Superconductivity and Other Collective Phenomena in a Hybrid Bose-Fermi Mixture Formed by a Polariton Condensate and an Electron System in Two Dimensions

Home , Roton

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

The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters

Citation Cotleţ, Ovidiu, Sina Zeytino#lu, Manfred Sigrist, Eugene Demler, and Ataç Imamo#lu. 2016. “Superconductivity and Other Collective Phenomena in a Hybrid Bose-Fermi Mixture Formed by a Polariton Condensate and an Electron System in Two Dimensions.” Physical Review B 93 (5). https://doi.org/10.1103/physrevb.93.054510.

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41412179

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Superconductivity and other phase transitions in a hybrid Bose-Fermi mixture formed by a polariton condensate and an electron system in two dimensions

Ovidiu Cotleţ,1, ∗ Sina Zeytinoˇglu,1, 2 Manfred Sigrist,2 Eugene Demler,3 and Ataç Imamoˇglu1 1Institute of Quantum Electronics, ETH Zürich, CH-8093, Zürich, Switzerland 2Institute for Theoretical Physics, ETH Zürich, CH-8093, Zürich, Switzerland 3Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA 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. 10, 104 (2010)] and excitonic supersolid formation [I.A. Shelykh, Phys. Rev. Lett. 14, 105 (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 wavevector, 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.

I. INTRODUCTION predicted that the 2DES can undergo a superconducting phase transition [3–5]. Our work unifies and extends the Interacting Bose-Fermi systems are regarded as a above mentioned prior work and shows that the predicted promising platform for investigating novel many-body phase transitions are closely intertwined. physics. Recent advances demonstrating mixtures of ul- The central finding of our work is that when screening tracold bosonic and fermionic atomic gases have inten- effects are taken into account, the long-range polariton- sified research efforts in this class of systems. Feshbach electron interaction is peaked at a wavevector q0 that is resonances in two-body atomic collisions can be used to determined by the distance between the 2DES and the tune the strength of interactions into the strong coupling quantum well (QW) hosting the polaritons. Remarkably, regime, allowing for the investigation of competition be- increasing the polariton condensate occupancy by in- tween various phase transitions such as supersolid for- creasing the resonant laser intensity leads to a substantial mation and superconductivity. Motivated by two recent softening of the polariton dispersion at qr (near q0) which proposals, we analyze a solid-state Bose-Fermi mixture in turn enhances the strength of the polariton-electron in- formed by an exciton-polariton Bose-Einstein condensate teraction, making it even more strongly peaked. Leaving (BEC) interacting with a two dimensional electron sys- a detailed analysis of competition between superconduc- tem (2DES). Unlike most solid-state systems, the inter- tivity and potential charge density wave (CDW) state as- action strength between the polaritons and electrons can sociated with polariton mode softening as an open prob- be controlled by adjusting the intensity of the laser that lem, we focus primarily on the superconducting phase drives the polariton system. We find that this system can transition. be used to reach the strong-coupling regime, evidenced After introducing the system composed of a bosonic by dramatic softening of the polariton dispersion at a polariton condensate interacting with a 2DES in Section tunable wavevector. II A, we investigate its strong coupling limit in Section Before proceeding, we remark that the interactions be- II B. Here, we summarize the effects of many body inter- tween a 2DES and an indirect-exciton BEC has been actions, leaving the more detailed calculations to the Ap- theoretically shown to lead to the formation of an exci- arXiv:1510.02001v1 [cond-mat.quant-gas] 7 Oct 2015 pendix A. In Section III we use the theoretical framework tonic supersolid [1, 2]. This prior work however, did not developed earlier and analyze the interactions between a take into account the effect of the exciton BEC on the polariton condensate and a 2DES self-consistently. We 2DES. Concurrently, the effect of a polariton condensate notice that the strong interactions can lead to instabili- on a 2DES has been investigated without considering the ties both in the condensate and in the 2DES. We inves- back-action of electrons on the polaritons and it has been tigate quantitatively the instability of the 2DES towards superconductivity [3–5] while also taking into account the effect of the 2DES on the BEC. We also comment briefly on the instability of the 2DES towards the forma- ∗ [email protected] tion of an unconventional CDW ordered state as a con- 2 sequence of the renormalized electron-polariton interaction becoming strongly peaked at wavevector qr. In Sec- tion IV we investigate how to 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 transition 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- tion 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 reflec- tors (DBRs), which confine light and and allow for a strong interaction between excitons and photons. Due to the non-perturbative light-matter coupling the new eigenstates are composite particles called polaritons. The polaritons can form a BEC either under non-resonant 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 k dipole of the exciton using a DC electric field. In this kph/10 ph scenario, an attractive interaction between electrons can q/kF be mediated by the polariton excitations of the BEC. As we will show below, the strength of the interaction is pro- Figure 1. Upper panel: The schematic of the semiconduc- portional to the number of polaritons in the condensate tor heterostructure that is analyzed. Lower panel: bare po- which can be tuned experimentally. The schematic de- lariton (blue) and electron (red) dispersion in logarithmic sign of the experimental setup is presented in the upper scale. The vertical dashed lines are at the photon wavevector panel of Figure 1. kph = 3.3Ec(0)/(¯hc) (corresponding to the maximal momenta In the following we will assume for simplicity that the that we can investigate optically; the 3.3 factor comes from polariton condensate is generated by a resonant laser field the GaAs index of refraction) and at kph/10 (roughly corre- for the k = 0 state of the lower polariton branch. The sponding to the momentum where the polariton dispersion switches from photonic to excitonic). The parameters used Hamiltonian describing the coupled exciton and cavity- are typical GaAs parameters: g0 = 2meV, me = 0.063m0, mode dynamics can be diagonalized through a canonical 11 −2 mh = 0.046m0, ne = 2×10 cm , Ec(0) = Ex(0) = 1.518eV. transformation and the resulting lower and upper polariton eigenstates are superpositions of exciton and photon states. Since the whole system is translationally invari- by: ant, in-plane momentum k is a good quantum number ! for polaritons. The annihilation operator for a lower po- 1 δE(k) X(k) 2 = 1 , (2) lariton of momentum k is: p 2 2 | | 2 − δE (k) + 4g0 (x) p 2 (c) b = X(k)a + 1 X(k) a , (1) where g0 is the light-matter coupling strength and k k − k δE(k) = Ex(k) Ec(k) is the energy difference be- where a(x) and a(c) are the exciton and cavity-photon tween the exciton− and the cavity mode. Denoting the annihilation operators. X(k) is the exciton fraction of k = 0 detuning between the photon and exciton disper- the lower polariton mode with wavevector k and is given sion by ∆ we express the energy difference: δE(k) = 3

2 2 −1 −1 ∆+¯h k (m m )/2 where m , m , mx = m +m , inducing a dipole in the excitons. This can be done in · x − c e h e h mc denote the electron, hole, exciton, and cavity effec- various ways depending on the choice of the material. tive masses. We also define the polariton mass mp by For example, one can use an electric field perpendicular −1 −1 −1 mp = mx + mc . to the exciton plane to polarise the excitons, however, Since the upper branch polaritons are unstable against this will result in small dipoles. A better approach is to relaxation into lower energy polariton states, we focus ex- use two tunnel-coupled QWs and bias the energy bands clusively on excitations within the lower polariton branch in such a way that holes can live only in the first QW, within which a BEC in the k=0 mode is formed. First while electrons can freely tunnel between the QWs. This we write the initial Hamiltonian of our system as leads to a new type of polariton which has both a large dipole (from the indirect exciton part) and a large light (e) (p) (e−e) (e−p) (p−p) H = H0 + H0 + HI + HI + HI , matter coupling (from the direct exciton part). In this (e) X † (p) X † way polaritons with dipoles of lengths comparable to the H = ε c c ,H = Ω b b , 0 k k k 0 k k k exciton Bohr radius, known as dipolaritons, can be pro- k k duced [13]. (e−e) 1 X † † H = V (q)c c 0 c + c 0− , (3) I 2 C k k k q k q Regardless of the mechanism, in the approximation of k,k0,q infinitely thin QWs, one can obtain an analytical expres- (p−p) 1 X 0 † † sion for the electron-polariton interaction VX as shown H = U(k, k , q)b b 0 b b 0 , I k k k+q k −q in Ref. 4. We will use this analytical expression in nu- 2 0 k,k ,q merical simulations but below we show how to obtain an (e−p) X † † HI = VX (k, q)bkck0 bk+qck0−q, approximate but simpler expression. First, VX is electro- k,k0,q static in origin and it will be proportional to VC (q). Since the interaction is due to the partially excitonic nature of † where ck (ck) denote the electron creation (destruction) polaritons it will be proportional to the exciton fraction 2 operators and VC (q) = e /(2Aq) is the usual Fourier of the polaritons involved. Finally, VX is proportional to transform of the Coulomb interaction between electrons. the exciton dipole length d. We expect that polaritons (A is the normalization area, e is the electron charge and will not be able to respond to momentum transfers larger is the dielectric constant of the medium.) than 1/aB (aB denotes the exciton Bohr radius), 1/d and The electron and (lower) polariton dispersions are 1/L, where L denotes the distance between the 2DES and given by: the position of the centre of mass of the polaritons in the direction orthogonal to the 2D planes. Analyzing the 2 2 ¯h k expression in Ref. 4 we see that the dominant momen- εk = εF , (4) 2me − tum cutoff is given by the distance L, so in the limit of 2 2 q q d, aB L it takes the following approximate form [14]: 1 2 2 ¯h k 2 2 Ωk = ∆ + 4g0 + δE (k) + 4g0 . ≤ 2 2mp − −qL VX (k, q) X(k)X(k + q)VC (q)qde . (6) where εF is the Fermi energy of the 2DES. Given the ≈ above parabolic dispersion, εF is proportional to the electron density ne. Unless otherwise stated, for the rest of the paper we will set ∆ = 0. For this case, the polariton Since we assume a polariton condensate in the k = 0 mode, we follow the Bogolyubov prescription and set dispersion is plotted in logarithmic scale in blue in the † lower panel of Figure 1. For comparison, we also super- b0 = b0 = √N0. We denote by N0 (n0) the number impose the electron dispersion in red in the same panel. (density) of polaritons in the BEC. We then make the Bo- The polariton-polariton interaction is given by golyubov approximation which consists in ignoring terms of lower order in N0. Leaving the details to Appendix U A, we remark that the polariton-polariton interaction U(k, k0, q) = X(k)X(k0)X(k + q)X(k0 q) . (5) − A Hamiltonian in this limit reduces to a quadratic Hamil- tonian with an interaction strength , As expected, this interaction is proportional to the ex- U(q) = U(0, 0, q) which can be eliminated through a canonical transfor- citon fraction X(k) of the polaritons involved. We as- mation. More importantly, the electron-polariton inter- sume a simple contact interaction between excitons even action Hamiltonian after the Bogolyubov approximation though this interaction is more complicated than the has the same structure as the electron-phonon Hamilto- above equation suggests [8–10]. We also emphasize that nian, with a tunable interaction strength: polariton-polariton interaction depends on the polariza- tion of the polaritons and is potentially tunable through the use of Feshbach resonances [11, 12]. In our analysis (e−p) X p † † H = N0VX (q)c ck(bq + b ), (7) we will treat U as a freely tunable parameter. I k+q −q k,q As mentioned above, excitons are neutral particles and therefore do not couple strongly to electrons but a significant electron-polariton interaction can be created by where VX (q) = VX (0, q). 4

