuning between the photon and exciton dispersion by = + 2 2 −1 − superpositions of exciton and photon states, called upper (UP) we express the energy difference δE(k) k (mx

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−1 = + mc )/2, where me, mh, mx me mh, mc denote the elec- ne. For comparison, we superimpose the electron (red) and tron, hole, exciton, and cavity effective masses. We also define lower polariton (blue) dispersions in the lower panel of Fig. 1. −1 = −1 + −1 the polariton mass mp by mp mx mc .Wehavealso The polariton-polariton interaction is given by introduced the polariton destruction operators, given by    U  U(k,k ,q) = X(k)X(k )X(k + q)X(k − q) . (9) 2 A bk,LP = X(k)ak,x + 1 − X(k) ak,c, (4)  As expected, this interaction is proportional to the exciton 2 fraction X(k) of the polaritons involved. We assume a simple bk,UP = 1 − X(k) ak,x − X(k)ak,c, (5) contact interaction between excitons even though this inter- where X(k) is the exciton fraction of the lower polariton mode action is more complicated than the above equation suggests with wave vector k and is given by [10–12]. We also emphasize that polariton-polariton interac-   1 δE(k) tion depends on the polarization of the polaritons and is poten- |X(k)|2 = 1 −  . (6) tially tunable through the use of Feshbach resonances [13,14]. 2 2 + 2 δE (k) 4g0 In our analysis we will treat U as a freely tunable parameter. As mentioned above, excitons are neutral particles and Since the upper branch polaritons are unstable against relax- therefore do not couple strongly to electrons but a significant ation into lower energy polariton states, we focus exclusively electron-polariton interaction can be created by inducing a on excitations within the lower polariton branch which we will dipole in the excitons. This can be done in various ways pump resonantly. For clarity we will also drop the LP index; depending on the choice of the material. For example, one ≡ (LP ) ≡ i.e., we will use the notation k k and bk bk,LP . can use an electric field perpendicular to the exciton plane to Unless otherwise stated, for the rest of the paper we will set polarize the excitons; however, this will result in small dipoles. = 0. For this case, the polariton dispersion is plotted in A better approach is to use two tunnel-coupled QWs and bias logarithmic scale in blue in the lower panel of Fig. 1. the energy bands in such a way that holes can live only in the first QW, while electrons can freely tunnel between the QWs. 2. Electron-polariton system This leads to a new type of polariton which has both a large Including the 2DES formed in the upper QW in Fig. 1,the dipole (from the indirect exciton part) and a large light-matter initial Hamiltonian of our system is coupling (from the direct exciton part). In this way polaritons with dipoles of lengths comparable to the exciton Bohr radius, = (e) + (p) + (e−e) + (e−p) + (p−p) known as dipolaritons, can be produced [15]. H H0 H0 HI HI HI ,   Regardless of the mechanism, in the approximation of (e) = † (p) = † H0 εkckck,H0 kbkbk, infinitely thin QWs, one can obtain an analytical expression k k for the electron-polariton interaction VX as shown in Ref. [4].  (e−e) 1 † † We will use this analytical expression in numerical simulations H = V (q)c c  c + c − , (7) I 2 C k k k q k q but below we show how to obtain an approximate but simpler k,k,q expression. First, VX is electrostatic in origin and it will be  (p−p) 1  † † proportional to VC (q). Since the interaction is due to the H = U(k,k ,q)b b  b + b − , I 2 k k k q k q partially excitonic nature of polaritons it will be proportional k,k,q  to the exciton fraction of the polaritons involved. Finally, VX (e−p) † † is proportional to the exciton dipole length d. We expect that H = V (k,q)b c  b + c − , I X k k k q k q polaritons will not be able to respond to momentum transfers k,k,q larger than 1/aB (aB denotes the exciton Bohr radius), 1/d, and † 1/L, where L denotes the distance between the 2DES and the where ck (ck) denote the electron creation (destruction) opera- 2 tors and VC (q) = e /(2Aq) is the usual Fourier transform position of the center of mass of the polaritons in the direction of the Coulomb interaction between electrons.1 (A is the orthogonal to the 2D planes. Analyzing the expression in normalization area, e is the electron charge, and  is the Ref. [4] we see that the dominant momentum cutoff is given  dielectric constant of the medium.) by the distance L, so in the limit of d,aB L it takes the The electron dispersion is given by following approximate form [16]: ≈ + −qL 2k2 VX(k,q) X(k)X(k q)VC (q)qde . (10) εk = − εF , (8) 2me 3. Driven-dissipative condensate where ε is the Fermi energy of the 2DES. Given the above F By pumping the lower polariton branch with a resonant laser parabolic dispersion, ε is proportional to the electron density F field we can sustain a polariton BEC at k = 0. In contrast to the classic BEC, the polariton condensate is a driven-dissipative condensate, in which the pump must compensate the polariton 1We do not take into account the spatial dependence of the dielectric losses. In our system the losses are mainly due to the leakage of function of the screened Coulomb potential, which is characteristic photons through the cavity mirrors. However, the effects due for 2D materials in vacuum, such as TMD monolayers. Instead, we to the driven-dissipative nature of the condensate are limited model the screened Coulomb potential in TMD monolayers with a to small momenta around k = 0 and for the purpose of our constant dielectric  = 40. paper we can just assume that the pump compensates losses

054510-3 OVIDIU COTLET¸ et al. PHYSICAL REVIEW B 93, 054510 (2016) to sustain a finite polariton number in the k = 0 state [7]. We do not discuss polarization effects although this extra degree of freedom might be used for our advantage. For example the polariton-polariton interaction is polarization dependent [14]. = Assuming a polariton condensate in the k 0 mode,√ we = † = follow the Bogolyubov prescription and set b0 b0 N0. We denote by N0 (n0) the number (density) of polaritons in the BEC. We then make the Bogolyubov approximation which consists in ignoring terms of lower order in N0. Leaving the details to Appendix A, we remark that the polariton-polariton interaction Hamiltonian in this limit reduces to a quadratic Hamiltonian with an interaction strength U(q) = U(0,0,q), which can be eliminated through a canonical transformation. More importantly, the electron-polariton interaction Hamil- tonian after the Bogolyubov approximation has the same structure as the electron-phonon Hamiltonian, with a tunable interaction strength:   (e−p) = † + † HI N0VX(q)ck+q ck(bq b−q ), (11) k,q where VX(q) = VX(0,q).

B. Theoretical investigation Given the formal correspondence between the electron- phonon and electron-polariton interaction Hamiltonians, we apply the well-known Migdal-Eliashberg theory [17,18] de- veloped for the electron-phonon Hamiltonian to analyze the electron-polariton interaction. This theory was developed by Eliashberg starting from Migdal’s theorem, which is the equivalent of the Born-Oppenheimer approximation in Green’s function language. We find that our system also satisfies Migdal’s theorem, which justifies the use of Migdal-Eliashberg theory. One of the important results of our theoretical analysis revolves around the substantial softening of the polariton FIG. 2. Upper panel: Bare polariton dispersion (blue) versus dispersion at a wave vector qr and the appearance of a roton-like minimum at this wave vector (see upper panel renormalized polariton dispersion (red). Lower√ panel: Screened M q = N V k / k of Fig. 2). We find that the electron-polariton interaction electron-polariton matrix element ( ) 0 X( ) ( ) (blue) versus screened and renormalized electron-polariton matrix element and consequently the effective electron-electron attractive M˜ (q) (red). The parameters used for the solid lines are typi- interaction mediated by polaritons increases significantly due cal GaAs parameters: d = aB = 10 nm, L = 20 nm, g0 = 2meV, the softening of the polaritons and is strongly peaked at the 11 −2  = 130, me = 0.063m0, mh = 0.046m0, ne = 2 × 10 cm , U = wave vector qr . In the strong-coupling regime, characterized 2 11 −2 0.209 μeV μm , n0 = 4 × 10 cm . The parameters used for the by a significant polariton softening, both the polariton BEC dashed lines are the same as for the solid lines except for L = 1.5L and the 2DES are susceptible to phase transitions.  = 2 and n0 2n0. The quantization area is taken to be 1μm . As far as we know, this strongly peaked interaction in momentum space at a tunable wave vector is unique to our system and stands in stark contrast to the contact interaction in neutral Bose-Fermi mixtures formed with cold atoms or the interaction has an exponential cutoff in momentum space due Kohn anomaly in solid-state systems that can result in large to the distance between the polariton and electron planes. At interactions at twice the Fermi wave vector. We will discuss the same time, V (q → 0) → constant. this strongly peaked interaction in more detail in Sec. III B.In X When screening is taken into account (the lower panel of the following we briefly summarize our results. For a detailed Fig. 2) we have to renormalize V as V˜ (q) → V (q)/(q) derivation we refer to Appendix A. X X X where (q) is the static Thomas-Fermi dielectric function given 2 2 by (q) = 1 + kTF/q, where kTF = mee /2π = 2/aB .As 1. Screening due to the electron system one can easily observe, the effect of screening is to cut off The reason for the strongly peaked interaction in the the contribution of small wave vectors such that VX(q → momentum space can be traced to the screening by the mobile 0) → 0. This small momentum cutoff together with the large carriers of the electron system. The bare electron-polariton momentum cutoff mentioned above leads to a maximum in the

