Setup: 2D material inside a cavity

In Fig. 1A, we show the setup for a 2D material inside a QED cavity environment with perfectly reflecting mirrors. The mirrors confine the photon modes inside the cavity and can lead to strong light-matter coupling even when only the vacuum of the electromagnetic field is considered (36, 37). Specifically, we propose a layered structure of a 2D material (e.g., monolayer FeSe) on a dielectric substrate with a large dielectric constant (e.g., SrTiO 3 ) that further helps confine the cavity photon modes of interest.

Fig. 1 Setup of 2D material in optical cavity, phonon polariton frequency dispersions, and momentum-dependent electron-phonon coupling vertices for the polariton branches. (A) We consider a setup with a 2D material on a dielectric substrate inside a small optical cavity with mirrors as shown. (B) Schematic phonon, photon, and upper and lower polariton dispersions versus 2D in-plane momentum q. The coupling of the phononic dipole current to the photonic vector potential leads to a splitting determined by the plasma frequency ω P . In the cavity, ω P is controlled by the cavity volume. (C) Momentum-dependent squared electron-boson vertex g2(q). For forward scattering, the squared bare electron-phonon vertex is peaked near q = 0. In the polaritonic case (ω P > 0), the upper polariton branch inherits some of the electron-phonon coupling at small q.

For the particular example of FeSe/SrTiO 3 , the effect of the cavity is to couple the electromagnetic field of the photons polarized along the z direction, perpendicular to the interfacial plane, to a cross-interfacial phonon mode. Here, we go beyond the often-used rotating-wave and dipole approximations for the light-matter interaction and use full minimal dipolar coupling including the J ⋅ A and A2 terms (see section S2), which makes the theory manifestly gauge invariant and avoids unphysical divergences. The phonon has a dipole moment along the z direction that involves motion of the O and Ti ions in the topmost layer of SrTiO 3 , spatially very close to the FeSe monolayer. Specifically, one quasi-dispersionless optical Fuchs-Kliewer phonon at 92 meV (29) was identified as the most relevant phonon mode that strongly couples to the FeSe electrons both in angle-resolved photoemission (27) and high-resolution electron energy loss spectroscopies (29). The influence of screening on this mode is not settled yet, particularly when it comes to phonon linewidths (30, 31). However, the experimental evidence for its influence on electronic properties (27, 29) is definitely present, suggesting use of this mode to build a simplified model Hamiltonian to address the impact of reaching strong light-matter coupling on the superconducting behavior of the material. We specifically use a single-band model for the electrons in two spatial dimensions in a partially filled band with filling n = 0.07 per spin, as previously used to model the relevant electronic structure fitting angle-resolved photoemission data (28). A bilinear electron-phonon scattering is introduced by a coupling vertex that is strongly peaked near momentum with a coupling range q 0 . The coupling strength g 0 is adjusted to keep a total dimensionless coupling strength λ ≈ 0.18 independent of q 0 , where λ is determined from the effective electronic mass renormalization m*/m = 1 + λ in the metallic normal state above the superconducting critical temperature in the absence of the cavity coupling. This conservative choice of λ is, for instance, below the value of 0.25 that was given in (29).

Through phonon-photon coupling, we study phonon-polariton formation in this setting. In Fig. 1B, we show schematically the resulting polariton branches that stem from a gauge-invariant coupling involving both J ⋅ A and A2 terms, where J is the current of phononic dipoles associated to an infrared-active phonon mode, and A is the electromagnetic gauge field of the photons. The relevant effective coupling strength between photons and phonons is given by the phononic plasma frequency , with M as the reduced mass of the phonon (see section S2). For the 2D system in the cavity, the plasma frequency is controlled by the length of the vacuum inside the cavity in the z direction, L z , and the 2D unit cell area ν 0,2D = L x L y /N x N y , with L i and N i as the length and number of unit cells of the system in i direction, respectively. The plasma frequency sets the splitting between the upper and lower polariton branches, reminiscent of the LO-TO splitting in bulk semiconductors. Obviously, this splitting is only relevant at very small momenta q, since the photon energies quickly become large compared to the phonon frequency as q increases because of the large magnitude of the speed of light.

The formation of phonon polaritons leads to a redistribution of the electron-phonon coupling vertex into the two polariton branches. In the following, we refer to this coupling between electrons and phonon polaritons as “electron-phonon coupling,” since the coupling originates from electron-phonon coupling in the free-space setting without cavity, and direct electron-photon coupling is not relevant in our setup. In Fig. 1C, we plot the squares of the coupling vertices between electrons and the respective polaritons as a function of q/k F , where k F is the Fermi momentum. A realistic value of the coupling range for FeSe/SrTiO 3 was estimated as q 0 /k F ≈ 0.1, as needed to create replica bands in angle-resolved photoemission that duplicate primary band features without substantial momentum smearing (27, 28). In a microscopic model, this value depends on the distance h 0 between the topmost TiO 2 layer and the FeSe monolayer as well as the anisotropy of in-plane and perpendicular dielectric constants via , with realistic estimates ϵ ∥ /ϵ ⊥ ≈ 100 and 1/(h 0 k F ) ≈ 1. This coupling range is larger than the momentum at which photon and phonon branches cross and mix most strongly in the polariton formation process. This implies that the modification of electron-phonon coupling due to the cavity only happens at very small momenta typically smaller than q 0 /k F . Thus, to investigate how the degree of forward scattering influences the way in which cavity coupling is able to modify the electronic properties, we use different values for q 0 /k F below, envisioning that cavity effects are enhanced when q 0 /k F becomes smaller, which would, in practice, be achieved by making the dielectric-constant anisotropy ratio larger. In Table 1, we summarize the relevant parameter values of the bare material used in our simulations.