B. Theoretical investigation as shown in the lower panel of Figure 2 in blue solid line. The broad maximum in the interaction be- Given the formal correspondence between the electron- comes very strongly peaked as the polaritons soften and phonon and electron-polariton interaction Hamiltonians, the electron-polariton interaction approaches the strong- we apply the well-known Migdal-Eliashberg theory [15, coupling regime. 16] developed for the electron-phonon Hamiltonian to an- The position of q0 depends only on the screening alyze the electron-polariton interaction. This theory was wavevector kTF and consequently on the electron Bohr developed by Eliashberg starting from Migdal’s theorem, radius aB, as well as the distance between the BEC and which is the equivalent of the Born-Oppenheimer approx- the 2DES. As a consequence, q0 can be tuned by chang- imation in Green’s function language. We find that our ing the distance between the BEC and the 2DES. We system also satisfies Migdal’s theorem, which justifies the show the tunability of this interaction by changing the use of Migdal-Eliashberg theory. distance L in the lower panel of Figure 2 in dashed blue One of the important results of our theoretical analysis line. Alternatively, one can tune the relative position of revolve around the substantial softening of the polariton q0 with respect to the Fermi wavevector kF by changing the 2DES Fermi energy. This latter method can provide dispersion at a wavevector qr and the appearance of a roton-like minimum at this wavevector (see upper panel real-time control of q0/kF . of Figure 2). We find that the electron-polariton interaction and consequently the effective electron-electron attractive interaction mediated by polaritons increases 2. Polariton softening significantly due the softening of the polaritons and is strongly peaked at the wavevector qr. In the strong- Remarkably, as the electron-polariton interaction in- coupling regime, characterised by a significant polariton creases the polaritons tend to soften due to interac- softening, both the polariton BEC and the 2DES are sus- tions with the electrons. Since the interaction is already ceptible to phase transitions. weakly peaked in momentum space this softening will be As far as we know, this strongly peaked interaction in most drastic around a certain wavevector qr which we momentum space at a tunable wavevector is unique to refer to as a roton-like minimum. As we show below, this our system and stands in stark contrast to the contact softening results in a significant increase in the electron- interaction in neutral Bose-Fermi mixtures formed with polariton interaction without which the strong-coupling cold-atoms or the Kohn anomaly in solid-state systems regime cannot be reached. that can result in large interactions at twice the Fermi The effect of electron-polariton interactions on the po- wavevector. We will discuss this strongly peaked inter- lariton spectrum can be understood as a renormalization action in more detail in Section III B. In the following we of the interaction between a polariton at momentum q briefly summarise our results. For a detailed derivation and a polariton in the condensate. In linear response the we refer to Appendix A. correction to the polariton-polariton interaction is given 2 by χ(q)VX (q) where χ(q) = χ0(q)/(q) is the response function in the random phase approximation (RPA) and 1. Screening due to the electron system χ0(q) is known as the Lindhard function and denotes the linear response of the electrons in the absence of electron- The reason for the strongly peaked interaction in the electron interactions. Since χ(q) is typically negative, one momentum space can be traced to the screening by the can imagine the density fluctuations in the 2DES mediat- mobile carriers of the electron system. The bare electron- ing attractive interactions between the polaritons in the polariton interaction has an exponential cutoff in mo- BEC. Intuitively, a polariton of momentum q creates a mentum space due to the distance between the polariton potential VX (q) in the 2DES which responds by creating a charge distribution δn(q) = χ(q)V (q). This in turn and electron planes. At the same time, VX (q 0) X constant. → → attracts a polariton in the condensate with the strength When screening is taken into account (the lower panel δn(q)VX (q). The interacting Bose-condensate can be exactly diag- of Figure 2) we have to renormalize V as V˜ (q) X X onalized in the Bogolyubov approximation which yields V (q)/(q) where (q) is the static Thomas-Fermi dielec-→ X the renormalized polariton dispersion: tric function given by (q) = 1 + kTF /q where kTF = 2 2 q m e /2π¯h = 2/a . As one can easily observe, the ef- 2 2 e B ωq Ω + 2N0Ωq [U(q) + χ(q)V (q)]. (9) fect of screening is to cut off the contribution of small → q X wavevectors such that VX (q 0) 0. This small mo- We plot the renormalized polariton dispersion in the mentum cutoff together with→ the large→ momentum cutoff upper panel of Figure 2 for some typical GaAs parame- mentioned above leads to a maximum in the interaction ters. Since the q dependences of U(q) and Ωq are negli- 2 in momentum space at the wavevector gible in comparison to that of χ(q)VX (q), by maximizing the latter we determine the roton minimum: "r # 1 2aB 1 r a q0 = 1 + 1 , (8) B a L − qr 1 + 1 (10) B ≈ aB L − 5

the renormalization factor encountered in the electron- 4 phonon interaction [15], it has a slightly diﬀerent physi-

Ωq cal origin. In the phonon case, this renormalization fac- ω tor is due to the quantisation of the position operator of 3 q the harmonic oscillator. In the polariton-electron system ωq0 this factor appears because of the condensate depletion due to interactions as shown in Appendix A. Therefore, V e 2 we can increase interactions by increasing the conden- m sate depletion. However, one must always make sure that the condensate depletion remains small compared to the 1 number of polaritons in the condensate to ensure that Bogolyubov’s approximation remain valid. The peak of the renormalized matrix element will be between q0 and 0 0.0 0.5 1.0 qr depending on the strength of this renormalisation factor. q/kF

60 3. Renormalized Hamiltonian Mq M˜ q We can gather all the results from the previous analysis 40 Mq0 and write down a renormalised Hamiltonian for the new

˜ 0 quasi-particles: V Mq e

µ (e) (p) (e−e) (e−p) H = H0 + H0 + HI + HI , 20 (e) X † (p) X ˜†˜ H0 = ε˜kc˜kc˜k,H0 = ωkbkbk, k k (e−e) 1 X ˜ † † H = VC (q)˜c c˜ 0 c˜k+qc˜k0−q, 0 I 2 k k 0.0 0.5 1.0 k,k0,q q/kF (e−p) X ˜ † ˜† ˜ HI = M(q)˜ckc˜k−q bq + b−q , (12) Figure 2. Upper panel: bare polariton dispersion (blue) k,q versus renormalised polariton dispersion (red). Lower where we denoted the new quasi-particles and quasi- panel: screened electron-polariton matrix element M(q) = √ particle interactions with a tilde. In the above ˜ N V (k)/(k) (blue) versus screened and renormalised VC (q) = 0 X and 2 2 ∗ . We remark that the ef- electron-polariton matrix element M˜ (q) (red). The parame- VC (q)/(q) ε˜k =h ¯ k /(2me) ters used for the solid lines are typical GaAs parameters: d = fect of polaritons on the electron dispersion is to increase the electron mass from to ∗ as shown in Eq. (A28). aB = 10nm, L = 20nm, g0 = 2meV, = 130, me = 0.063m0, me me 11 −2 2 mh = 0.046m0, ne = 2 × 10 cm , U = 0.209µeVµm , We caution that in using the above Hamiltonian in 11 −2 n0 = 4 × 10 cm . The parameters used for the dashed perturbative calculations one should be careful to avoid lines are the same as for the solid lines except for L0 = 1.5L double counting since electron-hole bubble diagrams have 0 and n0 = 2n0. already been taken into account.

Note the slight difference between qr and q0 which stems C. Justification of the renormalized Hamiltonian from the fact that the exponential cutoff is twice as ef- description fective for a second order interaction. This softening has been investigated theoretically for excitons in the context When using the diagrammatic techniques, we make of supersolidity [1, 2]. two important approximations. The first approximation The electron-polariton interaction also gets renor- is to ignore the finite linewidth of the polariton and elec- malised as the polariton dispersion softens. The renor- tron spectral functions, due to many-body interactions. malised electron-polariton interaction matrix element is: We investigate the validity of this approximation in Sec- s tion II C 1. The second and arguably the most important p VX (q) Ωq approximation that we make is to choose which diagrams M˜ (q) N0 . (11) → (q) ωq to discard and which diagrams to sum over. Our choice was motivated by the Migdal-Eliashberg theory. We present this renormalized interaction in the lower In order to investigate the validity of these approxima- p panel of Figure 2. Although the factor of Ωq/ωq which tions, we first introduce the Eliashberg function and the leads to an increase in interactions may look similar to electron-polariton coupling (EPC) constant. In analogy 6 to the Migdal-Eliashberg theory for phonons the electron- the electron quasi-particle picture remains valid since polariton interaction can be characterised by the EPC Γ(ω) ω2 for small frequencies. constant λ: In∝ addition to the above there are, of course, intrin- ∞ sic decoherence mechanisms that appear due to interac- Z dω λ = 2 α2F (ω), (13) tions that are neglected when writing down the initial 0 ω Hamiltonian in Eq. (3). The most important decoher- where we introduced the commonly used Eliashberg func- ence mechanism is the scattering by impurities in the tion α2F (ω), the only quantity we need to know in order system. Impurities will lead to a broadening of the po- to assess the effect of the polaritons on the electrons [15]. lariton dispersion due to localisation effects. This will The Eliashberg function is related to the scattering prob- limit how much the polaritons can soften. Since impu- ability of an electron on the Fermi surface through a vir- rity induced broadening is typically Gaussian, it can be tual polariton of frequency ω: neglected provided that the polariton energy exceeds the corresponding linewidth. Since the electrons are in a high 2 2 X mobility 2DES we do not expect any disorder/locaziation α F (ω) = M˜ − 0 δ(ω ω − 0 )δ(ε )δ(ε 0 )/N(0), k k − k k k k k,k0 effects to significantly affect them. (14) where N(0) is the electron density of states at the Fermi 2. A Debye energy for polaritons? surface. The EPC constant quantifies the strength of the In normal metals there is a frequency cutoff for electron-polariton coupling and the strong coupling phonons which is much lower than the Fermi energy. regime is characterised by large values of this parame- This energy scale separation is crucial for theoretical inter. Many properties of the interacting electron-polariton vestigations, because it allows one to make an adiabatic system depend on this constant (for example the electron approximation in which electrons instantaneously follow ∗ mass gets renormalised such that me = me(1 + λ)). the lattice motion. In our system there is no energy cutoff; however, not all polaritons interact as strongly with electrons. The polaritons that couple most strongly to 1. Quasi particle approximation electrons have energies bounded roughly by ωD, in analogy to the Debye frequency in the case of phonons. We In the strong coupling regime one has to check whether consider this to be the relevant energy scale of the po- the quasi-particle description remains valid for electrons, laritons. i.e. whether the quasi-particle linewidth is much smaller As mentioned above, the separation of electron and than the quasi-particle energy. polariton energy scales allows one to make a Born- At zero temperature, according to Eq. (A29), the elec- Oppenheimer approximation (known as Migdal’s theo- tron quasi particle linewidth Γ at energy ω above the rem in diagrammatic language) and obtain a perturba- Fermi surface is given by: tive expansion in the small parameter ¯hωD/εF . In some materials (as in GaAs) this condition is already satisfied, Z ω Γ(ω) = π dω0α2F (ω0). (15) without including renormalisation effects, due to the dif- 0 ferent electron/exciton masses. However, in other materials, the electron/exciton masses are comparable (as Generically, as long as the polariton energy scale is in TMD monolayers). In these materials renormalization much larger than the superconducting gap, the elec- effects are crucial in creating a small Debye frequency trons forming the Cooper pairs will be good quasi- and allowing the use of Migdal’s theorem for a theoreti- particles since they will interact only virtually with po- cal investigation of the system. laritons. While this is guaranteed to the extent that po- We notice that the polaritons at the roton minimum laritons themselves are good quasi-particles, in the limit will interact most strongly with the electrons, as shown of substantial polariton softening we need to ensure that in Figure 2. Furthermore, as we will see in the next Γ(k T ) k T , where T is the critical temperature of B c B c c section, the effective electron-electron attraction between the superconductor. the electrons on the FS is inversely proportional to the If we neglect cavity losses, the polariton excitations can frequency of the polaritons mediating the interaction, decay only by creating electron-hole pairs. Even if the po- which is what one would also expect from a second or- lariton excitations in the strong-coupling regime are not der perturbation theory. Therefore, we expect the typical well-defined excitations, our investigation of the transi- energy scale of the polaritons that dominate the contribu- tions of the 2DES are not affected as long as the electrons tions to attractive electron-electron interactions, to be of close to the Fermi surface remain good quasiparticles. the order of the polariton energy at the roton minimum. The finite polariton excitations’ linewidth can be incor- Quantitatively, we define the following Debye frequency: porated in our calculations by changing the Eliashberg Z ∞ function. We comment on this in Appendix A and show 2 ωD = 2 dω α F (ω)/λ. (16) that this in fact does not influence our results and that 0 7