054510-4 SUPERCONDUCTIVITY AND OTHER COLLECTIVE . . . PHYSICAL REVIEW B 93, 054510 (2016) interaction in momentum space at the wave vector 2 to that of χ(q)VX(q), by maximizing the latter we determine  the roton minimum: 1 2aB q0 = 1 + − 1 , (12)  aB L 1 aB qr ≈ 1 + − 1 . (14) as shown in the lower panel of Fig. 2 in blue solid line. The aB L broad maximum in the interaction becomes very strongly peaked as the polaritons soften and the electron-polariton Note the slight difference between qr and q0 which stems from interaction approaches the strong-coupling regime. the fact that the exponential cutoff is twice as effective for a second-order interaction. This softening has been investigated The position of q0 depends only on the screening wave theoretically for excitons in the context of supersolidity [1,2]. vector kTF and consequently on the electron Bohr radius aB , as well as the distance between the BEC and the 2DES. As The electron-polariton interaction also gets renormalized as the polariton dispersion softens. The renormalized electron- a consequence, q0 can be tuned by changing the distance between the BEC and the 2DES. We show the tunability of polariton interaction matrix element is this interaction by changing the distance L in the lower panel  of Fig. 2 in the dashed blue line. Alternatively, one can tune the VX(q) q M˜ (q) → N0 . (15) relative position of q0 with respect to the Fermi wave vector (q) ωq kF by changing the 2DES Fermi energy. This latter method can provide real-time control of q0/kF . We present this renormalized interaction in the lower panel of Fig. 2. Although the factor of q /ωq which leads to an 2. Polariton softening increase in interactions may look similar to the renormalization Remarkably, as the electron-polariton interaction increases factor encountered in the electron-phonon interaction [17], it the polaritons tend to soften due to interactions with the has a slightly different physical origin. In the phonon case, this electrons. Since the interaction is already weakly peaked renormalization factor is due to the quantization of the position in momentum space this softening will be most drastic operator of the harmonic oscillator. In the polariton-electron around a certain wave vector qr which we refer to as a system this factor appears because of the condensate deple- roton-like minimum. As we show below, this softening results tion due to interactions as shown in Appendix A. Therefore, in a significant increase in the electron-polariton interaction we can increase interactions by increasing the condensate without which the strong-coupling regime cannot be reached. depletion. However, one must always make sure that the The effect of electron-polariton interactions on the polariton condensate depletion remains small compared to the number spectrum can be understood as a renormalization of the of polaritons in the condensate to ensure that Bogolyubov’s interaction between a polariton at momentum q and a polariton approximation remains valid. The peak of the renormalized in the condensate. In linear response the correction to the matrix element will be between q0 and qr depending on the 2 strength of this renormalization factor. polariton-polariton interaction is given by χ(q)VX(q) where χ(q) = χ0(q)/(q) is the response function in the random phase approximation (RPA) and χ0(q) is known as the 3. Renormalized Hamiltonian Lindhard function and denotes the linear response of the electrons in the absence of electron-electron interactions. We can gather all the results from the previous analysis Since χ(q) is typically negative, one can imagine the density and write down a renormalized Hamiltonian for the new fluctuations in the 2DES mediating attractive interactions quasiparticles: between the polaritons in the BEC. Intuitively, a polariton = (e) + (p) + (e−e) + (e−p) of momentum q creates a potential V (q) in the 2DES which H H0 H0 HI HI , X   responds by creating a charge distribution δn(q) = χ(q)V (q). (e) † (p) † X H = ε˜ c˜ c˜ ,H = ω b˜ b˜ , This in turn attracts a polariton in the condensate with the 0 k k k 0 k k k strength δn(q)V (q). k k X  (16) The interacting Bose condensate can be exactly diago- (e−e) 1 † † H = V˜ (q)c˜ c˜  c˜ + c˜ − , nalized in the Bogolyubov approximation which yields the I 2 C k k k q k q k,k,q renormalized polariton dispersion:   (e−p) † † H = M˜ (q)c˜ c˜k−q (b˜ + b˜−q ), → 2 + + 2 I k q ωq q 2N0q U(q) χ(q)VX(q) . (13) k,q

We plot the renormalized polariton dispersion in the upper where we denoted the new quasiparticles and quasiparticle panel of Fig. 2 for some typical GaAs parameters.2 Since the interactions with a tilde. In the above V˜C (q) = VC (q)/(q) q dependencies of U(q) and q are negligible in comparison = 2 2 ∗ andε ˜k k /(2me ). We remark that the effect of polaritons on the electron dispersion is to increase the electron mass from ∗ me to me as shown in Eq. (A28). 2In plotting the polariton dispersion we do not consider the We caution that in using the above Hamiltonian in per- interaction of plasmons with polaritons because the polaritons at the turbative calculations one should be careful to avoid double roton minimum, that mediate superconductivity, are not affected by counting since electron-hole bubble diagrams have already this coupling. been taken into account.

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C. Justification of the renormalized Hamiltonian description the 2DES are not affected as long as the electrons close to the When using the diagrammatic techniques, we make two Fermi surface remain good quasiparticles. The finite polariton important approximations. The first approximation is to ignore excitations’ linewidth can be incorporated in our calculations the finite linewidth of the polariton and electron spectral by changing the Eliashberg function. We comment on this in functions, due to many-body interactions. We investigate the Appendix A and show that this in fact does not influence our validity of this approximation in Sec. II C 1. The second and results and that the electron quasiparticle picture remains valid ∝ 2 arguably the most important approximation that we make is to since (ω) ω for small frequencies. choose which diagrams to discard and which diagrams to sum In addition to the above there are, of course, intrinsic over. Our choice was motivated by the Migdal-Eliashberg decoherence mechanisms that appear due to interactions that theory. are neglected when writing down the initial Hamiltonian in In order to investigate the validity of these approxima- Eq. (7). The most important decoherence mechanism is the tions, we first introduce the Eliashberg function and the scattering by impurities in the system. Impurities will lead to electron-polariton coupling (EPC) constant. In analogy to the a broadening of the polariton dispersion due to localization Migdal-Eliashberg theory for phonons the electron-polariton effects. This will limit how much the polaritons can soften. interaction can be characterized by the EPC constant λ: Since impurity-induced broadening is typically Gaussian, it can be neglected provided that the polariton energy exceeds ∞ dω λ = 2 α2F (ω), (17) the corresponding linewidth. Since the electrons are in a high- 0 ω mobility 2DES we do not expect any disorder/localization where we introduced the commonly used Eliashberg function effects to significantly affect them. α2F (ω), the only quantity we need to know in order to assess the effect of the polaritons on the electrons [17]. The 2. A Debye energy for polaritons? Eliashberg function is related to the scattering probability of In normal metals there is a frequency cutoff for phonons an electron on the Fermi surface through a virtual polariton of which is much lower than the Fermi energy. This energy scale frequency ω: separation is crucial for theoretical investigations, because it  allows one to make an adiabatic approximation in which elec- 2 = | ˜  |2 −   α F (ω) Mk−k δ(ω ωk−k )δ(εk)δ(εk )/N(0), (18) trons instantaneously follow the lattice motion. In our system  k,k there is no energy cutoff; however, not all polaritons interact where N(0) is the electron density of states at the Fermi as strongly with electrons. The polaritons that couple most surface. strongly to electrons have energies bounded roughly by ωD, The EPC constant quantifies the strength of the electron- in analogy to the Debye frequency in the case of phonons. We polariton coupling and the strong-coupling regime is character- consider this to be the relevant energy scale of the polaritons. ized by large values of this parameter. Many properties of the As mentioned above, the separation of electron and polari- interacting electron-polariton system depend on this constant ton energy scales allows one to make a Born-Oppenheimer [for example the electron mass gets renormalized such that approximation (known as Migdal’s theorem in diagrammatic ∗ = + me me(1 λ)]. language) and obtain a perturbative expansion in the small pa- rameter ωD/εF . In some materials (as in GaAs) this condition 1. Quasiparticle approximation is already satisfied, without including renormalization effects, In the strong-coupling regime one has to check whether due to the different electron/exciton masses. However, in other the quasiparticle description remains valid for electrons, i.e., materials, the electron/exciton masses are comparable (as in whether the quasiparticle linewidth is much smaller than the TMD monolayers). In these materials renormalization effects quasiparticle energy. are crucial in creating a small Debye frequency and allowing At zero temperature, according to Eq. (A29), the electron the use of Migdal’s theorem for a theoretical investigation of quasiparticle linewidth at energy ω above the Fermi surface the system. is given by We notice that the polaritons at the roton minimum will interact most strongly with the electrons, as shown in Fig. 2. ω   Furthermore, as we will see in the next section, the effective (ω) = π dω α2F (ω ). (19) 0 electron-electron attraction between the electrons on the FS is inversely proportional to the frequency of the polaritons Generically, as long as the polariton energy scale is much mediating the interaction, which is what one would also expect larger than the superconducting gap, the electrons forming the from a second-order perturbation theory. Therefore, we expect Cooper pairs will be good quasiparticles since they will interact the typical energy scale of the polaritons that dominate the only virtually with polaritons. While this is guaranteed to the contributions to attractive electron-electron interactions, to be extent that polaritons themselves are good quasiparticles, in of the order of the polariton energy at the roton minimum. the limit of substantial polariton softening we need to ensure Quantitatively, we define the following Debye frequency: that (kB Tc) kB Tc, where Tc is the critical temperature of the superconductor. ∞ = 2 If we neglect cavity losses, the polariton excitations can ωD 2 dω α F (ω)/λ. (20) decay only by creating electron-hole pairs. Even if the 0 polariton excitations in the strong-coupling regime are not Notice that for weak coupling the Debye frequency ωD will well-defined excitations, our investigation of the transitions of be of the order of the light-matter coupling g0/.

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III. SUPERCONDUCTIVITY AND CHARGE DENSITY WAVES In the previous section we developed the theoretical frame- work needed to understand a system of fermions interacting with a bosonic condensate. The theory is general within the framework of Migdal’s theorem and up to the assumption that density-density interactions depend only on the momentum transfer. In this section, we restrict our analysis to the system introduced in Sec. II A: a 2DES interacting with a polariton BEC. We are interested in the phase transitions that are possible in this system. We find that the polariton BEC can undergo a phase transition into a supersolid, a superfluid with a spatially ordered structure similar to a crystal. At the same time, there are two closely intertwined instabilities in the 2DES: one towards a CDW phase and the other towards a superconducting phase. All of these transitions are possible due to the softening of the polaritons. We notice the similarity to strongly correlated electron systems such as high-temperature superconductors which exhibit a quantum critical point where many instabilities can occur. In Sec. III A we investigate quantitatively the supercon- ducting transition in the 2DES. This transition has been previously investigated [3–5] without taking into account either the screening effects due to the 2DES or the polariton FIG. 3. Upper-left panel: Critical temperature (solid purple line) softening, which we find to be crucial for reaching the for a typical TMD monolayer as a function of polariton density = strong-coupling regime. Moreover, we show in Appendix B n0. The dashed green line is at the critical polariton density nc − that the Frohlich¨ [19] type potential is not suitable for a reliable 2.169 × 1013 cm 2 and can be regarded as dividing the BEC (blue calculation of the critical temperature. region) and supersolid (green region) phases of the polaritons. Upper- In the next subsection we investigate qualitatively the right panel: The polariton spectral function at the roton minimum possibility of an unconventional CDW in the 2DES due to B(qr ,ω). Lower-left panel: Polariton dispersion renormalization (in the proximity of the BEC to a supersolid phase transition. purple is the bare dispersion). Lower-right panel: λq . The different colors of the lines in the upper-right, lower-left, and lower-right pan- els correspond to different values of n0 denoted by the colored dots in the upper-left panel. Expressed in percentages of nc these are A. Superconductivity n0 = 100%nc (blue), n0 = 99.7%nc (green), n0 = 99.3%nc (red), As mentioned above, as we reach the strong-coupling n0 = 98.8%nc (black).The rest of the parameters are typical TMD = = = = = regime the 2DES can become superconducting. Contrary to monolayer√ parameters: d aB 1 nm, L 1.5aB 1.5nm,g0 = = = previous assertions [3–5] we find that polariton-mediated 10 4 meV (4 exciton layers are used),  40, me mh 0.2m0, = 13 −2 = 2 superconductivity is not possible in the presence of electronic ne 10 cm , U 0.12 μeV μm . screening without taking into account the softening of the polaritons. As the polaritons soften and the polariton BEC approaches the supersolid transition, the electron-polariton polaritons at the roton minimum are relatively robust against interaction greatly increases and the system enters the strong- broadening by static disorder. coupling regime. Since materials with lower dielectric con- The quantity that we are most interested in here is the stants are more suitable for reaching the strong-coupling critical temperature of the superconductor. In Appendix B we regime (as we will show in Sec. IV), in this section we choose provide a short review on how to correctly calculate the critical to look at TMD monolayers. temperature of a superconductor. In the 2D polariton-electron system that we consider, the To calculate the superconducting critical temperature we biggest uncertainty originates from the polariton-polariton use the modified McMillan formula [20,21]: repulsion. We note however that the strength of this interaction  f f ω 1.04(1 + λ) can be tuned so as to reach a parameter range where our = 1 2 log − kB Tc exp ∗ ∗ , analysis is justified. The other unknown is the broadening 1.2 λ(1 − 0.62μ ) − μ of the polariton dispersion around q due to the impurities ∗ μ r μ = . (21) in the system, which in turn determines the lowest possible 1 + μ ln(εF /ωD) energy scale of polaritons at the roton minimum. However, we found out that the polaritons at qr , which contribute most This formula is only meaningful if the exponent is nega- to superconductivity, have an effective mass that is roughly 2 tive, and is roughly valid for λ<10. In the above ωlog = orders of magnitude lighter than the bare exciton mass (for exp [ ln(ω) ] [the average is taken with respect to the weight 2 the parameters used in Fig. 3 and n0 = nc). As a consequence, function α (ω)F (ω)/ω]. The screened Coulomb repulsion μ

054510-7 OVIDIU COTLET¸ et al. PHYSICAL REVIEW B 93, 054510 (2016) between electrons averaged over the Fermi surface is given by N V 2(q) γ = q |Im( (q,ω))| = 2 0 X q Im[χ (q,ω )],   q 12 2 0 q VC (k − k )  ω  (q) ω μ = δ(ε )δ(ε )/N(0). (22) q q (k − k) k k k,k (24) where D¯ (q,ω) is the propagator of the polariton quasiparticles The correction factors f1,f2 are given in Appendix B. In order to calculate the critical temperature we need to and γq , the polariton linewidth, is proportional to the imaginary know the strength of the exciton-exciton interaction U(q). part of the polariton self-energy 12 due to interactions with Given that the exciton-exciton interaction is generally repul- the electrons (see Appendix A1for details). sive it has the effect of pushing up the polariton dispersion, Starting from the definitions of the EPC constant and the stiffening the polaritons. In contrast, the electrons mediate an Eliashberg function from Eqs. (17) and (18) we can express attractive interaction between polaritons leading to softening. the EPC constant as The momentum dependence of the polariton-polariton inter- 1  λ = λ , (25) action around qr has not been investigated, since experiments N(0)ε q F q have been limited to small momentum values of the order of ε γ the photon momentum. Theoretical calculations [10–12] seem λ = F q . (26) q 2 to suggest that in our system the exciton-exciton interaction π ωq at large momenta q ≈ 1/a is about an order of magnitude B λ simply quantifies the attraction strength between two smaller that the interaction at q = 0 and might even be q electrons on the Fermi surface resulting from exchanging a attractive. virtual BEC excitation of momentum q. Fortunately, since we have a highly tunable system, the We see from Fig. 3 that in order to achieve critical highest critical temperatures that we can obtain do not depend temperatures of a few kelvins the roton-like minimum needs strongly on either the strength or the q-space dependence of to be lowered to energies of a few meV. As the polaritons U(q), as long as the polaritons can soften at some momentum q soften, the momentum-dependent electron-electron attraction, [i.e., there is a q such that U(q) + χ(q)V 2(q) < 0]. Therefore, X quantified by λ , develops a strong peak at q = q .This given the uncertainty, we choose U = 0.12 μeV μm2 for q r shows that mainly the soft polaritons are responsible for the our numerical simulations but emphasize that similar results superconducting phase transition of the 2DES, because they can be obtained by tuning other parameters as long as interact most strongly with the 2DES, for two reasons. First of U<0.5 μeV μm2. Furthermore, we reemphasize that tuning all the softening of the polaritons results in a depletion of the the polariton-polariton interaction using Feshbach resonances condensate which leads to an increase in the electron polariton has been proposed and demonstrated [13,14].  matrix element proportional to /ω . Second, the electron- In the upper-left panel of Fig. 3 we plot in solid purple the q q electron attraction mediated by the soft polaritons increases critical temperatures that can be achieved in a typical TMD even more because these polaritons are closer to resonance monolayer by tuning the polariton density n in a system with 4 0 with the electronic interactions which are confined to an energy exciton layers (which have the effect of doubling g compared 0 layer of width k T ω around the Fermi energy. Looking to the initial value). According to our mean-field calculation B c D at the spectral function of the quasiparticles at the roton in the blue region the 2DES should become superconducting minimum, denoted by B(q ,ω), we notice that the polaritons whereas we cannot apply our theory in the green region due r at the roton minimum become overdamped, similarly to para- to the breakdown of the Bogolyubov approximation. The magnons near a magnetic instability. At this point McMillan’s dashed green line dividing the two regions in phase space formula should still be valid but one should be careful about is at the critical polariton density n = 2.169 × 1013 cm−2.At c the broad polariton spectral function. We investigate the effects this point λ ≈ 1.5, the condensate depletion is less than 5% of the finite linewidth of this spectral function in Appendix A and the roton minimum is approximately at 1 meV. and show that the electrons remain good quasiparticles and that In the other three panels we investigate the polariton the broad polariton spectral function will not have a significant quasiparticles that mediate the electron pairing. Therefore we impact on the superconducting critical temperature. plot the polariton dispersion (lower left), the polariton spectral Another remarkable feature of the electron-electron attrac- function at q = q (upper right), and λ (lower right), which r q tion mediated by polaritons is that it favors p-wave pairing (or will be defined below. The different colors of the lines in these other higher symmetries) over s-wave pairing. This is easily three panels correspond to different values of n denoted by 0 understood if we look at the total electron-electron interaction the colored dots in the upper-left panel of Fig. 3. The blue line in real space, which we present in Fig. 5 in Appendix A.Notice corresponds to the case of highest polariton density (n = n ) 0 c that this interaction is formed by a strongly repulsive part at while the black line corresponds to the lowest polariton density the origin followed by a oscillatory part with the wavelength (n = 98.8%n ). 0 c 2π/q , due to the strongly peaked interaction in momentum The polariton spectral function is defined as r space. Because of this shape of the interaction, the s-wave   pair will feel the strongly repulsive interaction at the origin, 1 1 2ω while the p-wave pair will avoid this region due to the Pauli- B(q,ω) = Im[D¯ (q,ω)] = Im q , 2 2 exclusion principle. In accordance with this simple picture, π π ω − ω − 2iγq ωq q we find p-wave critical temperatures a few times higher than (23) the s-wave critical temperature. However, since the electrons