Notice that for weak coupling the Debye frequency ωD In the 2D polariton-electron system that we consider, will be of the order of the light matter coupling g0/¯h. the biggest uncertainty originates from the polariton- polariton repulsion. We note however that the strength of this interaction can be tuned so as to reach a parameter III. SUPERCONDUCTIVITY AND CHARGE range where our analysis is justified. The other unknown DENSITY WAVES is the broadening of the polariton dispersion around qr due to the impurities in the system, which in turn de- termines the lowest possible energy scale of polaritons In the previous section we developed the theoretical at the roton minimum. However, we found out that the framework needed to understand a system of fermions polaritons at q , which contribute most to superconduc- interacting with a bosonic condensate. The theory is gen- r tivity, have an effective mass that is roughly 2 orders of eral within the framework of the Migdal’s theorem and magnitude lighter than the bare exciton mass (for the up to the assumption that density-density interactions parameters used in Figure 3 and n0 = nc). As a conse- depend only on the momentum transfer. quence, polaritons at the roton minimum are relatively In this section, we restrict our analysis to the system robust against broadening by static disorder. introduced in Section II A: a 2DES interacting with a The quantity that we are most interested in here is the polariton BEC. We are interested in the phase transi- critical temperature of the superconductor. In Appendix tions that are possible in this system. We find that the B we provide a short review on how to correctly calculate polariton BEC can undergo a phase transition into a su- the critical temperature of a superconductor. persolid, a superfluid with a spatially ordered structure To calculate the superconducting critical temperature similar to a crystal. At the same time, there are two we use the modified McMillan formula [18, 19]: closely intertwined instabilities in the 2DES: one towards a CDW phase and the other towards a superconducting f f ω 1.04(1 + λ) k T = 1 2 log exp , phase. All of these transitions are possible due to the B c 1.2 −λ(1 0.62µ∗) µ∗ softening of the polaritons. We notice the similarity to µ − − µ∗ = . (17) strongly correlated electron systems such as high temper- 1 + µ ln(ε /¯hω ) ature superconductors which exhibit a quantum critical F D point where many instabilities can occur. This formula is only meaningful if the exponent is neg- In Section III A we investigate quantitatively the su- ative, and is roughly valid for λ < 10. In the above perconducting transition in the 2DES. This transition ωlog = exp [ ln(ω) ], (the average is taken with respect h i 2 has been previously investigated [3–5] without taking into to the weight function α (ω)F (ω)/ω). The screened account either the screening effects due to the 2DES or Coulomb repulsion µ between electrons averaged over the the polariton softening, which we find to be crucial for Fermi surface, is given by: reaching the strong coupling regime. Moreover, we show 0 X VC (k k ) in Appendix B that the Fröhlich [17] type potential is µ = − δ(ε )δ(ε0 )/N(0). (18) (k k0) k k not suitable for a reliable calculation of the critical tem- k,k0 − perature. In the next subsection we investigate qualitatively the The correction factors f1, f2 are given in Appendix B. possibility of an unconventional CDW in the 2DES due In order to calculate the critical temperature we need to the proximity of the BEC to a supersolid phase tran- to know the strength of the exciton-exciton interaction sition. U(q). Given that the exciton-exciton interaction is generally repulsive it has the effect of pushing up the polariton dispersion, stiffening the polaritons. In contrast, the electrons mediate an attractive interaction between A. Superconductivity polaritons leading to softening. The momentum dependence of the polariton-polariton interaction around qr has As mentioned above, as we reach the strong coupling not been investigated, since experiments have been lim- regime the 2DES can become superconducting. Contrary ited to small momentum values of the order of the photon to previous assertions [3–5] we find that polariton medi- momentum. Theoretical calculations [8–10]seem to sug- ated superconductivity is not possible in the presence of gest that in our system the exciton-exciton interaction at electronic screening without taking into account the soft- large momenta q 1/aB is about an order of magnitude ening of the polaritons. As the polaritons soften and the smaller that the interaction≈ at q = 0 and might even be polariton BEC approaches the supersolid transition, the attractive. electron-polariton interaction greatly increases and the Fortunately, since we have a highly tunable system, system enters the strong coupling regime. Since materi- the highest critical temperatures that we can obtain do als with lower dielectric constants are more suitable for not depend strongly on either the strength or the q- reaching the strong coupling regime (as we will show in space dependence of U(q), as long as the polaritons can Section IV), in this section we choose to look at TMD soften at some momentum q (i.e. there is a q such that 2 monolayers. U(q)+χ(q)VX (q) < 0). Therefore, given the uncertainty, 8

polariton density n = 2.169 1013cm−2. At this point 2 c λ 1.5, the condensate depletion× is less than 5% and the Tc (K) B(qr ,ω) ≈ 0.4 roton minimum is approximately at 1meV.

) In the other three panels we plot the polariton disper- )

ω q =qr 1 , K

r sion (lower-left), the polariton spectral function at q = qr ( q c (

T 0.2 (upper-right) and λq (lower-right), which quantiﬁes the B attraction strength between two electrons on the Fermi

0 surface resulting from exchanging a virtual BEC excita- 0.0 tion of momentum q. The exact definition of λq can be 2.14 2.15 2.16 2.17 0 2 4 6 8 found in Eq. (A27). The different colors of the lines in 13 2 n0 (10 cm− ) ω(meV) these three panels correspond to different values of n0 denoted by the colored dots in the upper-left panel Figure 40 30 3. The blue line correspond to the case of highest polari- ω (meV) λ q q ton density ( ) while the black line corresponds to 30 n0 = nc 20 the lowest polariton density (n0 = 98.8%nc).