054510-8 SUPERCONDUCTIVITY AND OTHER COLLECTIVE . . . PHYSICAL REVIEW B 93, 054510 (2016) in the p-wave pair are not time-reversal partners these pairs one can tune the wave vector where the polariton dispersion will be influenced by the disorder in the system. Therefore, touches zero and therefore can tune the nesting wave vector qr . an accurate calculation of the p-wave critical temperatures We reemphasize that in order to observe a superconducting requires an estimation of the randomness in our system. or CDW phase transition in the 2DES, the polaritons have Notice the strong dependence of the critical temperature on to soften and the BEC has to be close to the supersolid the polariton density, which indicates that some fine tuning phase transition. More generally, despite the differences in will be necessary in order to observe the superconducting structure and phenomenology, the phase diagram of many phase. Fortunately, the polariton density is proportional to unconventional superconductors exhibits the common trait the intensity of the laser generating the condensate, and the that superconductivity resides near the boundary of another intensity of a laser can be tuned with extreme accuracy. Laser symmetry-breaking phase. Examples are the superconducting intensities with less than 0.02% noise have been maintained phases appearing at magnetic quantum phase transitions as relatively easily in the context of polaritons by using a feedback found in many of the Ce-based heavy-fermion compounds loop [22]. We also emphasize that we tried to be conservative such as CeIn3 [26–29], or in iron pnictides accompanying in our choice of bare system parameters. It may for example spin density wave states [30], as well as magnetic, stripe, be possible to obtain higher Tc if the polariton density can be and nematic orders discussed for copper oxides [31]. Another increased further without reaching the Mott transition. example, similar in some respects to our system, is the TMD family, where charge density wave order competes with superconductivity and this feature has been attributed to a B. Supersolid and charge density waves softening of the finite-momentum phonon modes [23–25]. As we approach the regime of strong coupling characterized by a significant softening of the polaritons, the system becomes susceptible to other instabilities, in addition to the IV. MATERIALS SUITABLE FOR REACHING THE superconducting instability. When the polaritons soften to STRONG-COUPLING REGIME the degree that the polariton dispersion touches zero there In this section we do a systematic analysis of the materials will be an instability in the polariton BEC. A transition most suitable for reaching the strong-coupling regime where to a supersolid phase occurs in the green region in Fig. 3 the 2DES becomes unstable towards a new phase. We find that since the polariton dispersion touches zero at nc = 2.173 × 13 −2 materials with low dielectric constants are most suitable and 10 cm . Even though such a supersolid instability was conclude that semiconducting TMD monolayers are good can- proposed for indirect excitons [1,2], we remark that this phase didates for observing polariton-mediated superconductivity. transition can be more easily observed in a polariton BEC At first sight, it may seem that there are many parameters not least because the realization of an exciton BEC is still an that influence the electron-polariton interaction mediated experimental challenge. In our theoretical framework based instabilities that can change from material to material: , me, on the Bogolyubov approximation, the onset of this instability aB , d, L, g0, kF . However, these parameters are typically not can be observed as a dramatic increase in the BEC depletion independent. In fact, we argue that all of these material param- as the polariton dispersion approaches zero. Since our analysis eters scale with the dielectric constant  as shown in Table I. is only valid when the condensate depletion is small from now First, we emphasize that the mass dependence in the first on we assume that the polariton dispersion never touches zero. row is more complicated and one may even treat me as an As the polariton system approaches its BEC instability to independent parameter as well. For the dipole dependence d we a supersolid, the 2DES becomes susceptible to instabilities assumed that, regardless of the mechanism, the induced dipole mediated by the soft polaritons. In the previous section we can be of the order of the Bohr radius but not larger. Similarly, analyzed the susceptibility of the 2DES towards a supercon- we assumed that the distance L between the 2DES and polari- ducting phase. However, the strongly peaked electron-electron ton planes cannot be smaller than the exciton Bohr radius, to = attraction at q qr , as shown in the bottom-right panel of avoid tunneling between the two planes. We also assumed that Fig. 3, can result in a CDW state. −1 the light-matter coupling is proportional to aB . It turns out that A CDW order can also appear due to the phase transition because of the large momentum cutoff due to the finite distance of the BEC into a supersolid. This transition is analogous L between the 2DES and polariton planes we get better results to the case of “frozen phonons” which has been proposed with decreasing kF . However, we cannot lower kF arbitrarily as an explanation for the CDW order in materials such as since we still need RPA to be valid. Finally, in the last row, TMDs [23–25], where a finite-momentum phonon softening leads to a condensation resulting in a static distortion in the lattice which in turn leads to a modulation in the electron TABLE I. Parameter dependence on the dielectric constant . density. In our case this corresponds to the fact that the mean

field bqr becomes important (the momentum direction should −1 me ∝  be chosen spontaneously), which would require a careful 2 aB ∝ /me ∝  extension of our Bogolyubov approximation scheme. 2 d ∝ aB ∝  Remarkably, in contrast to the conventional behavior based 2 L ∝ aB ∝  on nesting features in the electron band structure, this type ∝ −1 ∝ −2 g0 aB  of singularity is not originating from the electronic response −1 −2 kF ∝ L ∝  function but due to a singular behavior in the electron-polariton ∝ −2 ∝ −4 n0 aB  interaction at some wave vector. In both cases mentioned above

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2 light-induced superconductivity, a very interesting open ques- we assumed that the maximum value of n0aB is a material- independent constant since it is set by phase space filling [32]. tion is the competition between superconductivity and po- The strong-coupling regime can be characterized by a large lariton supersolid/CDW phases. The boson-fermion system EPC constant λ and a small Coulomb repulsion constant that we consider allows for precise tuning of key param- μ. Therefore we need to investigate the dependence of eters and can be realized either in GaAs or TMD based these parameters on the dielectric constant. Introducing the 2DES/microcavity structures. variable u ≡ q/2kF and the material-independent constants Strictly speaking, the onset of superconductivity in the 0 0 L¯ = 2kF L ∝  and k¯TF = kTF/(2kF ) ∝  , 2DES will be due to a Berezinsky-Kosterlitz-Thouless tran- sition (BKT) transition. On the other hand, in the parameter 1 −2uL¯ − e λ() ∝  3 du√ 2kF u , range we consider, the Tc we estimate using the mean- − 2 + ¯ 2 ω2 0 1 u (1 kTF/u) 2kF u field approach should be comparable to the BKT transition temperature. 1 1 μ() = 0 du√ . (27) A very promising extension of our work is the realization 2 0 1 − u (1 + k¯TF/u) of photoinduced p-wave pairing of composite fermions in the Looking at the above expressions it is clear that μ remains quantum Hall regime [33]. Unlike the proximity effect due roughly constant from material to material which is not to an s-wave superconductor, finite-range polariton-mediated attractive interaction is more likely to be compatible with unexpected (if we chose me as an independent parameter we would then get some variance in μ for different materials). requirements for observing fragile fractional quantum Hall However, since the electron-polariton interaction is retarded states, enabling edge-state pairing that was proposed as a the relevant constant μ∗ given in Eq. (B1) indicates that method to realize parafermions [34]. we can decrease the effective electron-electron repulsion by In contrast to phonons, polaritons can be directly moni- choosing materials with large Fermi energies and small Debye tored by imaging the cavity output. This provides a unique frequency. possibility to simultaneously monitor changes in transport We also see that λ depends both on the dielectric constant properties of electrons and the spatial structure of polaritons,  and on the bare and renormalized polariton energies. All of as the strongly coupled system is driven through an instability. these three parameters can be tuned independently to some The driven-dissipative nature of the polariton condensate could extent through various methods. We notice that observation also be used to inject polaritons at a preferred wave vector q˜r . of polariton-mediated superconductivity requires materials Since q˜r need not be equal to the roton minimum qr , externally with smaller dielectric constants, because smaller dielectric imposing a spatial structure for the polariton condensate could constants favor the dipole interaction over the monopole alter the competition between superconductivity and CDW Coulomb repulsion. This explains why λ ∝ −3 while μ ∝ 0. instabilities. Moreover, the competition between the CDW and In addition to a small dielectric constant we want a large superconductivity can be investigated further by imposing a bare polariton energy and a small renormalized polariton periodic potential on photonic or excitonic degrees of freedom. Last but not least, the possibility to turn superconductivity dispersion. The dielectric constant also sets the value of g0 which gives the energy scale of the bare polariton energy. It in a semiconductor high-mobility electron system on or off should be emphasized though that the bare polariton energy using laser fields could form the basis of a new kind of can also be tuned through other methods. One method is to transistor that is compatible with the existing semiconductor tune the detuning between excitons and cavity and make the nanotechnology. A crucial development for such applications polaritons more excitonic or more photonic. This will also is a substantial increase of Tc, which may be achieved by change the electron-polariton coupling, an effect which is not properly choosing the properties of the semiconductor host captured in Eq. (27). Another method is the use of multiple material. quantum√ wells. By using N quantum wells one can increase g0 by N while leaving the other parameters unchanged. Another interesting quantity to investigate is what is the ACKNOWLEDGMENTS largest renormalization that can be obtained as a function of the dielectric constant. Looking at Eq. (13) and setting U(q) = 0 The authors acknowledge useful discussions with Sebastian we see that the largest renormalization that can be obtained is Huber and Alexey Kavokin. This work is supported by the ERC = 2 ∝ −3 Advanced Investigator grant POLTDES. given by ω 2N0χ(qr )VX(qr )  . To conclude, we note that under the conditions described in the preceding paragraphs, λ() ∝ −6. APPENDIX A: THEORETICAL ANALYSIS V. CONCLUSION To tackle the many-body problem we use a Green’s The striking feature of the coupled 2DES polariton that functions approach. After doing a mean-field approximation we analyzed is the rather unusual nature of the long-range the electron-polariton Hamiltonian has a similar structure to boson-fermion interaction which is peaked at a finite wave the well understood electron-phonon Hamiltonian. Therefore, vector q0. The latter can be tuned by choosing the system we will use Migdal-Eliashberg theory to analyze this system parameters and leads to the emergence of a roton-like theoretically. However, some of our results can also be minimum in polariton dispersion at qr ∼ q0 = kF . While we obtained through a more intuitive canonical transformation have primarily focused on the prospects for observation of and therefore we also present this method in Appendix A2.

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1. Migdal-Eliashberg theory We start from the initial Hamiltonian in Eq. (7). When q most of the polaritons are in the BEC ground state at k = 0 we can simplify this Hamiltonian using the Bogolyubov prescription [35], which is√ equivalent to making the following k k + q = † = replacement: b0 b0 N0 (N0 is the number of polaritons p p + q in the condensate). This is followed by a Bogolyubov ap- p − k proximation which consists of ignoring terms of lower order in N0. The resulting Hamiltonian is − − − H = H (e) + H (p) + H (e e) + H (e p) + H (p p), (A1) FIG. 4. First vertex correction diagram. In blue are the electron 0 0 I I I propagators while in red are the polariton propagators. where all the other terms remain the same as before except for  (p−p) U(k) † † † H = N (b b− + b b + 2b b ) I 0 2 k k k −k k k k=0 As we will show in the following sections most of the  polaritons that interact with the electrons on the Fermi surface + † N0U(0) bkbk, (A2) are found in a narrow energy interval. Therefore, we can k=0 associate an energy scale to the polaritons which we will denote   by ω , in analogy with the Debye energy in the phonon case. (e−p) = † † + D HI N0 VX(q)ck+q ck(bq b−q ). We will define this energy quantitatively below but for now we k,q will assume that such an energy scale can be associated with After the mean-field Bogolyubov approximation the the polaritons and furthermore we make the assumption that electron-polariton interaction has the same structure as the ωD εF . electron-phonon interaction and therefore we can analyze Let us consider the first correction to the electron-polariton it in analogy with the Migdal-Eliashberg theory, which is vertex, with the corresponding Feynman diagram presented in controlled by the small parameter ωD/εF , the ratio of Fig. 4. If this correction can be ignored then we can certainly the characteristic phonon/electron energy scales. Although ignore the higher order corrections. Suppose a polariton of in doing the many-body theory we treated all interactions momentum q decays and forms an electron-hole pair of simultaneously in order to avoid double counting, we choose momenta k + q and k respectively. This pair will be coherent to present our results in a more intuitive order. for a distance of 1/q. Since the electrons move at roughly We define the bare electron propagator as the Fermi velocity this gives us a coherence time scale of 1/qvF . Only in this time scale electrons can be scattered again. † 1 G(0)(p) =− T c (τ)c (0) = , (A3) Since the polaritons take much longer to respond we expect e τ k k iω − ε n k this vertex correction to be of order ωD/qvF . The average where p = (iωn,k). In light of the following analysis we define momentum of the phonon is of the order of kF so we expect the the bare polariton propagator as vertex correction to depend on the small parameter ωD/εF . (If we wish to be more accurate the first vertex correction is (0) † 1 G (p) =− T b (τ)b (0) = . (A4) of the order of λωD/εF , where λ is the electron-polariton 11 τ k k − iωn k coupling constant to be defined below). Thus, we can safely The analogy with phonons is made by introducing a phonon- conclude that vertex corrections can be ignored. like operator and propagator: The only nontrivial effect of many-body interactions between electrons is the renormalization of interaction be- ≡ + † Aq bq b−q , tween quasiparticles, i.e., screening [17]. In the following (A5) D(q,τ) ≡− Tτ Aq (τ)A−q (0) . we will explore the screening of both the electron-electron and the electron-polariton interactions in the random phase In the absence of interactions we have approximation (RPA). This effect appears as a renormalization 2k of the photon propagator, where, according to the RPA D(0)(p) = G(0)(p) + G(0)(−p) = . (A6) 11 11 2 − 2 approximation, the photon proper self-energy is approximated (iωn) k by the simplest polarization bubble. As mentioned above we will understand the electron- In the RPA framework, the screened electron-electron polariton interaction in terms of the Migdal-Eliashberg theory and electron-polariton interaction is expressed in terms of of the electron-phonon interaction. To make any progress, the dielectric function ε(q,iωn), which has the following we need to make a Born-Oppenheimer approximation, which analytical form: in the many-body physics language means that we ignore the electron-phonon vertex corrections due to phonons. This approach is justified by Migdal’s theorem [36]. We give a brief (k,iωn) = 1 − VC (k)χ0(k,iωn), argument, based on a phase space analysis, that summarizes 1  Migdal’s theorem as it applies to our system. For a more χ (k,iω ) = G(0)(k + q,iω + ik )G(0)(k,ik ) 0 n β n n n rigorous proof one should check Refs. [18,36]. k,ikn