20 In order to achieve critical temperatures of a few Kelvins the roton-like minimum needs to be lowered to 10 10 energies of a few meV. As the polaritons soften, the momentum dependent electron-electron attraction, quanti- 0 0 fied by λq, develops a strong peak at q = qr. This shows 0.0 0.2 0.4 0.20 0.25 0.30 0.35 0.40 that mainly the soft polaritons are responsible for the q/kF q/kF superconducting phase transition of the 2DES, because they interact most strongly with the 2DES, for two rea- Figure 3. Upper-left panel: critical temperature (solid purple sons. First of all the softening of the polaritons results in line) for a typical TMD monolayer as a function of polariton a depletion of the condensate which leads to an increase density n0. The dashed green line is at the critical polariton in the electron polariton matrix element proportional to 13 −2 density nc = 2.169 × 10 cm and can be regarded as di- p Ωq/ωq. Secondly, the electron-electron attraction me- viding the BEC (blue region) and supersolid (green region) diated by the soft polaritons increases even more because phases of the polaritons. Upper-right panel: The polariton these polaritons are closer to resonance with the elec- spectral function at the roton-minimum B(qr, ω). Lower-left panel: polariton dispersion renormalization (in purple is the tronic interactions which are confined to an energy layer of width k T ω around the Fermi energy. Looking bare dispersion). Lower-right panel: λq. The different col- B c D ors of the lines in the upper-right, lower-left and lower-right at the spectral function of the quasi-particles at the ro- panels correspond to different values of n0 denoted by the col- ton minimum, denoted by B(qr, ω), we notice that the ored dots in the upper-left panel. Expressed in percentages polaritons at the roton minimum become overdamped, of nc these are: n0 = 100%nc(blue), n0 = 99.7%nc(green), similar to paramagnons near a magnetic instability. At n0 = 99.3%nc(red), n0 = 98.8%nc(black).The rest of the pa- this point McMillan’s formula should still be valid but rameters are typical TMD monolayers√ parameters: d = aB = one should be careful about the broad polariton spectral 1nm, L = 1.5aB = 1.5nm, g0 = 10 4meV (4 exciton lay- 13 −2 function. We investigate the effects of the finite linewidth ers are used), = 40, me = mh = 0.2m0, ne = 10 cm , of this spectral function in Appendix A and show that the U = 0.12µeVµm2. electrons remain good quasi-particles and that the broad polariton spectral function will not have a significant im- pact on the superconducting critical temperature. 2 we choose U = 0.12µeVµm for our numerical simula- Another remarkable feature of the electron-electron at- tions but emphasize that similar results can be obtained traction mediated by polaritons is that it favors p-wave 2 by tuning other parameters as long as U < 0.5µeVµm . pairing (or other higher symmetries) to s-wave pairing. Furthermore, we reemphasize that tuning the polariton- This is easily understood if we look at the total electron- polariton interaction using Feshbach resonances has been electron interaction in real space, which we present in proposed and demonstrated [11, 12]. Figure 5 in the Appendix. Notice that this interaction is In the upper-left panel of Figure 3 we plot in solid formed by a strongly repulsive part at the origin followed purple the critical temperatures that can be achieved in by a oscillatory part with the wavelength 2π/qr, due to a typical TMD monolayer by tuning the polariton density the strongly peaked interaction in momentum space. Be- n0 in a system with 4 exciton layers (which have the effect cause of this shape of the interaction, the s-wave pair of doubling g0 compared to the initial value). According will feel the strongly repulsive interaction at the origin, to our mean-field calculation in the blue region the 2DES while the p-wave pair will avoid this region due to the should become superconducting whereas we cannot apply Pauli-exclusion principle. In accordance with this sim- our theory in the green region due to the breakdown of ple picture, we find p-wave critical temperatures a few the Bogolyubov approximation. The dashed green line times higher than the s-wave critical temperature. How- dividing the two regions in phase space is at the critical ever, since the electrons in the p-wave pair are not time- 9 reversal partners these pairs will be influenced by the dis- important (the momentum direction should be chosen order in the system. Therefore, an accurate calculation of spontaneously), which would require a careful extension the p-wave critical temperatures requires an estimation of our Bogolyubov approximation scheme. of the randomness in our system. Remarkably, in contrast to the conventional behavior Notice the strong dependence of the critical tempera- based on nesting features in the electron band structure, ture on the polariton density, which indicates that some this type of singularity is not originating from the elec- fine tuning will be necessary in order to observe the su- tronic response function but due to a singular behavior in perconducting phase. Fortunately, the polariton density the electron-polariton interaction at some wavevector. In is proportional to the intensity of the laser generating the both cases mentioned above one can tune the wavevector condensate, and the intensity of a laser can be tuned with where the polariton dispersion touches zero and therefore extreme accuracy. Laser intensities with less than 0.02% can tune the nesting wavevector qr. noise have been maintained relatively easily in the con- We reemphasize that in order to observe a supercon- text of polaritons by using a feedback loop [20]. We also ducting or CDW phase transition in the 2DES, the po- emphasize that we tried to be conservative in our choice laritons have to soften and the BEC has to be close to the of bare system parameters. It may for example be pos- super solid phase transition. More generally, despite the sible to obtain higher Tc if the polariton density can be differences in structure and phenomenology, the phase di- increased further without reaching the Mott transition. agram of many unconventional superconductors exhibit the common trait that superconductivity resides near the boundary of another symmetry breaking phase. Exam- B. Supersolid and Charge Density Waves ples are the superconducting phases appearing at magnetic quantum phase transtions as found in many of the As we approach the regime of strong coupling charac- Ce-based heavy fermion compounds such as CeIn3 [24– terized by a significant softening of the polaritons, the 27], or in iron pnictides accompanying spin density wave system becomes susceptible to other instabilities, in ad- states [28], as well as magnetic, stripe and nematic orders dition to the superconducting instability. When the po- discussed for copper oxides [29]. Another example, sim- laritons soften to the degree that the polariton dispersion ilar in some respects to our system, is the TMD family, touches zero there will be an instability in the polariton where charge density wave order competes with super- BEC. A transition to a supersolid phase occurs in the conductivity and this feature has been attributed to a green region in Figure 3 since the polariton dispersion softening of the finite momentum phonon modes [21–23]. 13 −2 touches zero roughly at nc 2.173 10 cm . Even though such a supersolid instability≈ was× proposed for indirect excitons [1, 2], we remark that this phase transi- IV. MATERIALS SUITABLE FOR REACHING tion can be more easily observed in a polariton BEC not THE STRONG COUPLING REGIME least because the realization of an exciton BEC is still an experimental challenge. In our theoretical framework In this section we do a systematic analysis of the ma- based on the Bogolyubov approximation, the onset of terials most suitable for reaching the strong coupling this instability can be observed as a dramatic increase in regime where the 2DES becomes unstable towards a new the BEC depletion as the polariton dispersion approaches phase. We find that materials with low dielectric con- zero. Since our analysis is only valid when the conden- stants are most suitable and conclude that semiconduct- sate depletion is small from now on we assume that the ing TMD monolayers are good candidates for observing polariton dispersion never touches zero. polariton mediated superconductivity. As the polariton system approaches its BEC instability At a first sight, it may seem that there are many pa- to a supersolid, the 2DES becomes susceptible to insta- rameters that influence the electron-polariton interaction bilities mediated by the soft polaritons. In the previous mediated instabilities that can change from material to section we analyzed the susceptibility of the 2DES to- material: , me, aB, d, L, g0, kF . However, these parame- wards a superconducting phase. However, the strongly ters are typically not independent. In fact, we argue that peaked electron-electron attraction at q = qr, as shown in all of these material parameters scale with the dielectric the bottom-right panel of Figure 3 can result in a CDW constant as shown in Table I. state. A CDW order can also appear due to the phase transition of the BEC into a supersolid. This transition is First, we emphasize that the mass dependence in the analogous to the case of ‘frozen phonons’ which has been first row is more complicated and one may even treat proposed as an explanation for the CDW order in ma- me as an independent parameter as well. For the dipole terials such as TMDs [21–23], where a finite-momentum dependence d we assumed that, regardless of the mecha- phonon softening leads to a condensation resulting in a nism, the induced dipole can be of the order of the Bohr static distortion in the lattice which in turn leads to a radius but not larger. Similarly, we assumed that the dis- modulation in the electron density. In our case this cor- tance L between the 2DES and polariton planes cannot responds to the fact that the mean field b becomes be smaller than the exciton Bohr radius, to avoid tunnel- h qr i 10

or more photonic. This will also change the electron- Table I. Parameter dependence on the dielectric constant polariton coupling, an effect which is not captured in −1 me ∝ Eq. ( 19). Another method is the use of multiple quan- 2 aB ∝ /me ∝ tum wells. By using N quantum wells one can increase 2 d ∝ aB ∝ by √ while leaving the other parameters unchanged. 2 g0 N L ∝ aB ∝ −1 −2 Another interesting quantity to investigate is what is g0 ∝ aB ∝ −1 −2 the largest renormalization that can be obtained as a kF ∝ L ∝ function of the dielectric constant. Looking at Eq. ( n ∝ a−2 ∝ −4 0 B 9) and setting U(q) = 0 we see that the largest renormalisation that can be obtained is given by ∆ω = 2N χ(q )V 2 (q ) −3. ing between the two planes. We also assumed that the 0 r X r To conclude, we∝ note that under the conditions de- light-matter coupling is proportional to a−1. It turns out B scribed in the preceding paragraphs, λ() −6. that because of the large momentum cutoff due to the fi- ∝ nite distance L between the 2DES and polariton planes we get better results with decreasing k . However, we F V. CONCLUSION cannot lower kF arbitrarily since we still need RPA to be valid. Finally, in the last row, we assumed that the max- 2 The striking feature of the coupled 2DES-polariton imum value of n0aB is a material independent constant since it is set by phase space filling [30]. that we analyzed is the rather unusual nature of the long- The strong coupling regime can be characterized by a range boson-fermion interaction which is peaked at a fi- large EPC constant λ and a small Coulomb repulsion con- nite wavevector q0. The latter can be tuned by choosing stant µ. Therefore we need to investigate the dependence the system parameters and leads to the emergence of a of these parameters on the dielectric constant. Introduc- roton-like minimum in polariton dispersion at qr q0 = . While we have primarily focused on the prospects∼ for6 ing the variable u q/2kF and the material independent kF ≡ 0 0 observation of light-induced superconductivity, a very in- constants L¯ = 2kF L and k¯TF = kTF /(2kF ) : ∝ ∝ teresting open question is the competition between super- ¯ Z 1 e−2uL Ω conductivity and polariton supersolid/CDW phases. The λ() −3 du 2kF u , 2 2 2 boson-fermion system that we consider allows for precise ∝ 0 √1 u (1 + k¯ /u) · ω TF 2kF u tuning of key parameters and can be realized either in 1 − Z 1 GaAs or TMD based 2DES/microcavity structures. µ() = 0 du . (19) 2 0 √1 u (1 + k¯TF /u) Strictly speaking, the onset of superconductivity in the − 2DES will be due to a Berezinsky-Kosterlitz-Thouless Looking at the above expressions it is clear that µ re- transition (BKT) transition. On the other hand, in the mains roughly constant from material to material which parameter range we consider, the Tc we estimate using is not unexpected (if we chose me as an independent pa- the mean-field approach should be comparable to the rameter we would then get some variance in µ for dif- BKT transition temperature. ferent materials). However, since the electron-polariton A very promising extension of our work is the re- interaction is retarded the relevant constant µ∗ given in alization of photo-induced p-wave pairing of composite Eq. (B1) indicates that we can decrease the effective fermions in the quantum Hall regime [31]. Unlike the electron-electron repulsion by choosing materials with proximity effect due to an s-wave superconductor, finite- large Fermi energies and small Debye frequency. range polariton-mediated attractive interaction is more We also see that λ depends both on the dielectric con- likely to be compatible with requirements for observ- stant and on the bare and renormalized polariton ener- ing fragile fractional quantum Hall states, enabling edge- gies. All of these three parameters can be tuned indepen- state pairing that was proposed as a method to realize dently to some extent through various methods. We no- parafermions [32]. tice that observation of polariton mediated superconduc- In contrast to phonons, polaritons can be directly mon- tivity requires materials with smaller dielectric constants, itored by imaging the cavity output. This provides a because smaller dielectric constants favor the dipole in- unique possibility to simultaneously monitor changes in teraction over the monopole Coulomb repulsion. This transport properties of electrons and the spatial structure explains why λ −3 while µ 0. of polaritons, as the strongly coupled system is driven In addition to∝ a small dielectric∝ constant we want a through an instability. The driven-dissipative nature of large bare polariton energy and a small renormalised po- the polariton condensate could also be used to inject polariton dispersion. The dielectric constant also sets the laritons at a preferred wavevector q˜r. Since q˜r need not value of g0 which gives the energy scale of the bare polari- be equal to the roton minimum qr, externally imposing a ton energy. It should be emphasized though that the bare spatial structure for the polariton condensate could alter polariton energy can also be tuned through other meth- the competition between superconductivity and CDW in- ods. One method is to tune the detuning between exci- stabilities. Moreover, the competition between the CDW tons and cavity and make the polaritons more excitonic and superconductivity can be investigated further by im- 11 posing a periodic potential on photonic or excitonic de- the electron-phonon interaction and therefore we can an- grees of freedom. alyze it in analogy with the Migdal-Eliashberg theory, Last but not least, the possibility to turn superconduc- which is controlled by the small parameter ¯hωD/εF , the tivity in a semiconductor high mobility electron system ratio of the characteristic phonon/electron energy scales. on or off using laser fields could form the basis of a new Although in doing the many body theory we treated kind of transistor that is compatible with the existing all interactions simultaneously in order to avoid double semiconductor nanotechnology. A crucial development counting, we choose to present our results in a more in- for such applications is a substantial increase of Tc, which tuitive order. may be achieved by properly choosing the properties of We define the bare electron propagator as the semiconductor host material. 1 (0)(p) = T c (τ)c† (0) = , (A3) Ge −h τ k k i iω ε n − k VI. ACKNOWLEDGEMENTS where p = (iωn, k). In light of the following analysis we define the bare polariton propagator as The Authors acknowledge useful discussions with Se- 1 bastian Huber and Alexey Kavokin. (0)(p) = T b (τ)b† (0) = . (A4) G11 −h τ k k i iω Ω n − k The analogy with phonons is made by introducing a Appendix A: Theoretical analysis phonon-like operator and propagator: To tackle the many-body problem we use a Green’s † Aq bq + b−q, functions approach. After doing a mean field approxi- ≡ (q, τ) T A (τ)A− (0) . (A5) mation the electron-polariton Hamiltonian has a similar D ≡ −h τ q q i structure to the well understood electron-phonon Hamil- In the absence of interactions we have: tonian. Therefore, we will use Migdal-Eliashberg theory 2Ω to analyze this system theoretically. However, some of (0)(p) = (0)(p) + (0)( p) = k . (A6) D G11 G11 − (iω )2 Ω2 our results can also be obtained through a more intuitive n − k canonical transformation and therefore we also present As mentioned above we will understand the electron- this method in Appendix A 2. polariton interaction in terms of the Migdal Eliashberg theory of the electron-phonon interaction. To make any progress, we need to make a Born-Oppenheimer approxi- 1. Migdal-Eliashberg theory mation, which in the many-body physics language means that we ignore the electron-phonon vertex corrections due We start from the initial Hamiltonian in Eq. (3). When to phonons. This approach is justified by Migdal’s the- most of the polaritons are in the BEC ground state at orem [34]. We give a brief argument, based on a phase k = 0 we can simplify this Hamiltonian using the Bo- space analysis, that summarizes Migdal’s theorem as it golyubov prescription [33], which is equivalent to mak- applies to our system. For a more rigorous proof one † ing the following replacement: b0 = b0 = √N0 (N0 is should check Refs. 16 and 34. the number of polaritons in the condensate). This is fol- As we will show in the following sections most of the lowed by a Bogolyubov approximation which consists in polaritons that interact with the electrons on the Fermi ignoring terms of lower order in N0. surface are found in a narrow energy interval. Therefore, The resulting Hamiltonian is: we can associate an energy scale to the polaritons which (e) (p) (e−e) (e−p) (p−p) we will denote by ωD, in analogy with the Debye energy H = H0 + H0 + HI + HI + HI , in the phonon case. We will define this energy quan- (A1) titatively below but for now we will assume that such where all the other terms remain the same as before ex- an energy scale can be associated to the polaritons and furthermore we make the assumption that ¯hω ε . cept for: D F