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 f (ε ) − f (ε + ) These propagators and the associated self-energies are not = k k q , iω + ε − ε + independent but satisfy the following identities: k n k k q G p = G −p , G p = G −p , χ0(k,iωn) 22( ) 11( ) 12( ) 21( ) χ(k,iωn) = . (A7) (A11) (k,iωn) 22(p) = 11(−p),12(p) = 21(−p). In the above, f (ε) is the Fermi distribution function. The Solving the Dyson-Beliaev equations in the Bogolyubov polarization bubble χ0 is the Lindhard function and denotes approximation and making the RPA approximation we obtain the linear response to a perturbation when electron-electron the renormalized propagators: interactions are neglected. In this case the perturbation disturbs iω + +  (p) the electron system and creates electron-hole pairs, thus G (p) = n k 12 , 11 2 − 2 − polarizing the system. In contrast. the screened polarization (iωn) k 2k12(p) bubble χ is the response function when electron-electron − (p) G (p) = 12 , (A12) interactions have been taken into account in the RPA. 12 2 2 (iωn) − − 2k12(p) The dielectric function for a 2D system at zero temperature k = + 2 has been calculated for the first time in Ref. [37]. The poles of 12(p) N0 U(k) χ(k,iωn)VX(k) . the dielectric function give us the collective excitations of the electron system, the plasmons. Unless otherwise indicated, for The dispersion of the renormalized polaritons, denoted by ωk, the rest of the paper we will take the static limit (also known is obtained from the zeros of the real part of the denominator; therefore, as the Thomas-Fermi limit) because it is easier to handle. This  limit is accurate as long as the frequencies involved are much ω = 2 + 2 N U(k) + χ(k)V 2(k) , (A13) smaller than the plasma frequency (in 2D, the only regime k k k 0 X where this limit is not satisfied is at very small momenta). In where we made the approximation Re[χ(k,iωn)] ≈ χ(k). the static limit we have Notice that the effect of electron-polariton interaction is to renormalize the polariton-polariton interaction. Because this ≈ = 1 (k,ω) (k,0) k , (A8) is a second-order interaction the effect is proportional to 1 + TF k the square of the bare electron-polariton interaction and the 2 2 response function χ which also contains the effects due to where kTF = mee /(2π ) = 2/aB is the Thomas-Fermi wave vector. screening. In terms of this dielectric function, the screened electron- The polariton spectral function linewidth comes from the electron and electron-polariton interaction is given by imaginary part of the response function, which is zero in the static limit, so we need to use the frequency-dependent VC (k) response function to calculate the polariton linewidth. Using V˜C (k) = , (k) k|Im[12(k,ω)]|=ωkγk (γk is the polariton linewidth) we (A9) obtain, in the limit |Im[(k,ω)]| |Re[(k,ω)]|, the following V (k) V˜ (k) = X . expression for the polariton linewidth: X (k) 2 N0VX(q) q The dilute Bose-condensed gas is another one of the few γq = 2 Im[χ0(q,ωq )] 2(q) ω many-body systems that are well understood. In this case q 2  the small parameter which allows a controlled expansion is N0VX(q) q 2 = 2π [f (ε ) − f (ε + )] given by n a , where n is the polariton density and a is the 2 k k q 0 0  (q) ωq scattering length of the bosonic repulsion. The field theoretical k treatment of the problem was first developed by Beliaev [38]. ×δ(ωq − εk + εk+q ). (A14) A more accessible exposition of this formalism is presented in As mentioned above, the polarization bubble has been Ref. [39]. evaluated exactly in Ref. [37] but to get a simpler analytical In addition to the normal polariton propagator introduced formula we make the following approximation. At low above, when interactions are turned on there is an additional temperatures the Fermi factors restrict the k integration to anomalous propagator that must be considered. Together, a narrow region about the Fermi surface of width ω ,sofor these propagators satisfy the Dyson-Beliaev equations. These q reasonably well behaved Fermi surfaces we can replace these equations can be written more compactly by introducing two factors with ω δ(ε ) to obtain additional propagators, which are, however, not independent q k of these two. In the end, the 4 propagators that need to be 2  N0VX(q) q γ ≈ 2π ω δ(ε )δ(ε + ). (A15) considered are q 2(q) ω q k k q q k † G (k,τ) =− b (τ)b (0) , 11 k k We remark that the same result can be obtained using Fermi’s G =− 12(k,τ) bk(τ)b+k(0) , golden rule if we consider the renormalized electron polariton (A10) N V 2 (q) † † ˜ = 0 X q interactions to be given by Mq 2(q) ω as shown in G21(k,τ) =− b− (τ)b (0) , q k k Refs. [40,41]. We will find out below that this is indeed the G =− † 22(k,τ) b−k(τ)b−k(0) . proper renormalized electron-polariton interaction.

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Evaluating the sum we obtain related to the polariton density of states:  ˜ 2 γq Mq N(0) 2kF F (ω) = δ(ω − ω ). (A22) ≈  , (A16) q 2 q ωq εF q 1 − (q/2kF ) where N(0) is the electron density of states at the the Fermi However, the connection is somewhat obscured in the usual definition: surface. In order to have long-lived quasiparticles we need to   satisfy γq /ωq 1. 2 = | ˜  |2 −  α F (ω) Mk−k δ(ω ωk−k )δ(εk)δ(εk)/N(0). (A23) Having discussed the condensate properties in a field theo- k,k retical formalism, we investigate the effect of the condensate on the electrons and we see that polariton excitations can The above function can be expressed in terms of the previously mediate an attractive interaction between electrons. investigated polariton linewidth γq : In the presence of interactions, the phonon-like propagator 1  introduced in Eq. (A5) has the form α2F (ω) = γ δ(ω − ω ). (A24) 2πN(0)ω q q q D(p) = G11(p) + G22(p) + G12(p) + G21(p) Most properties of the electron-polariton interaction can be 2 = k . (A17) expressed in terms of the EPC (electron-polariton coupling in 2 − 2 − (iωn) ωk 2iγq ωq our case) constant λ and averages ωn , where n are integers Notice that the propagator depends on both the bare and the and the average is taken with respect to the weight function 2 renormalized polariton spectrum. As usual [17], we define a α F (ω). For example, the Debye frequency that we defined reduced propagator D¯ which corresponds to the propagation above can be expressed quantitatively as of the new polariton quasiparticles and therefore depends only ∞ 2 on the renormalized quasiparticles’ spectrum: ωD ≡ ω =2 dω α F (ω)/λ. (A25) 0 D(p) = k D¯ (p). (A18) The definition of the EPC constant is ωk dωα2F (ω) The polariton-mediated electron-electron attraction can be λ = 2 . (A26) ω expressed in terms of this propagator: This constant can also be expressed in terms of the λq which V k N V 2 k (eff ) = C ( ) + 0 X( )D makes explicit the contribution of polaritons with different Ve−e (p) (p) (k) 2(k) momenta: V (k)   = C + ˜ 2D¯ 1 1 γq Mk (p), (A19) λ = λ = , (A27) (k) N q πN(0) ω2 q q q where we have introduced the renormalized electron-polariton matrix element where N is the total number of electrons, which in 2D is given by N = N(0)εF . N V 2(q) Returning to the electron self-energy due to interactions M˜ = 0 X q . (A20) q 2(q) ω with polaritons, the real part results in a mass renormalization q of the electron quasiparticle given by Notice that the term /ω from the initial propagator D(p) k k ∗ εk has been absorbed in the electron-polariton matrix element. In m = me(1 + λ) → ε˜k = . (A28) e 1 + λ the previous section we noticed that this is necessary in order to obtain the same polariton linewidth as the one calculated The imaginary part of the self-energy gives the electron using Fermi’s golden rule and now we have seen why. quasiparticle linewidth . At zero temperature [42],

We also investigate the electron self-energy acquired ω   through interactions with polaritons. This contribution is small (ω) = π dω α2F (ω ). (A29) 0 but its derivative with respect to energy is large within ωD of the Fermi surface. Therefore it will strongly affect the Clearly, the electron quasiparticles with energies close to the electrons within ωD of the Fermi surface. The main effects polariton energy scale ωD will be short-lived because electrons are a renormalized mass and a finite quasiparticle linewidth. will be able to lose their energy to excite polaritons. For The contribution of the polaritons to the electron self-energy these electrons, the quasiparticle picture fails. However, in has the following analytical form: our system, we have the following energy scale kB Tc ωD.

 2 Therefore, we expect that the superconducting electrons will 1 d q not be affected by dissipation due to polaritons, so we can still (k,iωn) =− M˜ q D¯ (q,iqn) β (2π)2 use the quasiparticle picture. iqn ×G(0) + + (k q,iωn iqn). (A21) Finite polariton spectral function linewidth The resulting effects are conventionally expressed in terms In the previous discussion we have treated the polaritons of the Eliashberg function α2F (ω). This function is closely as perfect quasiparticles. When the linewidth of the polariton

054510-13 OVIDIU COTLET¸ et al. PHYSICAL REVIEW B 93, 054510 (2016) spectral function becomes significant the Eliashberg function In conclusion we can safely neglect the effect of the needs to be modified and is broadened: polariton linewidth on the superconducting properties of the    2   2DES. 2 =  ˜   − α F (ω) Mk−k B(k k ,ω)δ(εk)δ(εk)/N(0), k,k   2. Canonical transformation 1 1 2ω B(q,ω) = Im[D¯ (q,ω)] = Im q , In this subsection we will show how some of the above 2 − 2 − π π ω ωq 2iγq ωq results can be obtained in a Hamiltonian formalism using canonical transformations. This approach might be preferred (A30) for its simplicity and because it yields information about the where the Lorentzian B(q,ω) is the polariton spectral function. quasiparticle wave function. We are interested in how the finite-polariton linewidth will In order to obtain the same results as in the diagrammatic influence the superconducting properties of the electron approach we must first better understand the approximations system. that we made in the diagrammatic approach and make the The first question that we need to ask is whether the elec- same approximations in this context. Notice that our choice of trons remain good quasiparticles. As shown in Refs. [40,41], the electron and polariton self-energies implies that we treat when the finite polariton linewidth is included the electron the electron-polariton interaction up to second order. This ap- lifetime scales as ε2 close to the Fermi surface. Therefore proximation was justified by Migdal’s theorem. However, the electrons close to the Fermi surface are well-defined quasi- polariton-polariton interactions are treated exactly. In contrast particles. In all our numerical simulations we checked that the electron-electron interactions are treated perturbatively in electrons are well-defined quasiparticles in a shell of the order the RPA approximation, which is an infinite sum containing of kB Tc, where Tc is the superconducting critical temperature. terms of all perturbative orders. According to McMillan’s formula in Eq. (B1) the super- Finally, McMillan’s formula is based on the approximation conducting critical temperature can be expressed in terms that the electrons that participate in superconductivity are in ∗ 2 1/2 of 4 constants: μ , λ, ωlog = exp [ ln(ω) ],ω ¯ 2 = ω a very thin layer around the Fermi surface much smaller than [the averages are taken with respect to the weight function the polariton energy scale which allows one to approximate α2(ω)F (ω)/ω]. Only the last 3 constants will be affected by the electrons as living on the Fermi surface. We will need this the broadening of the Eliashberg function.√ Furthermore, for approximation to avoid complications. large λ the critical temperature Tc ∝ λω¯ 2; therefore we will As before we start from the Hamiltonian in Eq. (A1) which only investigate how these constants are modified. has been obtained in the Bogolyubov approximation: We can rewrite λ as = (e) + (p) + (e−e) + (e−p) + (p−p)  ∞ H H0 H0 HI HI HI , (A35) 2 1 B(q,ω) λ = 2 |M˜ | δ(ε )δ(ε + ) dω . (A31) q k k q N(0) ω where, as before, k,q 0  (p−p) U(k) † † † H = N (b b− + b b + 2b b ) However, by definition I 0 2 k k k −k k k k=0 ∞ B(q,ω)  ¯ =  † D(q,ω) dω  . (A32) + −∞ ω − ω + iδ N0U(0) bkbk, k=0 Using the oddness of B(q,ω) we obtain  (e−p) = † † + ∞ HI Mq ck+q ck(bq b−q ). (A36) B(q,ω) D¯ (q,0) ωq dω =− = , (A33) k,q 2 0 ω 2 ωq (0) √ In the above Mq = N0VX(q). where ωq (0) denotes renormalized polariton energy obtained We will first show how one can investigate the effect of by using the static (Thomas-Fermi) dielectric function. Thus, electrons on polaritons by tracing out the electrons and then when the finite polariton lifetime is taken into account, λ we will show how one can investigate the electronic system by should be calculated using the polariton energies obtained from tracing out the polaritons. the static limit for the polarization bubble (which is actually what we already did to simplify calculations). a. Tracing out the electrons We now investigate the effect of the broadened Eliashberg function on λ ω2 : We first wish to investigate the effect of electrons on the polaritons. The effect of electrons on polaritons is to induce  ∞ 2 2 1 an effective attraction between polaritons. λ ω =2 |M˜ | δ(ε )δ(ε + ) dωωB(q,ω) q k k q N(0) To get the same results as in the diagrammatic approach we k,q 0  must treat the electron-polariton interaction to second order 2 1 in perturbation theory; however, the interaction must contain = 2 |M˜ | ω δ(ε )δ(ε + ) , (A34) q q k k q N(0) the screening effects due to electron-electron interactions. k,q Intuitively, a polariton of momentum k creates a potential = where we used the well-known [41]sumruleωq VX(k) in the 2DES, and the 2DES responds by creating a ∞ 2 = 0 dωωB(q,ω). Therefore λ ω is not affected at all by charge density δn(k) χ(k)VX(k) where χ(q) is the response the broadening of the Eliashberg function. function first introduced in Eq. (A7). This charge density

054510-14 SUPERCONDUCTIVITY AND OTHER COLLECTIVE . . . PHYSICAL REVIEW B 93, 054510 (2016) will create a potential VX(k) at the polariton condensate, thus As before, the product runs over half of momentum space to 2 resulting in an attraction χ(k)VX(k) between a polariton of avoid double counting. Notice that the ground state factorizes momentum k and a polariton in the condensate. After tracing between states of opposite momenta due to translational out the electrons and rearranging terms we obtain the following invariance. We mention that the condensate depletion due effective polariton Hamiltonian: to bare polariton excitations at momentum k given by  | † | = 2 (p) † † † † G bkbk G vk tells us how many polaritons of momentum H = [E (b b + b b− ) + g (b b + b b− )], k k k −k k k k −k k k k are contained in the ground state on average. One must k=0 2 always make sure that the total condensate depletion k vk (A37) is small compared to the number of polaritons in the ground state N . where the sum must be taken over half of k space to avoid 0 2 The elementary excitation of momentum k is easily found: double counting. In the above gk ≡ N0[U(k) + χ(k)V (k)] X   is the effective interaction between the condensate and a  n√ 1 −vk polariton of momentum k. Notice that the polariton energy | = | = + | + | − E k ak G k 2 n 1 n 1 k n k. u uk Ek ≡ k + gk contains the Hartree-Fock interaction gk. k n The above Hamiltonian is quadratic and therefore can (A43) be diagonalized by making a canonical transformation (also known as the Bogolyubov transformation) by introducing the We also express the electron-polariton interaction in the Fock new operators: basis: = + ∗ †   √ ak ukbk v−kb−k. (A38) (e−p) = † | − | HI ck+q ckMq n( n 1 q n q 2 2 This is a canonical transformation if |uk| −|v−k| = 1. Notice k,q n that the phases are irrelevant and therefore we choose u ,v ∈ k k +|n −q n − 1|−q ). (A44) Re. To diagonalize the Hamiltonian we must choose

 Notice that the interaction strength depends on the number 1 Ek uk,v−k =± ± 1, (A39) of polaritons at momentum q. Since both the ground state 2 ωk and the excited state contain many polaritons when the 2 where we introduced the excitation spectrum ω ≡ condensate depletion vq is large, the electrons will be able   k to scatter excitations at this momentum much more efficiently. 2 − 2 = 2 − Ek gk k 2kgk. The new Hamiltonian is diag- To illustrate this we calculate the amplitude for scattering a onal: bogolon of momentum q:  (p) †  H = ωka ak. (A40) − † k G| H (e p)|E = c c M (u − v ). (A45) k q I q k+q k q q q k,q At this point we have found the polariton excitations on top of the condensate. We wish to investigate the interaction between electrons and the new excitations. To do this we express the b. Tracing out the polaritons electron-polariton Hamiltonian H (e−p) in terms of the new We can also start our analysis by tracing out the polaritons. excitations: As in the diagrammatic approach we start from the Hamilto-

 nian in Eq. (A35) which has been obtained in the Bogolyubov (e−p) q † † H = M c c (a + a− ). (A41) approximation. In this section we are interested in the I q ω k+q k q q k,q q electronic part of the system and therefore we want to trace out the polaritons. We can achieve this by performing a Schrieffer- We see that the electron-polariton interaction is increased Wolff transformation and keeping only second-order terms. = − by the factor q /ωq uq vq which (not surprisingly) is There are different Schrieffer-Wolff transformations that can exactly what we found in the diagrammatic approach. be done as we explain in Appendix B. However, the situation is The reason for the increase of interactions is that the simplified if we consider scattering only on the Fermi surface elementary excitations on top of the condensate contain (which we also did in the diagrammatic approach by using many polaritons, and therefore they interact more strongly McMillan’s formula). Because this is an energy-conserving with the electrons. To understand this effect better we look process it could be measured experimentally and therefore in | at the interaction between the interacting ground state G this case all the different transformations must yield the same and an excited state of momentum k which we denote by second-order result. | | | = E k. Since G is defined by ak G 0 for all k, we can The Schrieffer-Wolff transformations mentioned above express the interacting ground state in the Fock basis of were derived in the context of electron-phonon interactions. | the initial/bare polaritons. We denote by n k the Fock state In that case phonon-phonon interactions were neglected containing n bare polaritons of momentum k. In this notation, because they are negligible. In order to apply these trans- |G = |G , where formations to our system we need get rid of the polariton- k k   n polariton interaction which we do with the help of the Bo- 1 −vk |G = |n |n − . (A42) golyubov transformation introduced previously in Eq. (A38). k u u k k k n k After doing the Bogolyubov transformation we obtain the

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Hamiltonian    = † + ˜ ˜† ˜ + † †  H εkckck kbkbk V (q)ck+q ck−q ckck k k k,k,q  + ˜ † ˜† + ˜ Mq ck+q ck(bq b−q ), (A46) k,q ˜ where we denote by bk the destruction operator for the ˜ = 2 + new noninteracting polaritons, k k 2kN0U(k)is ˜ = the spectrum of the noninteracting polaritons, and Mq