(p−p) X U(k) † † † H = N0 b b− + b b + 2b b I 2 k k k −k k k q k6=0 X † k k + q +N0U(0) bkbk, p p + q 6=0 p k k (e−p) p X † † HI = N0 VX (q)ck+qck bq + b−q . (A2) k,q Figure 4. First vertex correction diagram. In blue are the After the mean-ﬁeld Bogolyubov approximation the electron propagators while in red are the polariton propaga- electron-polariton interaction has the same structure as tor. 12

Let us consider the first correction to the electron- because it is easier to handle. This limit is accurate as polariton vertex, with the corresponding Feynman dia- long as the frequencies involved are much smaller than gram presented in Figure 4. If this correction can be the plasma frequency (in 2D, the only regime where this ignored than we can certainly ignore the higher order limit is not satisfied is at very small momenta). In the corrections. Suppose a polariton of momentum q decays static limit we have: and forms an electron-hole pair of momenta and k + q k 1 respectively. This pair will be coherent for a distance of (k, ω) (k, 0) = , (A8) ≈ kTF 1/q. Since the electrons move at roughly the Fermi ve- 1 + k locity this gives us a coherence timescale of 1/qvF . Only 2 2 in this timescale electrons can be scattered again. Since where kTF = mee /(2π¯h ) = 2/aB is the Thomas-Fermi the polaritons take much longer to respond we expect wavevector. In terms of this dielectric function, the screened this vertex correction to be of order ω /qv . The av- D F electron-electron and electron-polariton interaction is erage momentum of the phonon is of the order of kF so we expect the vertex correction to depend on the small given by: parameter ¯hω /ε . (If we wish to be more accurate the D F ˜ VC (k) first vertex correction is of the order of , where VC (k) = , λ¯hωD/εF (k) λ is the electron-polariton coupling constant to be defined below). Thus, we can safely conclude that vertex VX (k) V˜X (k) = . (A9) corrections can be ignored. (k) The only nontrivial effect of many-body interactions The dilute Bose-condensed gas is another one of the between electrons is the renormalisation of interaction few many body systems that are well understood. In between quasiparticles, i.e. screening [15]. In the follow- this case the small parameter which allows a controlled ing we will explore the screening of both the electron- expansion is given by n a2, where n is the polariton electron and the electron-polariton interactions in the 0 0 density and a is the scattering length of the bosonic re- random phase approximation (RPA). This effect appears pulsion. The field theoretical treatment of the problem as a renormalisation of the photon propagator, where, was first developed by Beliaev [36]. A more accessible according to the RPA approximation, the photon proper exposition of this formalism is presented in Ref. 37. self-energy is approximated by the simplest polarisation In addition to the normal polariton propagator intro- bubble. duced above, when interactions are turned on there is In the RPA framework, the screened electron-electron an additional anomalous propagator that must be con- and electron-polariton interaction is expressed in terms of sidered. Together, these propagators satisfy the Dyson- the dielectric function ε(q, iωn), which has the following Beliaev equations. These equations can be written more analytical form: compactly by introducing two additional propagators, which are, however, not independent of these two. In (k, iω ) = 1 V (k)χ0(k, iω ), n − C n the end, the 4 propagators that need to be considered 1 X (0) (0) χ0(k, iω ) = (k + q, iω + ik ) (k, ik ) are: n β G n n G n k,ikn † 11(k, τ) = bk(τ)bk(0) , X f(εk) f(εk+q) G −h i = − , 12(k, τ) = bk(τ)b+k(0) , iωn + εk εk+q G −h i k † † − 21(k, τ) = b (τ)b (0) , χ (k, iω ) G −h −k k i χ(k, iω ) = 0 n . (A7) † n 22(k, τ) = b (τ)b−k(0) . (A10) (k, iωn) G −h −k i In the above, f(ε) is the Fermi distribution function. These propagators and the associated self energies are not independent but satisfy the following identities: The polarisation bubble χ0 is the Lindhard function and denotes the linear response to a perturbation when 22(p) = 11( p), 12(p) = 21( p), electron-electron interactions are neglected. In this case G G − G G − Σ22(p) = Σ11( p), Σ12(p) = Σ21( p).. (A11) the perturbation disturbs the electron system and cre- − − ates electron-hole pairs and thus polarising the system. Solving the Dyson-Beliaev equations in the Bo- In contrast. the screened polarisation bubble χ is the re- golyubov approximation and making the RPA approx- sponse function when electron-electron interactions have imation we obtain the renormalised propagators: been taken into account in the RPA. The dielectric function for a 2D system at zero tem- iωn + Ωk + Σ12(p) perature has been calculated for the first time in Ref. 11(p) = 2 2 , G (iωn) Ω 2ΩkΣ12(p) 35. The poles of the dielectric function give us the col- − k − Σ (p) lective excitations of the electron system, the plasmons. 12 12(p) = 2 − 2 , G (iω ) Ω 2Ω Σ12(p) Unless otherwise indicated, for the rest of the paper we n − k − k 2 will take the static limit, (a.k.a. the Thomas-Fermi limit) Σ12(p) = N0 U(k) + χ(k, iωn)VX (k) . (A12) 13

The dispersion of the renormalised polaritons, denoted In the presence of interactions, the phonon like propa- by ωk, is obtained from the zeros of the real part of the gator introduced in Eq. ( A5) has the form: denominator, therefore: (p) = 11(p) + 22(p) + 12(p) + 21(p) q D G G G G ω = Ω2 + 2Ω N [U(k) + χ(k)V 2 (k)], (A13) 2Ω k k k 0 X = k . (A17) (iω )2 ω2 2iγ ω where we made the approximation [χ(k, iω )] χ(k). n − k − q q < n ≈ Notice that the effect of electron-polariton interaction is Notice that the propagator depends on both the bare to renormalise the polariton-polariton interaction. Be- and the renormalised polariton spectrum. As usual [15], cause this is a second order interaction the effect is pro- we define a reduced propagator ¯ which corresponds portional to the square of the bare electron-polariton in- to the propagation of the new polaritonD quasi-particles teraction and the response function χ which also contains and therefore depends only on the renormalized quasi- the effects due to screening. particles’ spectrum: The polariton spectral function linewidth comes from the imaginary part of the response function, which is Ωk ¯ zero in the static limit, so we need to use the frequency (p) = (p). (A18) D ωk D dependent response function to calculate the polariton linewidth. Using Ωk [Σ12(k, ω)] = ωkγk (γk is the po- The polariton mediated electron-electron attraction lariton linewidth) we|= obtain, in the| limit [(k, ω)] can be expressed in terms of this propagator: [(k, ω)] , the following expression for|= the polariton| |< | 2 linewidth: (eff) VC (k) N0VX (k) Ve−e (p) = + 2 (p) 2 (k) (k) D N0VX (q) Ωq γ = 2 [χ0(q, ω )] q 2(q) ω = q VC (k) ˜ 2 ¯ q = + Mk (p), (A19) 2 (k) D N0VX (q) Ωq X = 2π 2 (f(εk) f(εk+q)) (q) ωq − where we have introduced the renormalised electron- k polariton matrix element δ(ω ε + ε + ). (A14) · q − k k q s As mentioned above, the polarisation bubble has been N V 2 (q) Ω ˜ 0 X q (A20) evaluated exactly in Ref. 35 but to get a simpler ana- Mq = 2 . (q) ωq lytical formula we make the following approximation. At low temperatures the Fermi factors restrict the k integra- Notice that the term Ωk/ωk from the initial propagator tion to a narrow region about the Fermi surface of width (p) has been absorbed in the electron-polariton matrix ωq, so for reasonably well behaved Fermi surfaces we can element.D In the previous section we noticed that this is replace these factors with ωqδ(εk) to obtain: necessary in order to obtain the same polariton linewidth 2 as the one calculated using Fermi’s golden rule and now N0VX (q) Ωq X γq 2π 2 ωq δ(εk)δ(εk+q). (A15) we have seen why. ≈ (q) ωq k We also investigate the electron self energy acquired We remark that the same result can be obtained us- through interactions with polaritons. This contribution ing Fermi’s golden rule if we consider the renormalised is small but its derivative with respect to energy is large within ωD of the Fermi surface. Therefore it will strongly electron polariton interactions to be given by M˜ q = q 2 affect the electrons within ωD of the Fermi surface. The N0VX (q) Ωq 2 as shown in Refs. 38 and 39. We will find (q) ωq main effects are a renormalised mass and a finite quasi out below that this is indeed the proper renormalised particle linewidth. electron-polariton interaction. The contribution of the polaritons to the electron self- Evaluating the sum we obtain: energy has the following analytical form:

˜ 2 2 γq Mq N(0) 2kF 1 Z d q , (A16) X ˜ ¯ p 2 Σ(k, iωn) = 2 Mq (q, iqn) ωq ≈ εF q 1 (q/2kF ) −β (2π) D − iqn (0) where N(0) is the electron density of states at the the (k + q, iωn + iqn). (A21) Fermi surface. In order to have long lived quasi-particles ·G we need to satisfy γ /ω 1. The resulting effects are conventionally expressed in q q Having discussed the condensate properties in a field terms of the Eliashberg function α2F (ω). This function theoretical formalism, we investigate the effect of the con- is closely related to the polariton density of states: densate on the electrons and we see that polariton ex- X citations can mediate an attractive interaction between F (ω) = δ(ω ω ). (A22) − q electrons. q 14

However, the connection is somewhat obscured in the a. Finite polariton spectral function linewidth usual deﬁnition:

2 In the previous discussion we have treated the polari- 2 X 0 α F (ω) = M˜ k−k0 δ(ω ωk−k0 )δ(εk)δ(ε )/N(0). tons as perfect quasi-particles. When the linewidth of − k k,k0 the polariton spectral function becomes significant the (A23) Eliashberg function needs to be modified and is broadened: The above function can be expressed in terms of the pre- 2 2 X 0 0 viously investigated polariton linewidth γ : α F (ω) = M˜ − 0 B(k k , ω)δ(ε )δ(ε )/N(0), q k k − k k k,k0 2 2 X α F (ω) = N(0)ω γ δ(ω ω ). (A24) 1 1 Ωq 2ωq π q − q B(q, ω) = [D(q, ω)] = , q π = π = ω ω2 ω2 2iγ ω q − q − q q Most properties of the electron-polariton interaction (A30) can be expressed in terms of the EPC (electron-polariton where the Lorentzian B(q, ω) is the polariton spectral coupling in our case) constant λ and averages ωn , where h i function. We are interested in how the finite-polariton n are integers and the average is taken with respect to lie width will influence the superconducting properties of 2 the weight function α F (ω). For example, the Debye the electron system. frequency that we defined above can be expressed quan- The first question that we need to ask is whether the titatively as: electrons remain good quasi-particles. As shown in Refs. Z ∞ 38 and 39, when the finite polariton - linewidth is in- 2 2 ωD ω = 2 dω α F (ω)/λ. (A25) cluded the electron lifetime scales as ε close to the Fermi ≡ h i 0 surface. Therefore electrons close to the Fermi surface are well-defined quasi-particles. In all our numerical simula- The definition of the EPC constant is: tions we checked that electrons are well defined quasi- 2 Z dω α F (ω) particles in a shell of the order of kBTc, where Tc is the λ = 2 . (A26) ω superconducting critical temperature. According to McMillan’s formula in Eq. (B1) the su- This constant can also be expressed in terms of the λq perconducting critical temperature can be expressed in ∗ which makes explicit the contribution of polaritons with terms of 4 constants: µ , λ , ωlog = exp [ ln(ω) ], 2 1/2 h i different momenta: ω¯2 = ω (the averages are taken with respect to the weighth functioni α2(ω)F (ω)/ω). Only the last 3 constants 1 X 1 X γq λ = λ = , (A27) will be affected by the broadening of the Eliashberg func- N q πN(0) ω2 q q q tion. Furthermore, for large λ the critical temperature Tc √λω¯2, therefore we will only investigate how these where N is the total number of electrons, which in 2D is constants∝ are modified. given by N = N(0)εF . We can rewrite λ as: Returning to the electron self energy due to interac- 2 Z ∞ tions with polaritons, the real part results in a mass X 1 B(q, ω) λ = 2 M˜ δ(ε )δ(ε + ) dω . q k k q N(0) ω renormalisation of the electron quasi particle given by: k,q 0

∗ εk (A31) me = me(1 + λ) ε˜k = . (A28) → 1 + λ However, by definition: The imaginary part of the self energy gives the electron Z ∞ 0 ¯ 0 B(q, ω ) quasi particle linewidth Γ. At zero temperature [40]: D(q, ω) = dω 0 . (A32) −∞ ω ω + iδ Z ω − Γ(ω) = π dω0α2F (ω0). (A29) Using the oddness of B(q, ω) we obtain: 0 Z ∞ B(q, ω) D(q, 0) ω q (A33) Clearly, the electron quasi-particles with energies close dω = = 2 , 0 ω − 2 ωq (0) to the polariton energy scale ωD will be short lived because electrons will be able to lose their energy to excite where ωq(0) denotes renormalised polariton energy ob- polaritons. For these electrons, the quasi particle picture tained by using the static (Thomas-Fermi) dielectric fails. However, in our system, we have the following en- function. Thus, when the finite polariton lifetime is taken ergy scale kBTc ωD. Therefore, we expect that the into account, λ should be calculated using the polariton superconducting electrons will not be affected by dissi- energies obtained from the static limit for the polarisa- pation due to polaritons, so we can still use the quasi tion bubble (which is actually what we already did to particle picture. simplify calculations). 15

We now investigate the effect of the broadened Eliash- In the above Mq = √N0VX (q). berg function on λ ω2 : We will first show how the one can investigate the effect h i of electrons on polaritons by tracing out the electrons and 2 Z ∞ 2 X 1 then we will show how one can investigate the electronic λ ω = 2 M˜ δ(ε )δ(ε + ) dωωB(q, ω) h i q k k q N(0) k,q 0 system by tracing out the polaritons. X 2 1 = 2 M˜ ω δ(ε )δ(ε + ) , (A34) q q k k q N(0) k,q a. Tracing out the electrons where we used the well known [39] sum rule ωq = ∞ We first wish to investigate the effect of electrons on R dω ωB(q, ω). Therefore λ ω2 is not affected at all 0 h i the polaritons. The effect of electrons on polaritons is to by the broadening of the Eliashberg function. induce an effective attraction between polaritons. In conclusion we can safely neglect the effect of the To get the same results as in the diagrammatic ap- polariton linewidth on the superconducting properties of proach we must treat the electron-polariton interaction the 2DES. to second order in perturbation theory, however, the interaction must contain the screening effects due to electron-electron interactions. Intuitively, a polariton of 2. Canonical transformation momentum k creates a potential VX (k) in the 2DES, and the 2DES responds by creating a charge density In this subsection we will show how some of the above δn(k) = χ(k)VX (k) where χ(q) is the response function results can be obtained in a Hamiltonian formalism using first introduced in Eq. (A7). This charge density will cre- canonical transformations. This approach might be pre- ate a potential VX (k) at the polariton condensate, thus ferred for it’s simplicity and because it yields information 2 resulting in an attraction χ(k)VX (k) between a polariton about the quasi-particle wavefunction. of momentum k and a polariton in the condensate. Af- In order to obtain the same results as in the diagram- ter tracing out the electrons and rearranging terms we matic approach we must first better understand the ap- obtain the following effective polariton Hamiltonian: proximations that we made in the diagramattic approach h i and make the same approximations in this context. No- (p) X † † † † H = Ek bkbk + b−kb−k + gk bkb−k + bkb−k , tice that our choice of the electron and polariton self- k6=0 energies implies that we treat the electron-polariton in- (A37) teraction up to second order. This approximation was justified by Migdal’s theorem. However, the polariton- where the sum must be taken over half of k-space polariton interactions are treated exactly. In contrast the to avoid double counting. In the above gk 2 ≡ electron-electron interactions are treated perturbatively N0 U(k) + χ(k)VX (k) is the effective interaction be- in the RPA approximation, which is an infinite sum con- tween the condensate and a polariton of momentum k. taining terms of all perturbative orders. Notice that the polariton energy E Ω + g contains k ≡ k k Finally, McMillan’s formula is based on the approxima- the Hartree-Fock interaction gk. tion that the electrons that participate in superconduc- The above Hamiltonian is quadratic and therefore can tivity are in a very thin layer around the Fermi surface be diagonalized by making a canonical transformation much smaller than the polariton energy scale which al- (a.k.a. the Bogolyubov transformation) by introducing lows one to approximate the electrons as living on the the new operators:

Fermi surface. We will need this approximation to avoid ∗ † complications. ak = ukbk + v−kb−k. (A38) As before we start from the Hamiltonian in Eq. (A1) 2 2 This is a canonical transformation if u v = 1. which has been obtained in the Bogolyubov approxima- k −k Notice that the phases are irrelevant| and| − therefore | | we tion: choose uk, vk . To diagonalize the Hamiltonian we (e) (p) (e−e) (e−p) (p−p) must choose: ∈ < H = H0 + H0 + HI + HI + HI , r r (A35) 1 Ek uk, v−k = 1, (A39) where, as before: ± 2 ωk ± where we introduced the excitation spectrum ωk (p−p) X U(k) † † † p 2 2 p 2 . The new Hamiltonian≡ is H = N0 b b− + b b + 2b b Ek gk = Ωk 2Ωkgk I 2 k k k −k k k − − k6=0 diagonal: X † (p) X † +N0U(0) bkbk, H = ωkakak. (A40) k6=0 k (e−p) X † † At this point we have found the polariton excitations HI = Mqck+qck bq + b−q . (A36) k,q on top of the condensate. We wish to investigate the 16 interaction between electrons and the new excitations. b. Tracing out the polaritons To do this we express the electron-polariton Hamiltonian (e−p) H in terms of the new excitations: We can also start our analysis by tracing out the polari- s tons. As in the diagrammatic approach we start from the (e−p) X Ωq † † HI = Mq ck+qck(aq + a−q). (A41) Hamiltonian in Eq. (A35) which has been obtained in the ωq k,q Bogolyubov approximation. In this section we are inter- We see that the electron-polariton interaction is increased ested in the electronic part of the system and therefore we p want to trace out the polaritons. We can achieve this by by the factor Ωq/ωq = uq vq which, (not surprisingly) is exactly what we found in− the diagrammatic approach. performing a Schrieffer-Wolff transformation and keeping The reason for the increase of interactions is that the only second order terms. There are different Schrieffer- elementary excitations on top of the condensate con- Wolff transformations that can be done as we explain in tain many polaritons, and therefore they interact more Appendix B. However, the situation is simplified if we strongly with the electrons. To understand this effect consider scattering only on the Fermi surface (which we better we look at the interaction between the interact- also did in the diagrammatic approach by using McMil- ing ground state G and an excited state of momen- lan’s formula). Because this is an energy conserving pro- tum k which we denote| i by E . Since G is defined cess it could be measured experimentally and therefore | ik | i in this case all the different transformations must yield by ak G = 0 for all k, we can express the interacting ground| statei in the Fock basis of the initial/bare polari- the same second order result. The Schrieffer-Wolff transformations mentioned above tons. We denote by n k the Fock state containing n bare polaritons of momentum| i k. In this notation: were derived in the context of electron-phonon interactions. In that case phonon-phonon interactions were ne- G = Πk G k , where, glected because they are negligible. In order to apply | i | i n 1 X vk these transformations to our system we need get rid of G k = − n k n −k . (A42) | i uk uk | i | i the polariton-polariton interaction which we do with the n help of the Bogolyubov transformation introduced previ- As before, the product runs over half of momentum space ously in Eq. (A38). After doing the Bogolyubov trans- to avoid double counting. Notice that the ground state formation we obtain the Hamiltonian: factorizes between states of opposite momenta due to X † X ˜ ˜†˜ X † † 0 translational invariance. We mention that the conden- H = εkckck + Ωkbkbk + V (q)ck+qck0−qckck sate depletion due to bare polariton excitations at mo- k k k,k0,q † 2 mentum k given by G b b G = v tells us how many X † † k k k + M˜ c c ˜b + ˜b , (A46) polaritons of momentumh | k are| i contained in the ground q k+q k q −q state on average. One must always make sure that the k,q total condensate depletion P v2 is small compared to k k where we denote by ˜bk the destruction opera- the number of polaritons in the ground state N0. tor for the new non-interacting polaritons, Ω˜ k = The elementary excitation of momentum k is easily p 2 Ω + 2ΩkN0U(k) is the spectrum of the non- found: k q ˜ ˜ n interacting polaritons and Mq = Mq Ωq/Ωq is the new 1 X vk E = a G = − √n + 1 n + 1 n electron-polariton interaction matrix element. So far, we | ik k | ik u2 u | ik | i−k k n k made only the Bogolyubov approximation. (A43) At this point we can trace out the polaritons using a We also express the electron-polariton interaction in the Schrieffer-Wolff transformation and obtain the following Fock basis: effective Hamiltonian: (e−p) X † X † X † † 0 H = ε c c + V (q)c c 0 c c , HI = ck+qckMq (A44) e k k k k+q k −q k k k,q k k,k0,q X ˜ 2 √n n 1 n + n n 1 . 2 Mq · | − iq h |q | i−q h − |−q V (q) = VC (q) | | . (A47) n − Ω˜ q Notice that the interaction strength depends on the num- When taking screening into account in the RPA ap- ber of polaritons at momentum q. Since both the ground proximation just as in the diagrammatic approach we find state and the excited state contain many polaritons when that the screened electron-electron interaction is given the condensate depletion 2 is large, the electrons will be vq by: able to scatter excitations at this momentum much more efficiently. To illustrate this we calculate the amplitude ˜ V (q) for scattering a bogolon of momentum : V (q) = . (A48) q 1 V (q)χ0(q) ( − ) X − G H e p E = c† c M (u v ). (A45) h |q I | iq k+q k q q − q The above expression can be rewritten in a more intuitive k,q form by separating V˜ (q) into a screened Coulomb repul- 17 sion and an effective attraction mediated by polaritons: Appendix B: Superconductivity

2 ˜ ! In this section we briefly review the methods that ˜ VC (q) Mq 2 V (q) = + ,(A49) can be used to calculate the critical temperature of (q) (q) Ω˜ + 2χ(q)M˜ 2(q) q a polariton-mediated superconductor. This discussion is necessary in the polariton community in order to where as before we have introduced the dielectric func- make clear the connection to superconductivity and to tion (q) = 1 VC (q)χ0(q) and the response function see in which ways our system behaves as a conven- − χ(q) = χ0(q)/(q). The effective electron-electron in- tional/unconventional superconductor. teraction can be rewritten in a simpler form by in- We mention that it is notoriously difficult to make troducing the renormalised polariton energies ωq q ≡ quantitative theoretical predictions of the superconduct- 2 2 Ωq + 2N0 [U(q) + χ(q)VX (q)]: ing properties of a new material. However, this limitation is due to the lack of knowledge of the normal state of the material and not due to the accuracy of the BCS the- V (q) M 2 Ω 2 V˜ (q) = C + q q . (A50) ory. In metals, many complications arise which do not (q) (q) ωq ωq concern us, i.e. the choice of the bare pseudopotential describing electron-ion interaction, phonon polarisation This is exactly the result we found using a diagrammatic vectors, Umklapp processes, distortions of the Fermi sur- approach. The second term in the LHS is the effective face, etc. However, in our system the normal state can attraction between electrons when exchanging momen- be more readily investigated, because the bosons and the tum q through virtual polaritons. Notice that taking the fermions can be separated and investigated separately. In average on the Fermi surface of the first term in V (q) this respect, this type of superconductivity is most sim- yields the Coulomb repulsion constant µ (this constant ilar to the superconductivity in doped semiconductors, will be defined in Appendix B). Also, taking the average which are, in this sense, the best understood supercon- over the Fermi surface of the second term in V (q) yields ductors [41]. the EPC constant λ. In previous work on polariton mediated superconductivity, not only renormalisation effects have been ignored,

200 but also the method used to calculate the critical temperature is not valid. Therefore, in this section we review V(r) 150 the methods that can be used to make reliable predictions ˜ VC (r) about a new superconductor given that the normal state 100 is known and we point out the reason why the predictions

) made in previous work cannot be taken seriously. V e 50 In Section B 1 we present McMillan’s equation, which m

( is the simplest method to obtain the critical temperature 0 of a superconductor given some system paramters[42]. Then, is Section B 2 we introduce the BCS gap equation, 50 which was initially used by Bardeen, Cooper, and Schri- eﬀer to theoretically explain superconductivity, in order 100 0 5 10 15 20 25 30 35 40 to explain the discrepancy between our results and the results obtained in previous work [3] in Section B 3. r/aB

Figure 5. The total interaction V˜ (r) (blue) compared to the ˜ screened Coulomb repulsion VC (r)(green) for the parameters 1. McMillan equation used in Figure 3 with n0 = nc.

The state of the art in the theory of superconductivity Since it might be useful to have an idea of the shape of are the Eliashberg equations, obtained in a Green’s Func- the interaction in real space, we look at the Fourier trans- tion formalism. In certain limits they can be reduced to form of the total interaction compared to the screened a set of two coupled integral equations which must be Coulomb interaction for some typical TMD monolayer solved self-consistently. In some limits, which we will parameters in Figure 5. Notice the oscillatory behaviour present below, these equations can be solved analytically of the interaction at the wavelength 2π/qr that appears to obtain the critical temperature of the superconduc- due to the softening of the polaritons. In comparison tor [18]. Further correction factors can be introduced by to the long range attractive interaction, the screened ﬁtting to the exact results obtained by numerically solv- Coulomb interaction looks like a contact interaction. ing the Eliashberg equations, to obtain, as shown in Ref. 18

19: the polaritons to obtain an electron-electron attractive interaction between Cooper pairs such that: f1f2ωlog 1.04(1 + λ) kBTc = exp ∗ ∗ 0 † † 1.2 −λ(1 0.62µ ) µ H = V (k, k )c 0 c 0 c c . (B5) − − eff k −k k −k h i1/3 2 3/2 (¯ω2/ωlog 1) λ Supposing that this Hamiltonian can be obtained, then f1 = 1 + (λ/Λ1) , f2 = 1 + 2 −2 λ + Λ2 one can apply the methods first introduced by Bardeen, ∗ ∗ Λ1 = 2.46(1 + 3.8µ ), Λ2 = 1.82(1 + 6.3µ )(¯ω2/ωlog) Cooper and Schrieffer [43] to obtain the following BCS µ gap equation (at zero temperature): µ∗ = (B1) 1 + µ ln(εF /¯hωD) 0 X 0 ∆(k ) ∆(k) = V (k, k ) p . (B6) 2 1/2 0 2 0 In the above ωlog = exp [ ln(ω) ], ω¯2 = ω (the k0 2 ∆(k ) + εk averages are taken with respecth toi the weighth i function α2(ω)F (ω)/ω), and µ, the screened Coulomb repulsion Going to a continuum and changing variables from k to between electrons averaged over the Fermi surface, is ε, θ we obtain the above equation in a more convenient given by: form: Z εF 0 0 0 ∆(ε ) 0 X VC (k k ) 0 ∆(ε) = dε p V (ε ε ). (B7) µ = − δ(εk)δ(ε )/N(0). (B2) 0 2 0 − (k k0) k −εF 2 ∆(ε ) + ε k,k0 − It is not at all obvious how to correctly trace out the According to Ref. 19, the above formula, known as polaritons to obtain an electron-electron effective interac- McMillan’s formula, is accurate for µ∗ ranging between tion, mainly due to the retarded nature of this interaction 0 < µ∗ < 0.2 and 0.3 < λ < 10. Therefore, in order [44]. We briefly present three methods and comment on to know the critical temperature of a superconductor, their validity: we need to know four material constants λ, ωlog, ω¯2 and ∗ 2 µ . The difficulty lies in accurately determining these F 0 2 Mq ωq V (k, k ) = | 2 | 2 , parameters. In our case, due to the simplicity of our ∆ε ωq − 2 system, we expect these parameters to be close to the ( 2|Mq | 0 , if ∆ε < ωD theoretical predictions. V BCS(k, k ) = − ωq | | , It is useful to express the parameters λ and µ as mo- 0 , otherwise mentum integrals, rather than frequency integrals. In 2 2 Mq this case simple analytical expressions can be obtained: V S(k, k0) = | | , (B8) − ∆ε + ω | | q −1/2 2kF 2 " 2# 0 2N(0) Z M˜ (q) q where q = k k and ∆ε = εk0 εk. λ = dq 1 , The first effective− potential V −F was initially derived by πkF 0 ωq − 2kF Fröhlich [17] through a Schrieffer-Wolff transformation " 2#−1/2 N(0) Z 2kF q to leading order in the electron-phonon coupling. Notice µ = dqVC (q) 1 . (B3) that it has a resonance singularity, which means that πk − 2k F 0 F at that point higher order terms in the Schrieffer-Wolff transformation become important. However, the singularity is eliminated when performing the (principal value) 2. Superconducting gap equation angular integral which appears from changing variables in going from Eq. (B6) to Eq. (B7). We wish to compare our method to the method used in To eliminate the singularity of the Fröhlich potential, previous work on polariton-mediated superconductivity. Bardeen et.al. approximated the Fröhlich Hamiltonian In order to make this comparison, we show how the super- by a box potential V BCS . Such a simplification is pos- conducting critical temperature can be obtained from a sible because the potential is integrated over in the gap Hamiltonian formalism. Since this section is only meant equation, making the details of the potential insignifi- for comparison we do not consider any renormalisation cant. However, the price to be paid is the introduction effects, or the Coulomb repulsion. Therefore, our starting of a fitting parameter ωD. This means, that the BCS Hamiltonian will be: potential can be used to explain superconductivity but not to predict it, because of the unknown . X † X † ωD H = εkckck + ωqbqbq The last approach involves a more suitable renor- k q malization procedure, which involves continuous unitary X † † transformations. In this regard we mention the sim- + Mq bq + b−q ck+qck. (B4) k,q ilarity renormalization first introduced by Glazek and Wilson [45] and the flow equations introduced by Weg- In a Hamiltonian formalism one can obtain an integral ner [46]. It has been shown [47] that the potential ob- equation for the gap function, provided one can trace out tained through similarity renormalisation techniques V S 19

1.5 potentials V (ε) on top of each other in Figure 6. Notice that at the FS (i.e. = 0) all the potentials agree with 1.0 each other, as they should since at this point the potential corresponds to real processes. 0.5

0.0 3. Comparison to previous work

0.5 In the previous work on polariton-mediated superconductivity [3–5], the authors used the Fröhlich poten- 1.0 tial V F (ε). Notice that, although this potential is non- singular after being integrated, it still develops two large 1.5 shoulders close to the Debye energy. The dependence 0.6 0.4 0.2 0.0 0.2 0.4 0.6 of the critical temperature on the size and width of the ε/E F shoulders has been investigated in Ref. 4 and they have been used to predict a large critical temperatures ob- tainable in polariton-mediated superconductivity. As we Figure 6. Comparison between V F (ε) (blue), V BCS (ε) (red) and V S (ε) (green). For the BCS potential we choose ω = discussed in the main text, we ﬁnd much smaller Tc for D similar system parameters. Another consequence of the g0. For simulations we used typical GaAs parameters, the same parameters used for the solid lines in Figure 2. use of the Fröhlich potential is the appearance of an oscillatory gap, which again has been treated as a peculiarity of polariton-mediated superconductivity. We argue on the other hand that the peculiarities mentioned above can predict accurately the superconducting critical tem- are by no means unique to polariton-mediated supercon- perature. ductivity. Instead, their appearance is due to the use of To compare the diﬀerent approaches we plot the three the Fröhlich potential.

[1] Ivan A Shelykh, Thomas Taylor, and Alexey V Kavokin, tion and condensation of dipolaritons in coupled quan- “Rotons in a hybrid bose-fermi system,” Physical review tum wells,” Physical Review B 90, 125314 (2014). letters 105, 140402 (2010). [11] Iacopo Carusotto, Thomas Volz, and A Imamoğlu, “Fes- [2] Michał Matuszewski, Thomas Taylor, and Alexey V Ka- hbach blockade: Single-photon nonlinear optics using res- vokin, “Exciton supersolidity in hybrid bose-fermi sys- onantly enhanced cavity polariton scattering from biexci- tems,” Physical review letters 108, 060401 (2012). ton states,” EPL (Europhysics Letters) 90, 37001 (2010). [3] Fabrice P Laussy, Alexey V Kavokin, and Ivan A She- [12] Naotomo Takemura, Stéphane Trebaol, Michiel Wouters, lykh, “Exciton-polariton mediated superconductivity,” Marcia T Portella-Oberli, and Benoît Deveaud, “Polari- Physical review letters 104, 106402 (2010). tonic feshbach resonance,” Nature Physics (2014). [4] Fabrice P Laussy, Thomas Taylor, Ivan A Shelykh, and [13] Peter Cristofolini, Gabriel Christmann, Simeon I Alexey V Kavokin, “Superconductivity with excitons and Tsintzos, George Deligeorgis, George Konstantinidis, polaritons: review and extension,” Journal of Nanopho- Zacharias Hatzopoulos, Pavlos G Savvidis, and Jeremy J tonics 6, 064502–1 (2012). Baumberg, “Coupling quantum tunneling with cavity [5] ED Cherotchenko, T Espinosa-Ortega, AV Nalitov, photons,” Science 336, 704–707 (2012). IA Shelykh, and AV Kavokin, “Superconductivity in [14] We mention that this expression has been obtained by semiconductor structures: the excitonic mechanism,” assuming that the in-plane exciton wavefunction is not arXiv preprint arXiv:1410.3532 (2014). modified by the induced dipole. A better exciton wave- [6] VL Ginzburg, “On surface superconductivity,” Physics function is the one used in Ref. 48, unfortunately, this Letters 13, 101–102 (1964). expression is no longer analytical. [7] Iacopo Carusotto and Cristiano Ciuti, “Quantum fluids [15] Gerald D Mahan, Many-particle physics (Springer Sci- of light,” Reviews of Modern Physics 85, 299 (2013). ence & Business Media, 2000). [8] C Ciuti, V Savona, C Piermarocchi, A Quattropani, [16] Douglas J Scalapino, The electron-phonon interaction and P Schwendimann, “Role of the exchange of carriers and strong-coupling superconductors, Vol. 1 (Marcel in elastic exciton-exciton scattering in quantum wells,” Dekker, New York, 1969). Physical Review B 58, 7926 (1998). [17] H Frohlich, “Interaction of electrons with lattice vibra- [9] Tim Byrnes, Patrik Recher, and Yoshihisa Yamamoto, tions,” Proceedings of the Royal Society of London. Se- “Mott transitions of exciton polaritons and indirect ex- ries A. Mathematical and Physical Sciences 215, 291–298 citons in a periodic potential,” Physical Review B 81, (1952). 205312 (2010). [18] WL McMillan, “Transition temperature of strong- [10] Tim Byrnes, German V Kolmakov, Roman Ya Kez- coupled superconductors,” Physical Review 167, 331 erashvili, and Yoshihisa Yamamoto, “Effective interac- (1968). 20

[19] Ph B Allen and RC Dynes, “Transition temperature 362 – 396 (1991). of strong-coupled superconductors reanalyzed,” Physical [32] David J. Clarke, Jason Alicea, and Kirill Shtengel, Review B 12, 905 (1975). “Exotic non-abelian anyons from conventional fractional [20] H Abbaspour, G Sallen, S Trebaol, F Morier-Genoud, quantum hall states,” Nat Commun 4, 1348 (2013). MT Portella-Oberli, and B Deveaud, “The effect of a [33] L.P. Pitaevskii and S. Stringari, Bose-Einstein Conden- noisy driving field on a bistable polariton system,” arXiv sation, International Series of Monographs on Physics preprint arXiv:1505.00932 (2015). (Clarendon Press, 2003). [21] F Weber, S Rosenkranz, J-P Castellan, R Osborn, [34] AB Migdal, “Interaction between electrons and lattice G Karapetrov, R Hott, R Heid, K-P Bohnen, and vibrations in a normal metal,” Sov. Phys. JETP 7, 996– A Alatas, “Electron-phonon coupling and the soft phonon 1001 (1958). mode in tise 2,” Physical review letters 107, 266401 [35] Frank Stern, “Polarizability of a two-dimensional electron (2011). gas,” Physical Review Letters 18, 546 (1967). [22] Matteo Calandra and Francesco Mauri, “Charge-density [36] ST Beliaev, “Application of the methods of quantum field wave and superconducting dome in tise 2 from electron- theory to a system of bosons,” SOVIET PHYSICS JETP- phonon interaction,” Physical review letters 106, 196406 USSR 7, 289–299 (1958). (2011). [37] Hua Shi and Allan Griffin, “Finite-temperature excita- [23] Young Il Joe, XM Chen, P Ghaemi, KD Finkelstein, tions in a dilute bose-condensed gas,” Physics Reports GA de La Peña, Y Gan, JCT Lee, S Yuan, J Geck, 304, 1–87 (1998). GJ MacDougall, et al., “Emergence of charge density [38] Philip B Allen, “Neutron spectroscopy of superconduc- wave domain walls above the superconducting dome in tors,” Physical Review B 6, 2577 (1972). 1t-tise2,” Nature Physics 10, 421–425 (2014). [39] Philip B Allen, “Effect of soft phonons on superconduc- [24] ND Mathur, FM Grosche, SR Julian, IR Walker, DM Fr- tivity: A re-evaluation and a positive case for nb 3 sn,” eye, RKW Haselwimmer, and GG Lonzarich, “Magneti- Solid State Communications 14, 937–940 (1974). cally mediated superconductivity in heavy fermion com- [40] Göran Grimvall, The electron-phonon interaction in met- pounds,” Nature 394, 39–43 (1998). als, Vol. 8 (North-Holland Amsterdam, 1981). [25] Ph Monthoux and GG Lonzarich, “p-wave and d-wave su- [41] Marvin L Cohen, “Superconductivity in modified semi- perconductivity in quasi-two-dimensional metals,” Phys- conductors and the path to higher transition temper- ical Review B 59, 14598 (1999). atures,” Superconductor Science and Technology 28, [26] Philippe Monthoux and GG Lonzarich, “Magnetically 43001–43008 (2015). mediated superconductivity in quasi-two and three di- [42] Usually, the McMillan formula is used the other way mensions,” Physical Review B 63, 054529 (2001). around: given a critical temperature, system parameters [27] P Monthoux and GG Lonzarich, “Magnetic interactions like the EPC constant λ can be calculated. in a single-band model for the cuprates and ruthenates,” [43] John Bardeen, Leon N Cooper, and J Robert Schrief- Physical Review B 71, 054504 (2005). fer, “Theory of superconductivity,” Physical Review 108, [28] Paul C Canfield and Sergey L Bud’Ko, “Feas-based 1175 (1957). superconductivity: a case study of the effects of [44] PB Allen and B Mitrović, “Solid state physics, vol. 37,” transition metal doping on bafe2as2,” arXiv preprint Academic, New York , 1 (1982). arXiv:1002.0858 (2010). [45] Stanislaw D Glazek and Kenneth G Wilson, “Pertur- [29] JM Tranquada, BJ Sternlieb, JD Axe, Y Nakamura, and bative renormalization group for hamiltonians,” Physi- S Uchida, “Evidence for stripe correlations of spins and cal Review-Section D-Particles and Fields 49, 4214–4218 holes in copper oxide superconductors,” Nature 375, 561– (1994). 563 (1995). [46] Franz Wegner, “Flow-equations for hamiltonians,” An- [30] G Rochat, C Ciuti, V Savona, C Piermarocchi, A Quat- nalen der physik 506, 77–91 (1994). tropani, and P Schwendimann, “Excitonic bloch equa- [47] Andreas Mielke, “Calculating critical temperatures of su- tions for a two-dimensional system of interacting exci- perconductivity from a renormalized hamiltonian,” EPL tons,” Physical Review B 61, 13856 (2000). (Europhysics Letters) 40, 195 (1997). [31] Gregory Moore and Nicholas Read, “Nonabelions in the [48] RP Leavitt and JW Little, “Excitonic effects in the opti- fractional quantum hall effect,” Nuclear Physics B 360, cal spectra of superlattices in an electric field,” Physical Review B 42, 11784 (1990).