Mq q /˜ q is the new electron-polariton interaction matrix element. So far, we made only the Bogolyubov approximation. At this point we can trace out the polaritons using a Schrieffer-Wolff transformation and obtain the following effective Hamiltonian:   † † †  FIG. 5. The total interaction V˜ (r) (blue) compared to the screened He = εkc ck + V (q)c + c − ckc , k k q k q k Coulomb repulsion V˜ (r) (green) for the parameters used in Fig. 3 k k,k,q C with n0 = nc. 2 2|M˜ q | V (q) = VC (q) − . (A47) ˜ q APPENDIX B: SUPERCONDUCTIVITY When taking screening into account in the RPA approxi- mation just as in the diagrammatic approach we find that the In this section we briefly review the methods that can screened electron-electron interaction is given by be used to calculate the critical temperature of a polariton- mediated superconductor. This discussion is necessary in the V (q) V˜ (q) = . (A48) polariton community in order to make clear the connection 1 − V (q)χ0(q) to superconductivity and to see in which ways our system The above expression can be rewritten in a more intuitive form behaves as a conventional/unconventional superconductor. by separating V˜ (q) into a screened Coulomb repulsion and an We mention that it is notoriously difficult to make effective attraction mediated by polaritons: quantitative theoretical predictions of the superconducting   properties of a new material. However, this limitation is V (q) M˜ 2 2 due to the lack of knowledge of the normal state of the V˜ (q) = C + q , (A49) 2 material and not due to the accuracy of the BCS theory. (q) (q) ˜ q + 2χ(q)M˜ (q) In metals, many complications arise which do not concern where as before we have introduced the dielectric function us, i.e., the choice of the bare pseudopotential describing (q) = 1 − VC (q)χ0(q) and the response function χ(q) = electron-ion interaction, phonon polarization vectors, umklapp χ0(q)/(q). The effective electron-electron interaction can be processes, distortions of the Fermi surface, etc. However, in rewritten in a simpler form by introducing the renormalized our system the normal state can be more readily investigated, ≡ 2 + + 2 because the bosons and the fermions can be separated and polariton energies ωq q 2N0[U(q) χ(q)VX(q)]: investigated separately. In this respect, this type of supercon-   2 ductivity is most similar to the superconductivity in doped VC (q) Mq q 2 V˜ (q) = + . (A50) semiconductors, which are, in this sense, the best understood (q) (q) ω ω q q superconductors [43]. This is exactly the result we found using a diagrammatic In previous work on polariton-mediated superconductivity, approach. The second term on the left-hand side is the effective not only renormalization effects have been ignored, but also the attraction between electrons when exchanging momentum method used to calculate the critical temperature is not valid. q through virtual polaritons. Notice that taking the average Therefore, in this section we review the methods that can be on the Fermi surface of the first term in V (q) yields the used to make reliable predictions about a new superconductor Coulomb repulsion constant μ (this constant will be defined in given that the normal state is known and we point out the Appendix B). Also, taking the average over the Fermi surface reason why the predictions made in previous work cannot be of the second term in V (q) yields the EPC constant λ. taken seriously. Since it might be useful to have an idea of the shape of the In Sec. B1 we present McMillan’s equation, which is interaction in real space, we look at the Fourier transform of the the simplest method to obtain the critical temperature of total interaction compared to the screened Coulomb interaction a superconductor given some system paramters [44]. Then, for some typical TMD monolayer parameters in Fig. 5. Notice is Sec. B2 we introduce the BCS gap equation, which the oscillatory behavior of the interaction at the wavelength was initially used by Bardeen, Cooper, and Schrieffer to 2π/qr that appears due to the softening of the polaritons. theoretically explain superconductivity, in order to explain the In comparison to the long-range attractive interaction, the discrepancy between our results and the results obtained in screened Coulomb interaction looks like a contact interaction. previous work [3] in Sec. B3.

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1. McMillan equation In a Hamiltonian formalism one can obtain an integral The state of the art in the theory of superconductivity equation for the gap function, provided one can trace out the is the Eliashberg equations, obtained in a Green’s function polaritons to obtain an electron-electron attractive interaction formalism. In certain limits they can be reduced to a set between Cooper pairs such that of two coupled integral equations which must be solved  † † Heff = V (k,k )c  c−  ckc−k. (B5) self-consistently. In some limits, which we will present below, k k these equations can be solved analytically to obtain the critical Supposing that this Hamiltonian can be obtained, then one can temperature of the superconductor [20]. Further correction apply the methods first introduced by Bardeen, Cooper, and factors can be introduced by fitting to the exact results obtained Schrieffer [45] to obtain the following BCS gap equation (at by numerically solving the Eliashberg equations, to obtain, as zero temperature): shown in Ref. [21],  (k)  (k) = V (k,k)  . (B6) +  2 f1f2ωlog 1.04(1 λ)  2 (k ) + ε  k T = exp − , k k B c 1.2 λ(1 − 0.62μ∗) − μ∗ Going to a continuum and changing variables from k to ε,θ (¯ω /ω − 1)λ2 we obtain the above equation in a more convenient form: f = [1 + (λ/ )3/2]1/3,f= 1 + 2 log , 1 1 2 2 2 λ +  εF  2  (ε )  ∗ ∗ (ε) = dε  V (ε − ε ). (B7)  = 2.46(1 + 3.8μ ),= 1.82(1 + 6.3μ )(ω ¯ /ω ),  2  1 2 2 log −εF 2 (ε ) + ε μ μ∗ = . It is not at all obvious how to correctly trace out the +  (B1) 1 μ ln(εF / ωD) polaritons to obtain an electron-electron effective interaction, 2 1/2 mainly due to the retarded nature of this interaction [46]. We In the above ωlog = exp [ ln(ω) ],ω ¯ 2 = ω [the averages are taken with respect to the weight function α2(ω)F (ω)/ω], briefly present three methods and comment on their validity: and μ, the screened Coulomb repulsion between electrons 2|M |2ω V F (k,k) = q q , averaged over the Fermi surface, is given by 2 − 2 ε ωq    V (k − k ) | |2 C  − 2 Mq | | μ = δ(εk)δ(ε )/N(0). (B2) BCS  , if ε <ωD, (k − k) k V (k,k ) = ωq (B8) k,k 0, otherwise, According to Ref. [21], the above formula, known as | |2 S  2 Mq McMillan’s formula, is accurate for μ∗ ranging between V (k,k ) =− , |ε|+ω 0 <μ∗ < 0.2 and 0.3 <λ<10. Therefore, in order to know q  the critical temperature of a superconductor, we need to know where q = k − k and ε = εk − εk. ∗ F four material constants λ,ωlog,ω¯ 2, and μ . The difficulty lies The first effective potential V was initially derived by in accurately determining these parameters. In our case, due Frohlich¨ [19] through a Schrieffer-Wolff transformation to to the simplicity of our system, we expect these parameters to leading order in the electron-phonon coupling. Notice that it be close to the theoretical predictions. has a resonance singularity, which means that at that point It is useful to express the parameters λ and μ as momentum higher order terms in the Schrieffer-Wolff transformation integrals, rather than frequency integrals. In this case simple analytical expressions can be obtained:    − / 2N(0) 2kF M˜ 2(q) q 2 1 2 λ = dq 1 − , πkF 0 ωq 2kF    − / N(0) 2kF q 2 1 2 μ = dqVC (q) 1 − . (B3) πkF 0 2kF

2. Superconducting gap equation We wish to compare our method to the method used in previous work on polariton-mediated superconductivity. In order to make this comparison, we show how the superconduct- ing critical temperature can be obtained from a Hamiltonian formalism. Since this section is only meant for comparison we do not consider any renormalization effects, or the Coulomb repulsion. Therefore, our starting Hamiltonian will be    F BCS = † + † + + † † FIG. 6. Comparison between V (ε) (blue), V (ε) (red), and H εkckck ωq bq bq Mq (bq b−q )ck+q ck. S V (ε) (green). For the BCS potential we choose ωD = g0.For k q k,q simulations we used typical GaAs parameters, the same parameters (B4) used for the solid lines in Fig. 2.

054510-17 OVIDIU COTLET¸ et al. PHYSICAL REVIEW B 93, 054510 (2016) become important. However, the singularity is eliminated at the FS (i.e.,  = 0) all the potentials agree with each other, when performing the (principal value) angular integral which as they should since at this point the potential corresponds to appears from changing variables in going from Eq. (B6)to real processes. Eq. (B7). To eliminate the singularity of the Frohlich¨ potential, Bardeen et al. approximated the Frohlich¨ Hamiltonian by a 3. Comparison to previous work box potential V BCS. Such a simplification is possible because In the previous work on polariton-mediated superconduc- the potential is integrated over in the gap equation, making the tivity [3–5], the authors used the Frohlich¨ potential V F (ε). details of the potential insignificant. However, the price to be Notice that although this potential is nonsingular after being paid is the introduction of a fitting parameter ωD. This means integrated, it still develops two large shoulders close to the that the BCS potential can be used to explain superconductivity Debye energy. The dependence of the critical temperature on but not to predict it, because of the unknown ωD. the size and width of the shoulders has been investigated in The last approach involves a more suitable renormalization Ref. [4] and they have been used to predict the large critical procedure, which involves continuous unitary transformations. temperatures obtainable in polariton-mediated superconduc- In this regard we mention the similarity renormalization tivity. As we discussed in the main text, we find much smaller first introduced by Glazek and Wilson [47] and the flow Tc for similar system parameters. Another consequence of equations introduced by Wegner [48]. It has been shown [49] the use of the Frohlich¨ potential is the appearance of an that the potential obtained through similarity renormalization oscillatory gap, which again has been treated as a peculiarity of techniques V S can predict accurately the superconducting polariton-mediated superconductivity. We argue on the other critical temperature. hand that the peculiarities mentioned above are by no means To compare the different approaches we plot the three unique to polariton-mediated superconductivity. Instead, their potentials V (ε) on top of each other in Fig. 6. Notice that appearance is due to the use of the Frohlich¨ potential.

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